interactive student edition

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A Glencoe Program

Visit the Physics Web site physicspp.com You’ll find: Problem of the Week, Standardized Test Practice, Section Self-Check Quizzes, Chapter Review Tests, Online Student Edition, Web Links, Internet Physics Labs, Alternate CBL™ Lab Instructions, Vocabulary PuzzleMaker, In the News, Textbook Updates, Teacher Forum, Teaching Today—Professional Development and much more!

Cover Images Each cover image features a major concept taught in physics. The runner and the colliding spheres represent motion. In addition, the spheres demonstrate the conservation of momentum. Fire represents thermodynamics—the study of thermal energy—and lightning, which is composed of negative electric charges, represents electricity and magnetism.

Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database retrieval system, without prior written permission of the publisher. The term CBL 2 is a trademark of Texas Instruments, Inc. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 0-07-845813-7 Printed in the United States of America. 2 3 4 5 6 7 8 9 10 055/027 10 09 08 07 06 05 04

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Chapter 1

A Physics Toolkit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Mechanics

Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter

2 3 4 5 6 7 8 9 10 11

Representing Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Accelerated Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Forces in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Forces in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Motion in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 146 Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Rotational Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Momentum and Its Conservation . . . . . . . . . . . . . . . . . . . . 228 Energy, Work, and Simple Machines . . . . . . . . . . . . . . . . . 256 Energy and Its Conservation . . . . . . . . . . . . . . . . . . . . . . . . 284

States of Matter

Chapter 12 Chapter 13

Thermal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 States of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

Waves and Light

Chapter Chapter Chapter Chapter Chapter Chapter

14 15 16 17 18 19

Vibrations and Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 Fundamentals of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 Reflection and Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 Refraction and Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 Interference and Diffraction . . . . . . . . . . . . . . . . . . . . . . . . 514

Electricity and Magnetism

Chapter Chapter Chapter Chapter Chapter Chapter Chapter

20 21 22 23 24 25 26

Static Electricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 Current Electricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 Series and Parallel Circuits . . . . . . . . . . . . . . . . . . . . . . . . . 616 Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 Electromagnetic Induction . . . . . . . . . . . . . . . . . . . . . . . . . 670 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696

Modern Physics

Chapter Chapter Chapter Chapter

27 28 29 30

Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722 The Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 Solid-State Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 iii

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Paul W. Zitzewitz, lead author, is a professor of physics at the University of Michigan–Dearborn. He received his B.A. from Carleton College, and his M.A. and Ph.D. from Harvard University, all in physics. Dr. Zitzewitz has taught physics to undergraduates for 32 years, and is an active experimenter in the field of atomic physics with more than 50 research papers. He was named a Fellow of the American Physical Society for his contributions to physics and science education for high school and middle school teachers and students. He has been the president of the Michigan section of the American Association of Physics Teachers and chair of the American Physical Society’s Forum on Education. Todd George Elliott C.E.T., C.Tech., teaches in the Electrotechnology Department at Mohawk College of Applied Arts and Technology, Hamilton, Ontario, Canada. He received technology diplomas in electrical and electronics engineering technology from Niagara College. Todd has held various positions in the fields of semiconductor manufacturing, optical encoding, and electrical design. He is a pioneer in the field of distance education and is a developer of electrical/electronic technology courses, and works closely with major community colleges. David G. Haase is an Alumni Distinguished Undergraduate Professor of Physics at North Carolina State University. He earned a B.A. in physics and mathematics at Rice University and an M.A. and a Ph.D. in physics at Duke University where he was a J.B. Duke Fellow. He has been an active researcher in experimental low temperature and nuclear physics. He teaches undergraduate and graduate physics courses and has worked many years in K-12 teacher training. He is the founding director of The Science House at NC State which annually serves over 3000 teachers and 20,000 students across North Carolina. He has co-authored over 100 papers in experimental physics and in science education. He is a Fellow of the American Physical Society. He received the Alexander Holladay Medal for Excellence, NC State University; the Pegram Medal for Physics Teaching Excellence; and was chosen 1990 Professor of the Year in the state of North Carolina by the Council for the Advancement and Support of Education (CASE). Kathleen A. Harper is an instructional consultant with Faculty & TA Development and an instructor in physics at The Ohio State University. She received her M.A. in physics and B.S. in electrical engineering and applied physics from Case Western Reserve University, and her Ph.D. in physics from The Ohio State University. Her research interests include the teaching and learning of problem-solving skills and the development of alternative problem formats.

iv

Michael R. Herzog consults for the New York State Education Department on physics curriculum and test development, after having taught physics for 27 years at Hilton Central High School. He holds a B.A. in physics from Amherst College and M.S. and M.A. degrees in engineering and education from the University of Rochester. He serves on the executive committee of the New York State Section of AAPT and is a founding member of the New York State Physics Mentors organization. Jane Bray Nelson teaches at University High School in Orlando, Florida. She received her bachelor’s degree from Florida State University, and her M.A. from Memphis State University. She is a National Board Certified Teacher in Adolescents and Young Adults–Science. She has received a Toyota TAPESTRY Award and a Tandy Scholars Award. In addition, she has received the Disney American Teacher Award in Science, the National Presidential Award for Science Teaching, the Florida High School Science Teacher Award, and been inducted into the National Teacher’s Hall of Fame. Jim Nelson teaches at University High School in Orlando, Florida. He received his bachelor’s degree in physics from Lebanon Valley College and M.A.’s in secondary education from Temple University and in physics from Clarkson University. He has received the AAPT Distinguished Service Award, the AAPT Excellence in Pre-College Physics Teaching award, and the National Presidential Award for Science Teaching. Jim is the PI of the Physics Teaching Resource Agent program, and has served on the executive board of AAPT as high school representative and as president. Charles A. Schuler is a writer of textbooks about electricity, electronics, industrial electronics, ISO 9000, and digital signal processing. He taught electronics and electrical engineering technology at California University of Pennsylvania for 30 years. He also developed a course for the Honors Program at California called “Scientific Inquiry.” He received his B.S. from California University of Pennsylvania and his Ed.D. from Texas A&M University, where he was an NDEA Fellow. Margaret K. Zorn is a science and mathematics writer from Yorktown, Virginia. She received an A.B. from the University of Georgia and an M.Sc. in physics from the University of Florida. Ms. Zorn previously taught algebra and calculus. She was a laboratory researcher in the field of detector development for experimental particle physics.

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Contributing Writers Contributing writers provided labs, standardized test practice, features, as well as problems for the additional problems appendix.

Christa Bedwin

David Kinzer

Steven F. Moeckel

Science Writer Montreal, Canada

Optical Engineer, Science Writer Baraboo, WI

Science Writer Troy, OH

Thomas Bright

Craig Kramer

David J. Olney

Physics Teacher Concord High School Concord, NC

Physics Teacher Bexley High School Bexley, OH

Science Writer Mattapoisett, MA

David C. Haas

Suzanne Lyons

Science Writer Granville, OH

Science Writer Auburn, CA

Science Writer Spartanburg, SC

Pat Herak

Jack Minot

Science Teacher Westerville City Schools Westerville, OH

Physics Teacher Bexley High School Bexley, OH

Julie A.O. Smallfield

Amiee Wagner Physics Teacher Columbus State Community College Columbus, OH

Mark Kinsler, Ph.D. Science/Engineering Writer Lancaster, OH

Teacher Advisory Board The Teacher Advisory Board gave the editorial staff and design team feedback on the content and design of the 2005 edition of Physics: Principles and Problems.

Kathleen M. Bartley

Stan Hutto, M.S.

Gregory MacDougall

Physics Teacher Westville High School Westville, IN

Science Department Chair Alamo Heights High School San Antonio, TX

Wayne Fisher, NBCT

Martha S. Lai

Science Specialist Central Savannah River Area University of South Carolina–Aiken Aiken, SC

Physics Teacher Myers Park High School Charlotte, NC

Physics Teacher Massey Hill Classical High School Fayetteville, NC

Stan Greenbaum

Jane Bray Nelson Physics Teacher University High School Orlando, FL

Jim Nelson

Physics Teacher Gorton High School Yonkers, NY

Physics Teacher University High School Orlando, FL

Safety Consultant Kenneth Russell Roy, Ph.D. Director of Science and Safety Glastonbury Public Schools Glastonbury, CT

Teacher Reviewers Maria C. Aparicio

Tom Bartik

Patti R. Boles

Physics Teacher Spanish River High School Boca Raton, FL

Department Chairman Southside High School Chocowinity, NC

Physics Teacher East Rowan High School Salisbury, NC

Daniel Barber

Bob Beebe

Julia Bridgewater

Physics Teacher Klein Forest High School Houston, TX

Physics Teacher Elbert High School Elbert, CO

Physics Teacher Ramona High School Ramona, CA

v

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Jim Broderick

Shirley Hartnett

Megan Lewis-Schroeder

Physics Teacher Antelope Valley High School Lancaster, CA

Physics Teacher JC Birdlebough High School Phoenix, NY

Physics Teacher Bellaire High School Bellaire, MI

Hobart G. Cook

Mary S. Heltzel

Mark Lutzenhiser

Physics Teacher Cummings High School Burlington, NC

Physics Teacher Salina High School Salina, OK

Physics Teacher Sequim High School Sequim, WA

Jason Craigo

Tracy Hood

Jill McLean

Physics Teacher Oberlin High School Oberlin, OH

Physics Teacher Plainfield High School Plainfield, IN

Physics Teacher Centennial High School Champaign, IL

Gregory Cruz

Pam Hughes

Bradley E. Miller, Ph.D.

Physics Teacher Vanguard High School Ocala, FL

Physics Teacher Cherokee High School Cherokee, IA

Physics Teacher East Chapel Hill High School Chapel Hill, NC

Sue Cundiff

Kathy Jacquez

Don Rotsma

Physics Teacher Gulf Breeze High School Gulf Breeze, FL

Physics Teacher Fairmont Senior High School Fairmont, WV

Physics Teacher Galena High School Reno, NV

Terry Elmer

Wilma Jones

David Shoemaker

Physics Teacher Red Creek Central High School Red Creek, NY

Physics Teacher Taft Alternative Academy Lawton, OK

Physics Teacher Mechanicsburg Area High School Mechanicsburg, PA

Hank Grizzle

Gene Kutscher

Physics Teacher Quemado High School Quemado, NM

Science Chairman Roslyn High School Roslyn Heights, NY

Solomon Bililign, Ph.D.

Lewis E. Johnson, Ph.D.

David W. Peakheart, Ph.D.

Professor and Chair Department of Physics North Carolina A&T State University Greensboro, NC

Assistant Professor Department of Physics Florida A&M University Tallahassee, FL

Lecturer Department of Physics and Engineering University of Central Oklahoma Edmond, OK

Juan R. Burciaga, Ph.D.

Sally Koutsoliotas, Ph.D.

Toni D. Sauncy, Ph.D.

Visiting Professor Department of Physics and Astronomy Vassar College Poughkeepsie, NY

Associate Professor Department of Physics Bucknell University Lewisburg, PA

Assistant Professor of Physics Department of Physics Angelo State University San Angelo, TX

Valentina French, Ph.D.

Jun Qing Lu, Ph.D.

Sally Seidel, Ph.D.

Associate Professor of Physics Department of Physics Indiana State University Terre Haute, IN

Assistant Professor East Carolina University Greenville, NC

Associate Professor of Physics Department of Physics and Astronomy University of New Mexico Albuquerque, NM

Godfrey Gumbs, Ph.D.

Assistant Professor of Physics and Engineering Department of Physics and Engineering Jacksonville University Jacksonville, FL

Sudha Srinivas, Ph.D.

Jesús Pando, Ph.D.

Alma C. Zook, Ph.D.

Assistant Professor of Physics Department of Physics DePaul University Chicago, IL

Professor of Physics Department of Physics and Astronomy Pomona College Claremont, CA

Consultants

Chianta-Stoll Professor of Physics Department of Physics and Astronomy Hunter College of the City University of New York New York, NY

Ruth Howes, Ph.D. Professor of Physics Department of Physics Marquette University Milwaukee, WI

vi

William A. Mendoza, Ph.D.

Associate Professor of Physics Department of Physics Central Michigan University Mt. Pleasant, MI

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1

Chapter

A Physics Toolkit . . . . . . . . . . . . . . . . 2 Section 1.1

Section 1.2 Section 1.3

Launch Lab Do all objects fall at the same rate? . . . . . . . . . . . . . . . . 3 Mathematics and Physics . . . . . . . 3 Mini Lab Measuring Change . . . . . . . . . . . . 8 Measurement . . . . . . . . . . . . . . . . 11 Graphing Data . . . . . . . . . . . . . . . 15 Physics Lab Exploring Objects in Motion . . . 20

Chapter

Section 4.1 Section 4.2 Section 4.3

Representing Motion

Launch Lab Which car is faster? . . . . . . . . . . . 31 Picturing Motion . . . . . . . . . . . . . 31 Where and When? . . . . . . . . . . . . 34 Position-Time Graphs . . . . . . . . . 38 How Fast? . . . . . . . . . . . . . . . . . . . 43 Mini Lab Instantaneous Velocity Vectors . . . . . . . . . . . . . . . . . . . . . . 46 Physics Lab Creating Motion Diagrams . . . . . 48

3

Accelerated Motion . . . . . . . . . . . 56 Section 3.1

Section 3.2 Section 3.3

Launch Lab Which force is stronger? . . . . . . . 87 Force and Motion . . . . . . . . . . . . 87 Using Newton’s Laws . . . . . . . . . 96 Interaction Forces . . . . . . . . . . . 102 Mini Lab Tug-of-War Challenge . . . . . . . . 103 Physics Lab Forces in an Elevator . . . . . . . . 108

5

. . . . . . . . 30 Section 5.1 Section 5.2 Section 5.3

Chapter Chapter

. . . . . . . . . . . . . . . . . 86

Forces in Two Dimensions . . . . . . . . . . . . . . . 118

2

Section 2.1 Section 2.2 Section 2.3 Section 2.4

Forces in One Dimension

Chapter

Mechanics

4

Launch Lab Do all types of motion look the same when graphed? . . . . . . 57 Acceleration . . . . . . . . . . . . . . . . . 57 Mini Lab A Steel Ball Race . . . . . . . . . . . . . 58 Motion with Constant Acceleration . . . . . . . . . . . . . . . . . 65 Free Fall . . . . . . . . . . . . . . . . . . . . . 72 Physics Lab Acceleration Due to Gravity . . . .76

Launch Lab Can 2 N 2 N 2 N? . . . . . . . 119 Vectors . . . . . . . . . . . . . . . . . . . . 119 Friction . . . . . . . . . . . . . . . . . . . . 126 Force and Motion in Two Dimensions . . . . . . . . . . 131 Mini Lab What’s Your Angle? . . . . . . . . . . 135 Physics Lab The Coefficient of Friction . . . . 136

6

Motion in Two Dimensions Section 6.1

Section 6.2 Section 6.3

. . . . . . . . . . . . . . 146 Launch Lab How can the motion of a projectile be described? . . . . . . 147 Projectile Motion . . . . . . . . . . . . 147 Mini Lab Over the Edge . . . . . . . . . . . . . . 148 Circular Motion . . . . . . . . . . . . . 153 Relative Velocity . . . . . . . . . . . . . 157 Physics Lab On Target . . . . . . . . . . . . . . . . . . 160

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Chapter

Gravitation Section 7.1 Section 7.2

Chapter

. . . . . . . . . . . . . . . . . . . . . . 170 Launch Lab Can you model Mercury’s motion? . . . . . . . . . . . 171 Planetary Motion and Gravitation . . . . . . . . . . . . . 171 Using the Law of Universal Gravitation . . . . . . . . . 179 Mini Lab Weightless Water . . . . . . . . . . . . 182 Physics Lab Modeling the Orbits of Planets and Satellites . . . . . . . . 186

Rotational Motion . . . . . . . . . . . . 196 Section 8.1 Section 8.2 Section 8.3

Energy, Work, and Simple Machines . . . . . . . . . . . . . . 256 Launch Lab What factors affect energy? . . . . . . . . . . . . . . 257 Section 10.1 Energy and Work . . . . . . . . . . . . 257 Section 10.2 Machines . . . . . . . . . . . . . . . . . . 266 Mini Lab Wheel and Axle . . . . . . . . . . . . . 270 Physics Lab Stair Climbing and Power . . . . . 274

Chapter

8 Launch Lab How do different objects rotate as they roll? . . . . . . . . . . 197 Describing Rotational Motion . . . . . . . . . . . . . . . . . . . . . 197 Rotational Dynamics . . . . . . . . . 201 Equilibrium . . . . . . . . . . . . . . . . . 211 Mini Lab Spinning Tops . . . . . . . . . . . . . . 213 Physics Lab Translational and Rotational Equilibrium . . . . . . . . . . . . . . . . . 218

10

11

Energy and Its Conservation

. . . . . . . . . . . . . . 284 Launch Lab How can you analyze a bouncing basketball? . . . . . . 285 Section 11.1 The Many Forms of Energy . . . 285 Section 11.2 Conservation of Energy . . . . . . 293 Mini Lab Energy Exchange . . . . . . . . . . . . 301 Physics Lab Conservation of Energy . . . . . . 302

States of Matter

Chapter

9

Momentum and Its Conservation

Section 9.1 Section 9.2

viii

Chapter . . . . . . . . . . . . . . 228

Launch Lab What happens when a hollow plastic ball strikes a bocce ball? . . . . . . . . . . . . . . . 229 Impulse and Momentum . . . . . 229 Conservation of Momentum . . . . . . . . . . . . . . . . . 236 Mini Lab Rebound Height . . . . . . . . . . . . 239 Physics Lab Sticky Collisions . . . . . . . . . . . . . 246

12

Thermal Energy

. . . . . . . . . . . . . . . 312

Launch Lab What happens when you provide thermal energy by holding a glass of water? . . 313 Section 12.1 Temperature and Thermal Energy . . . . . . . . . . . . . . . . . . . . . 313 Section 12.2 Changes of State and the Laws of Thermodynamics . . . . 323 Mini Lab Melting . . . . . . . . . . . . . . . . . . . . 324 Physics Lab Heating and Cooling . . . . . . . . . 332

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13

Chapter

States of Matter . . . . . . . . . . . . . . . 340 Section 13.1

Section 13.2 Section 13.3 Section 13.4

Launch Lab Does it float or sink? . . . . . . . . 341 Properties of Fluids . . . . . . . . . . 341 Mini Lab Pressure . . . . . . . . . . . . . . . . . . . 345 Forces Within Liquids . . . . . . . . 349 Fluids at Rest and in Motion . . . . . . . . . . . . . . . . . . 352 Solids . . . . . . . . . . . . . . . . . . . . . . 359 Physics Lab Evaporative Cooling . . . . . . . . . 364

Fundamentals of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 Launch Lab How can you determine the path of light through air? . . . . . 431 Section 16.1 Illumination . . . . . . . . . . . . . . . . . 431 Section 16.2 The Wave Nature of Light . . . . 439 Mini Lab Color by Temperature . . . . . . . . 441 Physics Lab Polarization of Light . . . . . . . . . 448

Chapter

14

Vibrations and Waves

. . . . . . 374

Launch Lab How do waves behave in a coiled spring? . . . . . . . . . . . 375 Section 14.1 Periodic Motion . . . . . . . . . . . . . 375 Section 14.2 Wave Properties . . . . . . . . . . . . . 381 Section 14.3 Wave Behavior . . . . . . . . . . . . . . 387 Mini Lab Wave Interaction . . . . . . . . . . . . 389 Physics Lab Pendulum Vibrations . . . . . . . . . 392

Chapter

Sound

17

Reflection and Mirrors

Waves and Light

Chapter

16

15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 Launch Lab How can glasses produce musical notes? . . . . . . . . . . . . . . 403 Section 15.1 Properties and Detection of Sound . . . . . . . . . . 403 Section 15.2 The Physics of Music . . . . . . . . 411 Mini Lab Sounds Good . . . . . . . . . . . . . . . 418 Physics Lab Speed of Sound . . . . . . . . . . . . . 420

. . . . . . . . . . . . . . . . . . . . . 456 Launch Lab How is an image shown on a screen? . . . . . . . . . . . . . . . 457 Section 17.1 Reflection from Plane Mirrors . . . . . . . . . . . . . . . 457 Mini Lab Virtual Image Position . . . . . . . . 462 Section 17.2 Curved Mirrors . . . . . . . . . . . . . 464 Physics Lab Concave Mirror Images . . . . . . 474

Chapter

18

Refraction and Lenses . . . . . . 484 Launch Lab What does a straw in a liquid look like from the side view? . . . . . . . . . . 485 Section 18.1 Refraction of Light . . . . . . . . . . 485 Section 18.2 Convex and Concave Lenses . . . . . . . . . . . . 493 Mini Lab Lens Masking Effects . . . . . . . . 495 Section 18.3 Applications of Lenses . . . . . . . 500 Physics Lab Convex Lenses and Focal Length . . . . . . . . . . . . . . . 504

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19

Interference and Diffraction

Chapter

Current Electricity . . . . . . . . . . . 590 . . . . . . . . . . . . . . . . 514

Launch Lab Why does a compact disc reflect a rainbow of light? . . . . 515 Section 19.1 Interference . . . . . . . . . . . . . . . . 515 Section 19.2 Diffraction . . . . . . . . . . . . . . . . . . 524 Mini Lab Retinal Projection Screen . . . . . 531 Physics Lab Double-Slit Interference of Light . . . . . . . . . . . . . . . . . . . . 532

Launch Lab Can you get a lightbulb to light? . . . . . . . . . . . 591 Section 22.1 Current and Circuits . . . . . . . . . 591 Mini Lab Current Affairs . . . . . . . . . . . . . . 599 Section 22.2 Using Electric Energy . . . . . . . . 601 Physics Lab Voltage, Current, and Resistance . . . . . . . . . . . . . 606

Chapter Electricity and Magnetism

Chapter

22

20

Static Electricity . . . . . . . . . . . . . . 540 Launch Lab Which forces act over a distance? . . . . . . . . . . . . . . . . . 541 Section 20.1 Electric Charge . . . . . . . . . . . . . 541 Section 20.2 Electric Force . . . . . . . . . . . . . . . 546 Mini Lab Investigating Induction and Conduction . . . . . . . . . . . . . 549 Physics Lab Charged Objects . . . . . . . . . . . . 554

23

Series and Parallel Circuits

. . . . . . . . . . . . . . . 616 Launch Lab How do fuses protect electric circuits? . . . . . . . . . . . . . 617 Section 23.1 Simple Circuits . . . . . . . . . . . . . . 617 Mini Lab Parallel Resistance . . . . . . . . . . 623 Section 23.2 Applications of Circuits . . . . . . 627 Physics Lab Series and Parallel Circuits . . . . . . . . . . . . . . . . . . . . 632

Chapter

24

Magnetic Fields Chapter

21

Electric Fields

. . . . . . . . . . . . . . . . . . 562 Launch Lab How do charged objects interact at a distance? . . . . . . . 563 Section 21.1 Creating and Measuring Electric Fields . . . . . . . . . . . . . . . 563 Section 21.2 Applications of Electric Fields . . . . . . . . . . . . 569 Mini Lab Electric Fields . . . . . . . . . . . . . . . 573 Physics Lab Charging of Capacitors . . . . . . 580

x

. . . . . . . . . . . . . . . 642 Launch Lab In which direction do magnetic fields act? . . . . . . . . . 643 Section 24.1 Magnets: Permanent and Temporary . . . . 643 Mini Lab 3-D Magnetic Fields . . . . . . . . . 650 Section 24.2 Forces Caused by Magnetic Fields . . . . . . . . . . 652 Physics Lab Creating an Electromagnet . . . . . . . . . . . 660

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25

Electromagnetic Induction . . . . . . . . . . . . . . . . . . . . . . . . . 670 Launch Lab What happens in a changing magnetic field? . . . . . . . . . . . . . 671 Section 25.1 Electric Current from Changing Magnetic Fields . . . . . . . . . . . . . 671 Section 25.2 Changing Magnetic Fields Induce EMF . . . . . . . . . . . . . . . . 679 Mini Lab Motor and Generator . . . . . . . . 682 Physics Lab Induction and Transformers . . . 686

Chapter

26

Chapter

28

The Atom

. . . . . . . . . . . . . . . . . . . . . . . . . 746 Launch Lab How can identifying different spinning coins model types of atoms? . . . . . . . . . . . . . . . . . . 747 Section 28.1 The Bohr Model of the Atom . . . . . . . . . . . . . . . . . 747 Mini Lab Bright-Line Spectra . . . . . . . . . . 755 Section 28.2 The Quantum Model of the Atom . . . . . . . . . . . . . . . . . . . 760 Physics Lab Finding the Size of an Atom . . 766

Chapter

29

Electromagnetism . . . . . . . . . . . . 696

Solid-State Electronics . . . . . 774

Launch Lab From where do radio stations broadcast? . . . . . . . . . . . . . . . . . 697 Section 26.1 Interactions of Electric and Magnetic Fields and Matter . . 697 Mini Lab Modeling a Mass Spectrometer . . . . . . . . . . . . . . . 702 Section 26.2 Electric and Magnetic Fields in Space . . . . . . . . . . . . . . . . . . . 705 Physics Lab Electromagnetic Wave Shielding . . . . . . . . . . . . . 714

Launch Lab How can you show conduction in a diode? . . . . . . . . . . . . . . . . . 775 Section 29.1 Conduction in Solids . . . . . . . . . 775 Section 29.2 Electronic Devices . . . . . . . . . . . 784 Mini Lab Red Light . . . . . . . . . . . . . . . . . . 788 Physics Lab Diode Current and Voltage . . . 790

Chapter

30

Nuclear Physics Modern Physics

Chapter

27

Quantum Theory . . . . . . . . . . . . . . 722 Launch Lab What does the spectrum of a glowing lightbulb look like? . . . 723 Section 27.1 A Particle Model of Waves . . . . 723 Mini Lab Glows in the Dark . . . . . . . . . . . 724 Section 27.2 Matter Waves . . . . . . . . . . . . . . . 735 Physics Lab Modeling the Photoelectric Effect . . . . . . . . . . 738

. . . . . . . . . . . . . . . 798

Launch Lab How can you model the nucleus? . . . . . . . . . . . . . . . . 799 Section 30.1 The Nucleus . . . . . . . . . . . . . . . . 799 Section 30.2 Nuclear Decay and Reactions . . . . . . . . . . . . . . 806 Mini Lab Modeling Radioactive Decay . . . . . . . . . . . . . . . . . . . . . 813 Section 30.3 The Building Blocks of Matter . . . . . . . . . . . . . . . . . . . 815 Physics Lab Exploring Radiation . . . . . . . . . . 824

xi

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Page xii

Chapter 1 Do all objects fall at the same rate? . . . . . . . . . . . . . . 3 Chapter 2 Which car is faster? . . . . . . . . . . . 31

Chapter 24

In which direction do magnetic fields act? . . . . . . . . . . . . . . . . . . 643

Chapter 25

What happens in a changing magnetic field? . . . . . . . . . . . . . . 671

Chapter 26

From where do radio stations broadcast? . . . . . . . . . . . . . . . . . . 697

Chapter 3 Do all types of motion look the same when graphed? . . . . . . 57

Chapter 27 What does the spectrum of a glowing lightbulb look like? . . . 723

Chapter 4 Which force is stronger? . . . . . . . 87

Chapter 28

How can identifying different spinning coins model types of atoms? . . . . . . . . . . . . . . . . . . . 747

Chapter 29

How can you show conduction in a diode? . . . . . . . . . . . . . . . . . 775

Chapter 5 Can 2 N 2 N 2 N? . . . . . . . . 119 Chapter 6 How can the motion of a projectile be described? . . . . . 147 Chapter 7 What is the shape of the orbit of the planet Mercury? . . . . . . . 171 Chapter 8 How do different objects rotate as they roll? . . . . . . . . . . . 197

Chapter 30 How can you model the nucleus? . . . . . . . . . . . . . . . . 799

Chapter 9 What happens when a hollow plastic ball strikes a bocce ball? . . . . . . . . . . . . . . . 229 Chapter 10 What factors affect energy? . . . . . . . . . . . . . . . 257 Chapter 11

How can you analyze a bouncing basketball? . . . . . . . 285

Chapter 12

What happens when you provide thermal energy by holding a glass of water? . . . . . 313

Chapter 13

Does it float or sink? . . . . . . . . . 341

Chapter 14 How do waves behave in a coiled spring? . . . . . . . . . . . . . 375 Chapter 15

How can glasses produce musical notes? . . . . . . . . . . . . . . 403

Chapter 2 Physics Lab Creating Motion Diagrams

CBL

. . . . . . . . . 48

Chapter 3 Internet Physics Lab Acceleration Due to Gravity CBL . . . . . . . . . . . 76 Chapter 4 Internet Physics Lab Forces in an Elevator CBL . . . . 108

Chapter 16 How can you determine the path of light through air? . . . . . . . . . . 431

Chapter 5 Physics Lab The Coefficient of Friction CBL . . . . . . . . . . . . . . 136

Chapter 17 How is an image shown on a screen? . . . . . . . . . . . . . . . . . . . 457

Chapter 6 Design Your Own Physics Lab On Target . . . . . . . . . . . . . . . . . . . 160

Chapter 18 What does a straw in a liquid look like from the side view? . . 485

Chapter 7 Physics Lab Modeling the Orbits of Planets and Satellites . . . . . . . . . . . . . . . 186

Chapter 19 Why does a compact disc reflect a rainbow of light? . . . . . 515

xii

Chapter 1 Internet Physics Lab Exploring Objects in Motion . . . . . . . . . . . . . 20

Chapter 20

Which forces act over a distance? . . . . . . . . . . . . . . . . . 541

Chapter 21

How do charged objects interact at a distance? . . . . . . . . 563

Chapter 22

Can you get a lightbulb to light? . . . . . . . . . . . . . . . . . . . . 591

Chapter 23

How do fuses protect electric circuits? . . . . . . . . . . . . . . . . . . . . 617

Chapter 8 Physics Lab Translational and Rotational Equilibrium

CBL

. . . 218

Chapter 9 Internet Physics Lab Sticky Collisions CBL . . . . . . . . 246 Chapter 10 Physics Lab Stair Climbing and Power . . . . . 274 Chapter 11

Physics Lab Conservation of Energy

CBL

. . 302

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Chapter 12 Chapter 13

Page xiii

Physics Lab Heating and Cooling Physics Lab Evaporative Cooling

CBL

. . . . . 332 . . . . . 364

Chapter 1 Measuring Change . . . . . . . . . . . . . 8

Chapter 14 Design Your Own Physics Lab Pendulum Vibrations . . . . . . . . . 392

Chapter 2 Instantaneous Velocity Vectors . . 46

Chapter 15

Chapter 4 Tug-of-War Challenge . . . . . . . . 103

Physics Lab Speed of Sound

CBL

Chapter 16 Physics Lab Polarization of Light

CBL

. . . . . . . . . 420

Chapter 3 A Steel Ball Race . . . . . . . . . . . . . 58 Chapter 5 What’s Your Angle? . . . . . . . . . . 135

CBL

Chapter 17 Physics Lab Concave Mirror Images

. . . . . 448

Chapter 6 Over the Edge . . . . . . . . . . . . . . . 148 Chapter 7 Weightless Water . . . . . . . . . . . . 182

CBL

. . 474

Chapter 8 Spinning Tops . . . . . . . . . . . . . . . 213

Chapter 18 Physics Lab Convex Lenses and Focal Length . . . . . . . . . . . . . . . . 504

Chapter 9 Rebound Height . . . . . . . . . . . . . 239

Chapter 19 Design Your Own Physics Lab Double-Slit Interference of Light CBL . . . . . 532

Chapter 11

Energy Exchange . . . . . . . . . . . . 301

Chapter 12

Melting . . . . . . . . . . . . . . . . . . . . 324

Chapter 13

Pressure . . . . . . . . . . . . . . . . . . . . 345

Chapter 20 Chapter 21 Chapter 22

Chapter 23

Design Your Own Physics Lab Charged Objects . . . . . . . . . . . . 554 Physics Lab Charging of Capacitors

CBL

Chapter 14 Wave Interaction . . . . . . . . . . . . 389 Chapter 15

. . 580

Physics Lab Voltage, Current, and Resistance . . . . . . . . . . . . . . 606 Physics Lab Series and Parallel Circuits

Chapter 10 Wheel and Axle . . . . . . . . . . . . . 270

Sounds Good . . . . . . . . . . . . . . . 418

Chapter 16 Color by Temperature . . . . . . . . 441 Chapter 17 Virtual Image Position . . . . . . . . 462 Chapter 18 Lens Masking Effects . . . . . . . . 495 Chapter 19 Retinal Projection Screen . . . . . 531

. . . . . . . . . 632

Chapter 20

Investigating Induction and Conduction . . . . . . . . . . . . . . . . . 549

Chapter 24

Design Your Own Physics Lab Creating an Electromagnet . . . . 660

Chapter 21

Electric Fields . . . . . . . . . . . . . . . 573

Chapter 25

Physics Lab Induction and Transformers . . . 686

Chapter 22

Current Affairs . . . . . . . . . . . . . . 599

Chapter 23

Parallel Resistance . . . . . . . . . . . 623

Physics Lab Electromagnetic Wave Shielding CBL . . . . . . . . . . . . . . 714

Chapter 24

3-D Magnetic Fields . . . . . . . . . 650

Chapter 25

Motor and Generator . . . . . . . . . 682

Chapter 26

Modeling a Mass Spectrometer . . . . . . . . . . . . . . . 702

Chapter 26

CBL

Chapter 27 Physics Lab Modeling the Photoelectric Effect . . . . . . . . . . 738 Chapter 28

Physics Lab Finding the Size of an Atom . . . 766

Chapter 29

Physics Lab Diode Current and Voltage CBL . . . . . . . . . . . . 790

Chapter 27 Glows in the Dark . . . . . . . . . . . 724 Chapter 28

Bright-Line Spectra . . . . . . . . . . 755

Chapter 29

Red Light . . . . . . . . . . . . . . . . . . . 788

Chapter 30 Modeling Radioactive Decay . . 813

Chapter 30 Design Your Own Physics Lab Exploring Radiation CBL . . . . . . 824

xiii

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Chapter 5 Roller Coasters . . . . . . . . . . . . . . 138

Chapter 1 Computer History and Growth . . 22

Chapter 8 The Stability of Sport-Utility Vehicles . . . . . . . . . . . . . . . . . . . . 220

Chapter 6 Spinning Space Stations . . . . . . 162

Chapter 11

Chapter 17 Adaptive Optical Systems . . . . . 476

Running Smarter . . . . . . . . . . . . 304

Chapter 14 Earthquake Protection . . . . . . . . 394

Chapter 9 Solar Sailing . . . . . . . . . . . . . . . . 248

Chapter 20

Spacecraft and Static Electricity . . . . . . . . . . . . . 556 Atom Laser . . . . . . . . . . . . . . . . . 768

Chapter 16 Advances in Lighting . . . . . . . . . 450 Chapter 22

Hybrid Cars . . . . . . . . . . . . . . . . . 608

Chapter 28

Chapter 26

Cellular Phones . . . . . . . . . . . . . 716

Chapter 30 Thermonuclear Fusion . . . . . . . . 826

Chapter 4 Bathroom Scale . . . . . . . . . . . . . 110

Chapter 2 Accurate Time . . . . . . . . . . . . . . . . 50

Chapter 10 Bicycle Gear Shifters . . . . . . . . . 276 The Heat Pump . . . . . . . . . . . . . . 334

Chapter 3 Time Dilation at High Velocities . . . . . . . . . . . . . . . 78

Chapter 19 Holography . . . . . . . . . . . . . . . . . 534

Chapter 7 Black Holes . . . . . . . . . . . . . . . . . 188

Chapter 21

Lightning Rods . . . . . . . . . . . . . . 582

Chapter 13

A Strange Matter . . . . . . . . . . . . 366

Chapter 23

Ground Fault Circuit Interrupters (GFCI) . . . . . . . . . . . 634

Chapter 15

Sound Waves in the Sun . . . . . . 422

How a Credit-Card Reader Works . . . . . . . . . . . . . . . 688

Chapter 24

The Hall Effect . . . . . . . . . . . . . . 662

Chapter 29

Artificial Intelligence . . . . . . . . . 792

Chapter 12

Chapter 25

Chapter 27 Scanning Tunneling Microscope . . . . . . . . . . . . . . . . . 740

xiv

Chapter 18 Gravitational Lenses . . . . . . . . . 506

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Chapter 1 Plotting Line Graphs . . . . . . . . . . . 16

Chapter 1 Distance to the Moon . . . . . . . . . 13

Chapter 4 Force and Motion . . . . . . . . . . . . . 98

Chapter 2 Speed Records . . . . . . . . . . . . . . . 44

Interaction Pairs . . . . . . . . . . . . . 103

Chapter 3 Drag Racing . . . . . . . . . . . . . . . . . 68

Chapter 5 Vector Addition . . . . . . . . . . . . . . 123

Chapter 4 Shuttle Engine Thrust . . . . . . . . . . 95

Chapter 6 Motion in Two Dimensions . . . . 149

Chapter 5 Causes of Friction . . . . . . . . . . . 130

Chapter 10 Work . . . . . . . . . . . . . . . . . . . . . . . 260

Chapter 6 Space Elevators . . . . . . . . . . . . . 154

Chapter 11

Conservation of Energy . . . . . . . 295

Chapter 7 Geosynchronous Orbit . . . . . . . . 180

Chapter 17 Using Ray Tracing to Locate Images Formed by Curved Mirrors . . . . . . . . . . . . . . 466

Chapter 8 The Fosbury-Flop . . . . . . . . . . . . 212

Chapter 19 Thin-Film Interference . . . . . . . . 521 Chapter 20

Electric Force Problems . . . . . . . 550

Chapter 22

Drawing Schematic Diagrams . 599

Chapter 23

Series-Parallel Circuits . . . . . . . 629

Chapter 27 Units of hc and Photon Energy . . . . . . . . . . . . . . 728

Chapter 9 Running Shoes . . . . . . . . . . . . . . 231 Chapter 10 Tour de France . . . . . . . . . . . . . . 265 Chapter 11

Potential Energy of an Atom . . 289

Chapter 12

Steam Heating . . . . . . . . . . . . . . 317

Chapter 13

Plants . . . . . . . . . . . . . . . . . . . . . . 350

Chapter 14 Foucault Pendulum . . . . . . . . . . 380 Chapter 15

Hearing and Frequency . . . . . . . 413

Chapter 16 Illuminated Minds . . . . . . . . . . . 435 Age of the Universe . . . . . . . . . . 438 Chapter 17 Hubble Trouble . . . . . . . . . . . . . . 467 Chapter 18 Contacts . . . . . . . . . . . . . . . . . . . 501 Chapter 19 Nonreflective Eyeglasses . . . . . 520 Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Chapter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

Chapter 20

Conductor or Insulator? . . . . . . 544

Chapter 21

Static Electricity . . . . . . . . . . . . . 570

Chapter 22

Resistance . . . . . . . . . . . . . . . . . . 597

Chapter 23

Testing Resistance . . . . . . . . . . . 624

Chapter 24

Electromagnets . . . . . . . . . . . . . . 649

Chapter 25

Common Units . . . . . . . . . . . . . . 682

Chapter 26

Frequencies . . . . . . . . . . . . . . . . . 710

Chapter 27 Temperature of the Universe . . 725

Chapter 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

Chapter 28

Laser Eye Surgery . . . . . . . . . . . 764

Chapter 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

Chapter 29

Diode Laser . . . . . . . . . . . . . . . . . 787

Chapter 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

Chapter 30 Forces . . . . . . . . . . . . . . . . . . . . . 802 Radiation Treatment . . . . . . . . . . 811

Chapter 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728

xv

What You’ll Learn • You will use mathematical tools to measure and predict. • You will apply accuracy and precision when measuring. • You will display and evaluate data graphically.

Why It’s Important The measurement and mathematics tools presented here will help you to analyze data and make predictions. Satellites Accurate and precise measurements are important when constructing and launching a satellite— errors are not easy to correct later. Satellites, such as the Hubble Space Telescope shown here, have revolutionized scientific research, as well as communications.

Think About This Physics research has led to many new technologies, such as satellite-based telescopes and communications. What are some other examples of tools developed from physics research in the last 50 years?

physicspp.com 2 NASA

Do all objects fall at the same rate? Question How does weight affect the rate at which an object falls? Procedure

Analysis

The writings of the Greek philosopher Aristotle included works on physical science theories. These were a major influence in the late Middle Ages. Aristotle reasoned that weight is a factor governing the speed of fall of a dropped object, and that the rate of fall must increase in proportion to the weight of the object. 1. Tape four pennies together in a stack. 2. Place the stack of pennies on your hand and place a single penny beside them. 3. Observe Which is heaviest and pushes down on your hand the most? 4. Observe Drop the two at the same time and observe their motions.

According to Aristotle, what should be the rate of fall of the single penny compared to the stack? What did you observe? Critical Thinking Explain which of the following properties might affect the rate of fall of an object: size, mass, weight, color, shape.

1.1 Mathematics and Physics

W

hat do you think of when you see the word physics? Many people picture a chalkboard covered with formulas and mathematics: E mc2, I V/R, d 12at2 v0t d0. Perhaps you picture scientists in white lab coats, or well-known figures such as Marie Curie and Albert Einstein. Or, you might think of the many modern technologies created with physics, such as weather satellites, laptop computers, or lasers.

• Demonstrate scientific methods. • Use the metric system. • Evaluate answers using dimensional analysis. • Perform arithmetic operations using scientific notation.

What is Physics? Physics is a branch of science that involves the study of the physical world: energy, matter, and how they are related. Physicists investigate the motions of electrons and rockets, the energy in sound waves and electric circuits, the structure of the proton and of the universe. The goal of this course is to help you understand the physical world. People who study physics go on to many different careers. Some become scientists at universities and colleges, at industries, or in research institutes. Others go into related fields, such as astronomy, engineering, computer science, teaching, or medicine. Still others use the problem-solving skills of physics to work in business, finance, or other very different disciplines.

Objectives

Vocabulary physics dimensional analysis significant digits scientific method hypothesis scientific law scientific theory

Section 1.1 Mathematics and Physics

3

Horizons Companies

■ Figure 1-1 Physicists use mathematics to represent many different phenomena—a trait sometimes spoofed in cartoons.

©1998 Bill Amend/Dist. by Universal Press Syndicate

Mathematics in Physics Physics uses mathematics as a powerful language. As illustrated in Figure 1-1, this use of mathematics often is spoofed in cartoons. In physics, equations are important tools for modeling observations and for making predictions. Physicists rely on theories and experiments with numerical results to support their conclusions. For example, think back to the Launch Lab. You can predict that if you drop a penny, it will fall. But how fast? Different models of falling objects give different answers to how the speed of the object changes, or on what the speed depends, or which objects will fall. By measuring how an object falls, you can compare the experimental data with the results predicted by different models. This tests the models, allowing you to pick the best one, or to develop a new model.

Electric Current The potential difference, or voltage, across a circuit equals the current multiplied by the resistance in the circuit. That is, V (volts) I (amperes) R (ohms). What is the resistance of a lightbulb that has a 0.75 amperes current when plugged into a 120-volt outlet? 1

Analyze the Problem • Rewrite the equation. • Substitute values.

2

Known:

Unknown:

I 0.75 amperes V 120 volts

R?

Solve for the Unknown Rewrite the equation so the unknown is alone on the left. V IR Reflexive property of equality IR V V I

Divide both sides by I.

120 volts 0.75 amperes

Substitute 120 volts for V, 0.75 amperes for I.

160 ohms

Resistance will be measured in ohms.

R

3

Math Handbook Isolating a Variable page 845

Evaluate the Answer • Are the units correct? 1 volt 1 ampere-ohm, so the answer in volts/ampere is in ohms, as expected. • Does the answer make sense? 120 is divided by a number a little less than 1, so the answer should be a little more than 120.

4

Chapter 1 A Physics Toolkit

FOXTROT © 1998 Bill Amend. Reprinted with permission of UNIVERSAL PRESS SYNDICATE. All rights reserved.

For each problem, give the rewritten equation you would use and the answer.

1. A lightbulb with a resistance of 50.0 ohms is used in a circuit with a 9.0-volt battery. What is the current through the bulb? 2. An object with uniform acceleration a, starting from rest, will reach a speed of v in time t according to the formula v at. What is the acceleration of a bicyclist who accelerates from rest to 7 m/s in 4 s? 3. How long will it take a scooter accelerating at 0.400 m/s2 to go from rest to a speed of 4.00 m/s? 4. The pressure on a surface is equal to the force divided by the area: P F/A. A 53-kg woman exerts a force (weight) of 520 Newtons. If the pressure exerted on the floor is 32,500 N/m2, what is the area of the soles of her shoes?

Does it make sense? Sometimes you will work with unfamiliar units, as in Example Problem 1, and you will need to use estimation to check that your answer makes sense mathematically. At other times you can check that an answer matches your experience, as shown in Figure 1-2. When you work with falling objects, for example, check that the time you calculate an object will take to fall matches your experience—a copper ball dropping 5 m in 0.002 s, or in 17 s, doesn’t make sense. The Math Handbook in the back of this book contains many useful explanations and examples. Refer to it as needed.

SI Units

■ Figure 1-2 What is a reasonable range of values for the speed of an automobile?

To communicate results, it is helpful to use units that everyone understands. The worldwide scientific community and most countries currently use an adaptation of the metric system to state measurements. The Système International d’Unités, or SI, uses seven base quantities, which are shown in Table 1-1. These base quantities were originally defined in terms of direct measurements. Other units, called derived units, are created by combining the base units in various ways. For example, energy is measured in joules, where 1 joule equals one kilogram-meter squared per second squared, or 1 J 1 kgm2/s2. Electric charge is measured in coulombs, where 1 C 1 As.

Table 1-1 SI Base Units Base Quantity Length Mass Time Temperature Amount of a substance Electric current Luminous intensity

Base Unit meter kilogram second kelvin mole ampere candela

Symbol m kg s K mol A cd

Section 1.1 Mathematics and Physics

5 Mark E. Gibson

■ Figure 1-3 The standards for the kilogram and meter are shown. The International Prototype Meter originally was measured as the distance between two marks on a platinum-iridium bar, but as methods of measuring time became more precise than those for measuring length, the meter came to be defined as the distance traveled by light in a vacuum in 1/299 792 458 s.

Scientific institutions have been created to define and regulate measures. The SI system is regulated by the International Bureau of Weights and Measures in Sèvres, France. This bureau and the National Institute of Science and Technology (NIST) in Gaithersburg, Maryland keep the standards of length, time, and mass against which our metersticks, clocks, and balances are calibrated. Examples of two standards are shown in Figure 1-3. NIST works on many problems of measurement, including industrial and research applications. You probably learned in math class that it is much easier to convert meters to kilometers than feet to miles. The ease of switching between units is another feature of the metric system. To convert between SI units, multiply or divide by the appropriate power of 10. Prefixes are used to change SI units by powers of 10, as shown in Table 1-2. You often will encounter these prefixes in daily life, as in, for example, milligrams, nanoseconds, and gigabytes.

Table 1-2 Prefixes Used with SI Units Prefix

Symbol

Multiplier

Scientific Notation

Example

femto-

f

0.000000000000001

femtosecond (fs)

pico-

p

0.000000000001

1015 1012

0.000000001

109

nanometer (nm) microgram ( g)

nano-

n

picometer (pm)

micro-

0.000001

106

milli-

m

0.001

103

milliamps (mA)

centi-

c

0.01

102

centimeter (cm)

0.1

101

deciliter (dL)

1000

103

kilometer (km)

1,000,000

106

megagram (Mg)

1,000,000,000

109

gigameter (Gm)

1012

terahertz (THz)

decikilomegagigatera-

d k M G T

1,000,000,000,000

Dimensional Analysis

Math Handbook Dimensional Analysis page 847

You can use units to check your work. You often will need to use different versions of a formula, or use a string of formulas, to solve a physics problem. To check that you have set up a problem correctly, write out the equation or set of equations you plan to use. Before performing calculations, check that the answer will be in the expected units, as shown in step 3 of Example Problem 1. For example, if you are finding a speed and you see that your answer will be measured in s/m or m/s2, you know you have made an error in setting up the problem. This method of treating the units as algebraic quantities, which can be cancelled, is called dimensional analysis. Dimensional analysis also is used in choosing conversion factors. A conversion factor is a multiplier equal to 1. For example, because 1 kg 1000 g, you can construct the following conversion factors: 1 kg 1000 g

1 6

Chapter 1 A Physics Toolkit

National Institute of Standards and Technology

1000 g 1 kg

1

Choose a conversion factor that will make the units cancel, leaving the answer in the correct units. For example, to convert 1.34 kg of iron ore to grams, do as shown below.

1001 k0g g

1.34 kg 1340 g You also might need to do a series of conversions. To convert 43 km/h to m/s, do the following: 1h 43 km 1000 m 1 min 12 m/s 1 h 1 km 60 min 60 s

Use dimensional analysis to check your equation before multiplying.

5. 6. 7. 8.

How many megahertz is 750 kilohertz? Convert 5021 centimeters to kilometers. How many seconds are in a leap year? Convert the speed 5.30 m/s to km/h.

Significant Digits Suppose you use a meterstick to measure a pen, and you find that the end of the pen is just past 14.3 cm. This measurement has three valid digits: two you are sure of, and one you estimated. The valid digits in a measurement are called significant digits. The last digit given for any measurement is the uncertain digit. All nonzero digits in a measurement are significant. Are all zeros significant? No. For example, in the measurement 0.0860 m, the first two zeros serve only to locate the decimal point and are not significant. The last zero, however, is the estimated digit and is significant. The measurement 172,000 m could have 3, 4, 5, or 6 significant digits. This ambiguity is one reason to use scientific notation: it is clear that the measurement 1.7200105 m has five significant digits. Arithmetic with significant digits When you perform any arithmetic operation, it is important to remember that the result never can be more precise than the least-precise measurement. To add or subtract measurements, first perform the operation, then round off the result to correspond to the least-precise value involved. For example, 3.86 m 2.4 m 6.3 m because the least-precise measure is to one-tenth of a meter. To multiply or divide measurements, perform the calculation and then round to the same number of significant digits as the least-precise

Math Handbook Significant Digits pages 833—836

■ Figure 1-4 This answer to 3.9 7.2 should be rounded to two significant digits.

measurement. For example, 409.2 km/11.4 L 35.9 km/L, because the least-precise measure has three significant digits. Some calculators display several additional digits, as shown in Figure 1-4, while others round at different points. Be sure to record your answers with the correct number of digits. Note that significant digits are considered only when calculating with measurements. There is no uncertainty associated with counting (4 washers) or exact conversion factors (24 hours in 1 day). Section 1.1 Mathematics and Physics

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Horizons Companies

Solve the following problems.

9. a. 6.201 cm 7.4 cm 0.68 cm 12.0 cm b. 1.6 km 1.62 m 1200 cm 10. a. 10.8 g 8.264 g b. 4.75 m 0.4168 m 11. a. 139 cm 2.3 cm b. 3.2145 km 4.23 km 12. a. 13.78 g 11.3 mL b. 18.21 g 4.4 cm3

Measuring Change Collect five identical washers and a spring that will stretch measurably when one washer is suspended from it. 1. Measure the length of the spring with zero, one, two, and three washers suspended from it. 2. Graph the length of the spring versus the mass. 3. Predict the length of the spring with four and five washers. 4. Test your prediction. Analyze and Conclude 5. Describe the shape of the graph. How did you use it to predict the two new lengths?

Scientific Methods In physics class, you will make observations, do experiments, and create models or theories to try to explain your results or predict new answers, as shown in Figure 1-5. This is the essence of a scientific method. All scientists, including physicists, obtain data, make predictions, and create compelling explanations that quantitatively describe many different phenomena. The experiments and results must be reproducible; that is, other scientists must be able to recreate the experiment and obtain similar data. Written, oral, and mathematical communication skills are vital to every scientist. A scientist often works with an idea that can be worded as a hypothesis, which is an educated guess about how variables are related. How can the hypothesis be tested? Scientists conduct experiments, take measurements, and identify what variables are important and how they are related. For example, you might find that the speed of sound depends on the medium through which sound travels, but not on the loudness of the sound. You can then predict the speed of sound in a new medium and test your results. a

b

■ Figure 1-5 These students are conducting an experiment to determine how much power they produce climbing the stairs (a). They use their data to predict how long it would take an engine with the same power to lift a different load (b).

8 Laura Sifferlin

Chapter 1 A Physics Toolkit

■ Figure 1-6 In the mid-1960s, Arno Penzias and Robert Wilson were trying to eliminate the constant background noise in an antenna to be used for radio astronomy. They tested systems, duct-taped seams, and cleared out pigeon manure, but the noise persisted. This noise is now understood to be the cosmic microwave background radiation, and is experimental support for the Big Bang theory.

Models, laws, and theories An idea, equation, structure, or system can model the phenomenon you are trying to explain. Scientific models are based on experimentation. Recall from chemistry class the different models of the atom that were in use over time—new models were developed to explain new observations and measurements. If new data do not fit a model, both are re-examined. Figure 1-6 shows a historical example. If a very well-established model is questioned, physicists might first look at the new data: can anyone reproduce the results? Were there other variables at work? If the new data are born out by subsequent experiments, the theories have to change to reflect the new findings. For example, in the nineteenth century it was believed that linear markings on Mars showed channels, as shown in Figure 1-7a. As telescopes improved, scientists realized that there were no such markings, as shown in Figure 1-7b. In recent times, again with better instruments, scientists have found features that suggest Mars once had running and standing water on its surface, as shown in Figure 1-7c. Each new discovery has raised new questions and areas for exploration. A scientific law is a rule of nature that sums up related observations to describe a pattern in nature. For example, the law of conservation of charge states that in the various changes matter can undergo, the electric charge before and after stays the same. The law of reflection states that the angle of incidence for a light beam on a reflective surface equals the angle of reflection. Notice that the laws do not explain why these phenomena happen, they simply describe them. a

b

■ Figure 1-7 Drawings of early telescope observations (a) showed channels on Mars; recent photos taken with improved telescopes do not (b). In this photo of Mars’ surface from the Mars Global Surveyor spacecraft (c), these layered sedimentary rocks suggest that sedimentary deposits might have formed in standing water.

c

Section 1.1 Mathematics and Physics

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(t)Tom Pantages, (bc)NASA, (others)Photo Researchers

■ Figure 1-8 Theories are changed and modified as new experiments provide insight and new observations are made. The theory of falling objects has undergone many revisions.

Greek philosophers proposed that objects fall because they seek their natural places. The more massive the object, the faster it falls.

Revision Galileo showed that the speed at which an object falls depends on the amount of time it falls, not on its mass.

Revision Galileo’s statement is true, but Newton revised the reason why objects fall. Newton proposed that objects fall because the object and Earth are attracted by a force. Newton also stated that there is a force of attraction between any two objects with mass.

Revision Galileo’s and Newton’s statements still hold true. However, Einstein suggested that the force of attraction between two objects is due to mass causing the space around it to curve.

A scientific theory is an explanation based on many observations supported by experimental results. Theories may serve as explanations for laws. A theory is the best available explanation of why things work as they do. For example, the theory of universal gravitation states that all the mass in the universe is attracted to other mass. Laws and theories may be revised or discarded over time, as shown in Figure 1-8. Notice that this use of the word theory is different from the common use, as in “I have a theory about why it takes longer to get to school on Fridays.” In scientific use, only a very well-supported explanation is called a theory.

1.1 Section Review 13. Math Why are concepts in physics described with formulas? 14. Magnetism The force of a magnetic field on a charged, moving particle is given by F Bqv, where F is the force in kgm/s2, q is the charge in As, and v is the speed in m/s. B is the strength of the magnetic field, measured in teslas, T. What is 1 tesla described in base units? 15. Magnetism A proton with charge 1.601019 As is moving at 2.4105 m/s through a magnetic field of 4.5 T. You want to find the force on the proton. a. Substitute the values into the equation you will use. Are the units correct? 10

Chapter 1 A Physics Toolkit

b. The values are written in scientific notation, m10n. Calculate the 10n part of the equation to estimate the size of the answer. c. Calculate your answer. Check it against your estimate from part b. d. Justify the number of significant digits in your answer. 16. Magnetism Rewrite F Bqv to find v in terms of F, q, and B. 17. Critical Thinking An accepted value for the acceleration due to gravity is 9.801 m/s2. In an experiment with pendulums, you calculate that the value is 9.4 m/s2. Should the accepted value be tossed out to accommodate your new finding? Explain. physicspp.com/self_check_quiz

1.2 Measurement

W

hen you visit the doctor for a checkup, many measurements are taken: your height, weight, blood pressure, and heart rate. Even your vision is measured and assigned a number. Blood might be drawn so measurements can be made of lead or cholesterol levels. Measurements quantify our observations: a person’s blood pressure isn’t just “pretty good,” it’s 110/60, the low end of the good range. What is a measurement? A measurement is a comparison between an unknown quantity and a standard. For example, if you measure the mass of a rolling cart used in an experiment, the unknown quantity is the mass of the cart and the standard is the gram, as defined by the balance or spring scale you use. In the Mini Lab in Section 1.1, the length of the spring was the unknown and the centimeter was the standard.

Objectives • Distinguish between accuracy and precision. • Determine the precision of measured quantities.

Vocabulary measurement precision accuracy

Comparing Results As you learned in Section 1.1, scientists share their results. Before new data are fully accepted, other scientists examine the experiment, looking for possible sources of error, and try to reproduce the results. Results often are reported with an uncertainty. A new measurement that is within the margin of uncertainty confirms the old measurement. For example, archaeologists use radiocarbon dating to find the age of cave paintings, such as those from the Lascaux cave, in Figure 1-9, and the Chauvet cave. Radiocarbon dates are reported with an uncertainty. Three radiocarbon ages from a panel in the Chauvet cave are 30,940 610 years, 30,790 600 years, and 30,230 530 years. While none of the measurements exactly match, the uncertainties in all three overlap, and the measurements confirm each other.

■ Figure 1-9 Drawings of animals from the Lascaux cave in France. By dating organic material in the cave, such as pigments and torch marks, scientists are able to suggest dates at which these cave paintings were made. Each date is reported with an uncertainty to show how precise the measurement is.

Section 1.2 Measurement

11 PhotoEdit

Spring length (cm)

Mini Lab Data

15.0

14.5

14.0 13.7

1

2

3

Student ■

Figure 1-10 Three students took multiple measurements. Are the measurements in agreement? Is student 1’s result reproducible?

Suppose three students performed the Mini Lab from Section 1.1 several times, starting with springs of the same length. With two washers on the spring, student 1 made repeated measurements, which ranged from 14.4 cm to 14.8 cm. The average of student 1’s measurements was 14.6 cm, as shown in Figure 1-10. This result was reported as (14.6 0.2) cm. Student 2 reported finding the spring’s length to be (14.8 0.3) cm. Student 3 reported a length of (14.0 0.1) cm. Could you conclude that the three measurements are in agreement? Is student 1’s result reproducible? The results of students 1 and 2 overlap; that is, they have the lengths 14.5 cm to 14.8 cm in common. However, there is no overlap and, therefore, no agreement, between their results and the result of student 3.

Precision Versus Accuracy Both precision and accuracy are characteristics of measured values. How precise and accurate are the measurements of the three students? The degree of exactness of a measurement is called its precision. Student 3’s measurements are the most precise, within 0.1 cm. The measurements of the other two students are less precise because they have a larger uncertainty. Precision depends on the instrument and technique used to make the measurement. Generally, the device that has the finest division on its scale produces the most precise measurement. The precision of a measurement is one-half the smallest division of the instrument. For example, the graduated cylinder in Figure 1-11a has divisions of 1 mL. You can measure an object to within 0.5 mL with this device. However, the smallest division on the beaker in Figure 1-11b is 50 mL. How precise were your measurements in the MiniLab? The significant digits in an answer show its precision. A measure of 67.100 g is precise to the nearest thousandth of a gram. Recall from Section 1.1 the rules for performing operations with measurements given to different levels of precision. If you add 1.2 mL of acid to a beaker containing 2.4102 mL of water, you cannot say you now have 2.412102 mL of fluid, because the volume of water was not measured to the nearest tenth of a milliliter, but to 100 times that. a

■

Figure 1-11 The graduated cylinder contains 41 0.5 mL (a). The flask contains 325 mL 25 mL (b).

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Chapter 1 A Physics Toolkit

Horizons Companies

b

Accuracy describes how well the results of a measurement agree with the “real” value; that is, the accepted value as measured by competent experimenters. If the length of the spring that the three students measured had been 14.8 cm, then student 2 would have been most accurate and student 3 least accurate. How accurate do you think your measurements in the Mini Lab on page 8 were? What might have led someone to make inaccurate measurements? How could you check the accuracy of measurements? A common method for checking the accuracy of an instrument is called the two-point calibration. First, does the instrument read zero when it should? Second, does it give the correct reading when it is measuring an accepted standard, as shown in Figure 1-12? Regular checks for accuracy are performed on critical measuring instruments, such as the radiation output of the machines used to treat cancer.

■ Figure 1-12 Accuracy is checked by measuring a known value.

Techniques of Good Measurement To assure accuracy and precision, instruments also have to be used correctly. Measurements have to be made carefully if they are to be as precise as the instrument allows. One common source of error comes from the angle at which an instrument is read. Scales should be read with one’s eye directly above the measure, as shown in Figure 1-13a. If the scale is read from an angle, as shown in Figure 1-13b, you will get a different, and less accurate, value. The difference in the readings is caused by parallax, which is the apparent shift in the position of an object when it is viewed from different angles. To experiment with parallax, place your pen on a ruler and read the scale with your eye directly over the tip, then read the scale with your head shifted far to one side. ■ Figure 1-13 By positioning the scale head on (a), your results will be more accurate than if you read your measurements at an angle (b). How far did parallax shift the measurement in b?

a

b Distance to the Moon For over 25 years, scientists have been accurately measuring the distance to the Moon by shining lasers through telescopes. The laser beam reflects off reflectors placed on the surface of the Moon by Apollo astronauts. They have determined that the average distance between the centers of Earth and the Moon is 385,000 km, and it is known with an accuracy of better than one part in 10 billion. Using this laser technique, scientists also have discovered that the Moon is receding from Earth at about 3.8 cm/yr.

Section 1.2 Measurement

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Horizons Companies

■ Figure 1-14 A series of expeditions succeeded in placing a GPS receiver on top of Mount Everest. This improved the accuracy of the altitude measurement: Everest’s peak is 8850 m, not 8848 m, above sea level.

The Global Positioning System, or GPS, offers an illustration of accuracy and precision in measurement. The GPS consists of 24 satellites with transmitters in orbit and numerous receivers on Earth. The satellites send signals with the time, measured by highly accurate atomic clocks. The receiver uses the information from at least four satellites to determine latitude, longitude, and elevation. (The clocks in the receivers are not as accurate as those on the satellites.) Receivers have different levels of precision. A device in an automobile might give your position to within a few meters. Devices used by geophysicists, as in Figure 1-14, can measure movements of millimeters in Earth’s crust. The GPS was developed by the United States Department of Defense. It uses atomic clocks, developed to test Einstein’s theories of relativity and gravity. The GPS eventually was made available for civilian use. GPS signals now are provided worldwide free of charge and are used in navigation on land, at sea, and in the air, for mapping and surveying, by telecommunications and satellite networks, and for scientific research into earthquakes and plate tectonics.

1.2 Section Review 18. Accuracy Some wooden rulers do not start with 0 at the edge, but have it set in a few millimeters. How could this improve the accuracy of the ruler?

22. Precision A box has a length of 18.1 cm and a width of 19.2 cm, and it is 20.3 cm tall. a. What is its volume?

19. Tools You find a micrometer (a tool used to measure objects to the nearest 0.01 mm) that has been badly bent. How would it compare to a new, highquality meterstick in terms of its precision? Its accuracy?

b. How precise is the measure of length? Of volume?

20. Parallax Does parallax affect the precision of a measurement that you make? Explain.

23. Critical Thinking Your friend states in a report that the average time required to circle a 1.5-mi track was 65.414 s. This was measured by timing 7 laps using a clock with a precision of 0.1 s. How much confidence do you have in the results of the report? Explain.

21. Error Your friend tells you that his height is 182 cm. In your own words, explain the range of heights implied by this statement. 14 Bill Crouse

Chapter 1 A Physics Toolkit

c. How tall is a stack of 12 of these boxes? d. How precise is the measure of the height of one box? of 12 boxes?

physicspp.com/self_check_quiz

1.3 Graphing Data

A

well-designed graph can convey information quickly and simply. Patterns that are not immediately evident in a list of numbers take shape when the data are graphed. In this section, you will develop graphing techniques that will enable you to display, analyze, and model data.

Objectives • Graph the relationship between independent and dependent variables. • Interpret graphs.

Identifying Variables When you perform an experiment, it is important to change only one factor at a time. For example, Table 1-3 gives the length of a spring with different masses attached, as measured in the Mini Lab. Only the mass varies; if different masses were hung from different types of springs, you wouldn’t know how much of the difference between two data pairs was due to the different masses and how much to the different springs.

Table 1-3

• Recognize common relationships in graphs.

Vocabulary independent variable dependent variable line of best fit linear relationship quadratic relationship inverse relationship

Length of a Spring for Different Masses Mass Attached to Spring (g)

Length of Spring (cm)

0 5 10 15 20 25 30 35

13.7 14.1 14.5 14.9 15.3 15.7 16.0 16.4

■ Figure 1-15 The independent variable, mass, is on the horizontal axis. The graph shows that the length of the spring increases as the mass suspended from the spring increases.

Length (cm)

A variable is any factor that might affect the behavior of an experimental setup. The independent variable is the factor that is changed or manipulated during the experiment. In this experiment, the mass was the independent variable. The dependent variable is the factor Length of a Spring for Different Masses that depends on the independent variable. In this experiment, the amount that the spring stretched depended on 16.5 the mass. An experiment might look at how radioactivity varies with time, how friction changes with weight, 16.0 or how the strength of a magnetic field depends on the 15.5 distance from a magnet. One way to analyze data is to make a line graph. This 15.0 shows how the dependent variable changes with the inde14.5 pendent variable. The data from Table 1-3 are graphed in black in Figure 1-15. The line in blue, drawn as close to 14.0 all the data points as possible, is called a line of best fit. The line of best fit is a better model for predictions than 13.5 any one point that helps determine the line. The problem0 solving strategy on the next page gives detailed instruc5 10 15 20 25 30 35 tions for graphing data and sketching a line of best fit. Mass (g) Section 1.3 Graphing Data

15

Plotting Line Graphs Use the following steps to plot line graphs from data tables. 1. Identify the independent and dependent variables in your data. The independent variable is plotted on the horizontal axis, the x-axis. The dependent variable is plotted on the vertical axis, the y-axis. 2. Determine the range of the independent variable to be plotted.

Math Handbook Graphs of Relations pages 848—852

3. Decide whether the origin (0, 0) is a valid data point. 4. Spread the data out as much as possible. Let each division on the graph paper stand for a convenient unit. This usually means units that are multiples of 2, 5, or 10. 5. Number and label the horizontal axis. The label should include the units, such as Mass (grams). 6. Repeat steps 2–5 for the dependent variable. 7. Plot the data points on the graph. 8. Draw the best-fit straight line or smooth curve that passes through as many data points as possible. This is sometimes called eyeballing. Do not use a series of straight line segments that connect the dots. The line that looks like the best fit to you may not be exactly the same as someone else’s. There is a formal procedure, which many graphing calculators use, called the least-squares technique, that produces a unique best-fit line, but that is beyond the scope of this textbook. 9. Give the graph a title that clearly tells what the graph represents.

■ Figure 1-16 To find an equation of the line of best fit for a linear relationship, find the slope and y-intercept.

Length of a Spring for Different Masses 17.0

Length (cm)

16.0

15.0

14.0

rise

b 13.7 5

run

10 15 20 25 30 35 Mass (g)

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Chapter 1 A Physics Toolkit

Linear Relationships Scatter plots of data may take many different shapes, suggesting different relationships. (The line of best fit may be called a curve of best fit for nonlinear graphs.) Three of the most common relationships will be shown in this section. You probably are familiar with them from math class. When the line of best fit is a straight line, as in Figure 1-15, the dependent variable varies linearly with the independent variable. There is a linear relationship between the two variables. The relationship can be written as an equation. Linear Relationship Between Two Variables

y mx b

Find the y-intercept, b, and the slope, m, as illustrated in Figure 1-16. Use points on the line—they may or may not be data points.

The slope is the ratio of the vertical change to the horizontal change. To find the slope, select two points, A and B, far apart on the line. The vertical change, or rise, y, is the difference between the vertical values of A and B. The horizontal change, or run, x, is the difference between the horizontal values of A and B. y rise run x The slope of a line is equal to the rise divided by the run, which also can be expressed as the change in y divided by the change in x.

Slope

m

(16.0 cm 14.1 cm) In Figure 1-16: m (30 g 5 g)

0.08 cm/g If y gets smaller as x gets larger, then y/x is negative, and the line slopes downward. The y-intercept, b, is the point at which the line crosses the y-axis, and it is the y-value when the value of x is zero. In this example, b 13.7 cm. When b 0, or y mx, the quantity y is said to vary directly with x.

■ Figure 1-17 This graph indicates a quadratic, or parabolic, relationship.

Nonlinear Relationships Figure 1-17 shows the distance a brass ball falls versus time. Note that the graph is not a straight line, meaning the relationship is not linear. There are many types of nonlinear relationships in science. Two of the most common are the quadratic and inverse relationships. The graph in Figure 1-17 is a quadratic relationship, represented by the following equation. Quadratic Relationship Between Two Variables y ax2 bx c A quadratic relationship exists when one variable depends on the square of another.

Math Handbook Quadratic Graphs page 852 Quadratic Equations page 846

A computer program or graphing calculator easily can find the values of the constants a, b, and c in this equation. In this case, the equation is d 5t2. See the Math Handbook in the back of the book for more on making and using line graphs.

An object is suspended from spring 1, and the spring’s elongation (the distance it stretches) is X1. Then the same object is removed from the first spring and suspended from a second spring. The elongation of spring 2 is X2 . X2 is greater than X1. 1. On the same axes, sketch the graphs of the mass versus elongation for both springs. 2. Is the origin included in the graph? Why or why not? 3. Which slope is steeper? 4. At a given mass, X2 1.6 X1. If X2 5.3 cm, what is X1?

Section 1.3 Graphing Data

17

■ Figure 1-18 This graph shows the inverse relationship between resistance and current. As resistance increases, current decreases.

Current v. Resistance at 120 V

30

Current (A)

25 20 15 10 5 0

10

20

30

40

Resistance (ohms)

The graph in Figure 1-18 shows how the current in an electric circuit varies as the resistance is increased. This is an example of an inverse relationship, represented by the following equation. Inverse Relationship

a x

y

A hyperbola results when one variable depends on the inverse of the other.

The three relationships you have learned about are a sample of the simple relations you will most likely try to derive in this course. Many other mathematical models are used. Important examples include sinusoids, used to model cyclical phenomena, and exponential growth and decay, used to study radioactivity. Combinations of different mathematical models represent even more complex phenomena.

24. The mass values of specified volumes of pure gold nuggets are given in Table 1-4. a. Plot mass versus volume from the values given in the table and draw the curve that best fits all points. b. Describe the resulting curve. c. According to the graph, what type of relationship exists between the mass of the pure gold nuggets and their volume? d. What is the value of the slope of this graph? Include the proper units. e. Write the equation showing mass as a function of volume for gold. f. Write a word interpretation for the slope of the line.

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Chapter 1 A Physics Toolkit

Table 1-4 Mass of Pure Gold Nuggets Volume (cm3)

Mass (g)

1.0 2.0 3.0 4.0 5.0

19.4 38.6 58.1 77.4 96.5

■ Figure 1-19 Computer animators use mathematical models of the real world to create a convincing fictional world. They need to accurately portray how beings of different sizes move, how hair or clothing move with a character, and how light and shadows fall, among other physics topics.

Predicting Values When scientists discover relations like the ones shown in the graphs in this section, they use them to make predictions. For example, the equation for the linear graph in Figure 1-16 is as follows: y (0.08 cm/g)x 13.7 cm Relations, either learned as formulas or developed from graphs, can be used to predict values you haven’t measured directly. How far would the spring in Table 1-3 stretch with 49 g of mass? y (0.08 cm/g)(49 g) 13.7 cm 18 cm It is important to decide how far you can extrapolate from the data you have. For example, 49 kg is a value far outside the ones measured, and the spring might break rather than stretch that far. Physicists use models to accurately predict how systems will behave: what circumstances might lead to a solar flare, how changes to a circuit will change the performance of a device, or how electromagnetic fields will affect a medical instrument. People in all walks of life use models in many ways. One example is shown in Figure 1-19. With the tools you have learned in this chapter, you can answer questions and produce models for the physics questions you will encounter in the rest of this textbook.

1.3 Section Review 25. Make a Graph Graph the following data. Time is the independent variable. Time (s)

5

Speed (m/s) 12 10

10 15 20 25 30 35 8

6

4

2

2

2

26. Interpret a Graph What would be the meaning of a nonzero y-intercept to a graph of total mass versus volume? physicspp.com/self_check_quiz

27. Predict Use the relation illustrated in Figure 1-16 to determine the mass required to stretch the spring 15 cm. 28. Predict Use the relation in Figure 1-18 to predict the current when the resistance is 16 ohms. 29. Critical Thinking In your own words, explain the meaning of a shallower line, or a smaller slope than the one in Figure 1-16, in the graph of stretch versus total mass for a different spring. Section 1.3 Graphing Data

19 0017_3949.ps

Physics is a science that is based upon experimental observations. Many of the basic principles used to describe and understand mechanical systems, such as objects in linear motion, can be applied later to describe more complex natural phenomena. How can you measure the speed of the vehicles in a video clip?

QUESTION What types of measurements could be made to find the speed of a vehicle?

Objectives

Procedure

■ Observe the motion of the vehicles seen in

1. Visit physicspp.com/internet_lab to view the Chapter 1 lab video clip.

the video. ■ Describe the motion of the vehicles. ■ Collect and organize data on the vehicle’s motion. ■ Calculate a vehicle’s speed.

Safety Precautions

Materials Internet access is required. watch or other timer

2. The video footage was taken in the midwestern United States at approximately noon. Along the right shoulder of the road are large, white, painted rectangles. These types of markings are used in many states for aerial observation of traffic. They are placed at 0.322-km (0.2-mi) intervals. 3. Observe What type of measurements might be taken? Prepare a data table, such as the one shown on the next page. Record your observations of the surroundings, other vehicles, and markings. On what color vehicle is the camera located, and what color is the pickup truck in the lane to the left? 4. Measure and Estimate View the video again and look for more details. Is the road smooth? In what direction are the vehicles heading? How long does it take each vehicle to travel two intervals marked by the white blocks? Record your observations and data.

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Benjamin Coifman

Exploring Objects in Motion

Data Table Marker

Distance (km)

White Vehicle Time (s)

Gray Pickup Time (s)

Analyze

Real-World Physics

1. Summarize your qualitative observations.

When the speedometer is observed by a front-seat passenger, the driver, and a passenger in the rear driver’s-side seat, readings of 90 km/h, 100 km/h, and 110 km/h, respectively, are observed. Explain the differences.

2. Summarize your quantitative observations. 3. Make and Use Graphs Graph both sets of data on one pair of axes. 4. Estimate What is the speed of the vehicles in km/s and km/h? 5. Predict How far will each vehicle travel in 5 min?

Conclude and Apply

Design an Experiment Visit physicspp.com/ internet_lab to post your experiment for

1. Measure What is the precision of the distance and time measurements?

measuring speed in the classroom using remote-controlled cars. Include your list of materials, your procedure, and your predictions for the accuracy of your lab. If you actually perform your lab, post your data and results as well.

2. Measure What is the precision of your speed measurement? On what does it depend? 3. Use Variables, Constants, and Controls Describe the independent and the dependent variables in this experiment. 4. Compare and Contrast Which vehicle’s graph has a steeper slope? What is the slope equal to? 5. Infer What would a horizontal line mean on the graph? A line with a steeper slope?

Going Further Speed is distance traveled divided by the amount of time to travel that distance. Explain how you could design your own experiment to measure speed in the classroom using remote-controlled cars. What would you use for markers? How precisely could you measure distance and time? Would the angle at which you measured the cars passing the markers affect the results? How much? How could you improve your measurements? What units make sense for speed? How far into the future could you predict the cars’ positions? If possible, carry out the experiment and summarize your results.

To find out more about measurement, visit the Web site: physicspp.com

21

Computer History and Growth Each pixel of the animations or movies you watch, and each letter of the instant messages you send presents your computer with several hundred equations. Each equation must be solved in a few billionths of a second—if it takes a bit longer, you might complain that your computer is slow. Early Computers The earliest computers

Memory Third, electronic memories were extremely expensive. You may know that a larger memory lets your computer work faster. When one byte of memory required eight circuit boards, 1024 bytes (or 1 K) of memory was enormous. Because memory was so precious, computer programs had to be written with great cleverness. Astronants got to the Moon with 64 K of memory in Apollo’s on-board computers.

could solve very complex arrays of equations, just as yours can, but it took them a lot longer to do so. There were several reasons for this. First, the mathematics of algorithms (problemsolving strategies) still was new. Computer scientists were only beginning to learn how to arrange a particular problem, such as the conversion of a picture into an easily-transmittable form, so that it could be solved by a machine. Processor chips used in today’s computers are tiny compared to the old computer systems.

UNIVAC 1, an early computer, filled an entire room.

Machine Size Second, the machines were physically large. Computers work by switching patterns of electric currents that represent binary numbers. A 16-bit machine works with binary numbers that are 16 bits long. If a 64-bit number must be dealt with, the machine must repeat the same operation four times. A 32-bit machine would have to repeat the operation only twice, thus making it that much faster. But a 32-bit machine is four times the size of a 16-bit machine; that is, it has four times as many wires and transistor switches, and even 8-bit machines were the size of the old UNIVAC shown above. Moreover, current travels along wires at speeds no greater than about two-thirds the speed of light. This is a long time if the computer wires are 15 m long and must move information in less than 109 s. 22 CORBIS

Future Technology

When Gordon Moore and others invented the integrated circuit around 1960, the size and cost of computer circuitry dropped drastically. Physically smaller, and thus faster, machines could be built and very large memories became possible. Today, the transistors on a chip are now smaller than bacteria. The cost and size of computers have dropped so much that your cell phone has far more computing power than most big office machines of the 1970s.

Going Further 1. Research A compression protocol makes a computer file smaller and less prone to transmission errors. Look up the terms .jpg, .mp3, .mpeg, and .midi and see how they apply to the activities you do on your computer. 2. Calculate Using the example here, how long does it take for a binary number to travel 15 m? How many such operations could there be each second?

1.1 Mathematics and Physics Vocabulary

Key Concepts

• physics (p. 3) • dimensional analysis

• •

(p. 6)

• • • • •

significant digits (p. 7) scientific method (p. 8) hypothesis (p. 8) scientific law (p. 9) scientific theory (p. 10)

• • • •

Physics is the study of matter and energy and their relationships. Dimensional analysis is used to check that an answer will be in the correct units. The result of any mathematical operation with measurements never can be more precise than the least-precise measurement involved in the operation. The scientific method is a systematic method of observing, experimenting, and analyzing to answer questions about the natural world. Scientific ideas change in response to new data. Scientific laws and theories are well-established descriptions and explanations of nature.

1.2 Measurement Vocabulary

Key Concepts

• measurement (p. 11) • precision (p. 12) • accuracy (p. 13)

• • • •

New scientific findings must be reproducible; that is, others must be able to measure and find the same results. All measurements are subject to some uncertainty. Precision is the degree of exactness with which a quantity is measured. Scientific notation shows how precise a measurement is. Accuracy is the extent to which a measurement matches the true value.

1.3 Graphing Data Vocabulary

Key Concepts

• independent variable

•

(p. 15)

• dependent variable

•

(p. 15)

• line of best fit (p. 15) • linear relationship (p. 16) • quadratic relationship

•

Data are plotted in graphical form to show the relationship between two variables. The line that best passes through or near graphed data is called the line of best fit. It is used to describe the data and to predict where new data would lie on the graph. A graph in which data points lie on a straight line is a graph of a linear relationship. In the equation, m and b are constants. y mx b

(p. 17)

• inverse relationship (p. 18)

•

The slope of a straight-line graph is the vertical change (rise) divided by the horizontal change (run) and often has a physical meaning. y x

rise run

m

•

The graph of a quadratic relationship is a parabolic curve. It is represented by the equation below. The constants a, b, and c can be found with a computer or a graphing calculator; simpler ones can be found using algebra. y ax2 bx c

•

The graph of an inverse relationship between x and y is a hyperbolic curve. It is represented by the equation below, where a is a constant. a x

y

physicspp.com/vocabulary_puzzlemaker

23

Concept Mapping 30. Complete the following concept map using the following terms: hypothesis, graph, mathematical model, dependent variable, measurement.

39. What determines the precision of a measurement? (1.2)

40. How does the last digit differ from the other digits in a measurement? (1.2)

41. A car’s odometer measures the distance from home to school as 3.9 km. Using string on a map, you find the distance to be 4.2 km. Which answer do you think is more accurate? What does accurate mean? (1.2)

experiment

42. How do you find the slope of a linear graph? (1.3) 43. For a driver, the time between seeing a stoplight and

independent variable

stepping on the brakes is called reaction time. The distance traveled during this time is the reaction distance. Reaction distance for a given driver and vehicle depends linearly on speed. (1.3) a. Would the graph of reaction distance versus speed have a positive or a negative slope? b. A driver who is distracted has a longer reaction time than a driver who is not. Would the graph of reaction distance versus speed for a distracted driver have a larger or smaller slope than for a normal driver? Explain.

44. During a laboratory experiment, the temperature of the gas in a balloon is varied and the volume of the balloon is measured. Which quantity is the independent variable? Which quantity is the dependent variable? (1.3)

Mastering Concepts 31. Describe a scientific method. (1.1) 32. Why is mathematics important to science? (1.1)

45. What type of relationship is shown in Figure 1-20? Give the general equation for this type of relation. (1.3)

33. What is the SI system? (1.1) 34. How are base units and derived units related? (1.1)

y

35. Suppose your lab partner recorded a measurement as 100 g. (1.1) a. Why is it difficult to tell the number of significant digits in this measurement? b. How can the number of significant digits in such a number be made clear?

36. Give the name for each of the following multiples of the meter. (1.1) 1 a. m 100

1 b. m 1000

x

c. 1000 m ■

Figure 1-20

37. To convert 1.8 h to minutes, by what conversion factor should you multiply? (1.1)

38. Solve each problem. Give the correct number of significant digits in the answers. (1.1) a. 4.667104 g + 3.02105 g b. (1.70102 J) ÷ (5.922104 cm3)

24

46. Given the equation F mv2/R, what relationship exists between each of the following? (1.3) a. F and R b. F and m c. F and v

Chapter 1 A Physics Toolkit For more problems, go to Additional Problems, Appendix B.

Applying Concepts

55. You measure the dimensions of a desk as 132 cm,

47. Figure 1-21 gives the height above the ground of a ball that is thrown upward from the roof of a building, for the first 1.5 s of its trajectory. What is the ball’s height at t 0? Predict the ball’s height at t 2 s and at t 5 s. Height of Ball v. Time

Height (m)

25

of a week and spend $1.00 each day for lunch. You prepare a graph of the amount you have left at the end of each day for one week. Would the slope of this graph be positive, zero, or negative? Why? y-axis is the same for each value of the independent variable. What is the slope? Why? How does y depend on x?

15

58. Driving The graph of braking distance versus car

10 5 1

2

3

4

Time (s) ■

56. Money Suppose you receive $5.00 at the beginning

57. Data are plotted on a graph, and the value on the

20

83 cm, and 76 cm. The sum of these measures is 291 cm, while the product is 8.3105 cm3. Explain how the significant digits were determined in each case.

Figure 1-21

speed is part of a parabola. Thus, the equation is written d av2 bv c. The distance, d, has units in meters, and velocity, v, has units in meters/second. How could you find the units of a, b, and c? What would they be?

59. How long is the leaf in Figure 1-22? Include the uncertainty in your measurement.

48. Is a scientific method one set of clearly defined steps? Support your answer.

49. Explain the difference between a scientific theory and a scientific law.

50. Density The density of a substance is its mass per unit volume. a. Give a possible metric unit for density. b. Is the unit for density a base unit or a derived unit?

51. What metric unit would you use to measure each of the following? a. the width of your hand b. the thickness of a book cover c. the height of your classroom d. the distance from your home to your classroom

52. Size Make a chart of sizes of objects. Lengths should range from less than 1 mm to several kilometers. Samples might include the size of a cell, the distance light travels in 1 s, and the height of a room.

53. Time Make a chart of time intervals. Sample intervals might include the time between heartbeats, the time between presidential elections, the average lifetime of a human, and the age of the United States. Find as many very short and very long examples as you can.

54. Speed of Light Two students measure the speed of light. One obtains (3.001 0.001)108 m/s; the other obtains (2.999 0.006)108 m/s. a. Which is more precise? b. Which is more accurate? (You can find the speed of light in the back of this textbook.) physicspp.com/chapter_test

■

Figure 1-22

60. The masses of two metal blocks are measured. Block A has a mass of 8.45 g and block B has a mass of 45.87 g. a. How many significant digits are expressed in these measurements? b. What is the total mass of block A plus block B? c. What is the number of significant digits for the total mass? d. Why is the number of significant digits different for the total mass and the individual masses?

61. History Aristotle said that the speed of a falling object varies inversely with the density of the medium through which it falls. a. According to Aristotle, would a rock fall faster in water (density 1000 kg/m3), or in air (density 1 kg/m3)? b. How fast would a rock fall in a vacuum? Based on this, why would Aristotle say that there could be no such thing as a vacuum? Chapter 1 Assessment

25 Laura Sifferlin

62. Explain the difference between a hypothesis and a scientific theory.

63. Give an example of a scientific law. 64. What reason might the ancient Greeks have had not to question the hypothesis that heavier objects fall faster than lighter objects? Hint: Did you ever question which falls faster?

73. Gravity The force due to gravity is F mg where

g 9.80 m/s2. a. Find the force due to gravity on a 41.63-kg object. b. The force due to gravity on an object is 632 kgm/s2. What is its mass?

74. Dimensional Analysis Pressure is measured in pascals, where 1 Pa 1 kg/ms2. Will the following expression give a pressure in the correct units?

65. Mars Explain what observations led to changes in

(0.55 kg)(2.1 m/s) 9.8 m/s2

scientists’ ideas about the surface of Mars.

66. A graduated cylinder is marked every mL. How precise a measurement can you make with this instrument?

Mastering Problems 1.1 Mathematics and Physics 67. Convert each of the following measurements to meters. a. 42.3 cm b. 6.2 pm c. 21 km d. 0.023 mm e. 214 m f. 57 nm

1.2 Measurement 75. A water tank has a mass of 3.64 kg when it is empty and a mass of 51.8 kg when it is filled to a certain level. What is the mass of the water in the tank?

76. The length of a room is 16.40 m, its width is 4.5 m, and its height is 3.26 m. What volume does the room enclose?

77. The sides of a quadrangular plot of land are 132.68 m, 48.3 m, 132.736 m, and 48.37 m. What is the perimeter of the plot?

78. How precise a measurement could you make with the scale shown in Figure 1-23?

68. Add or subtract as indicated. a. b. c. d.

5.80109 s 3.20108 s 4.87106 m 1.93106 m 3.14105 kg 9.36105 kg 8.12107 g 6.20106 g

69. Rank the following mass measurements from least

to greatest: 11.6 mg, 1021 g, 0.000006 kg, 0.31 mg.

70. State the number of significant digits in each of the

a. b. c. d.

16.2 m 5.008 m 13.48 m 5.006 m 12.0077 m 8.0084 m 78.05 cm2 32.046 cm2 15.07 kg 12.0 kg

Figure 1-23

79. Give the measure shown on the meter in Figure 1-24 as precisely as you can. Include the uncertainty in your answer.

3

2

4

1

5A

71. Add or subtract as indicated.

■

following measurements. a. 0.00003 m b. 64.01 fm c. 80.001 m d. 0.720 g e. 2.40106 kg f. 6108 kg g. 4.071016 m

A

CLASS A

72. Multiply or divide as indicated. a. b. c. d.

26

(6.21018 m)(4.71010 m) (5.6107 m)/(2.81012 s) (8.1104 km)(1.6103 km) (6.5105 kg)/(3.4103 m3) Chapter 1 A Physics Toolkit For more problems, go to Additional Problems, Appendix B.

Horizons Companies

■

Figure 1-24

80. Estimate the height of the nearest door frame in centimeters. Then measure it. How accurate was your estimate? How precise was your estimate? How precise was your measurement? Why are the two precisions different?

81. Base Units Give six examples of quantities you might measure in a physics lab. Include the units you would use.

82. Temperature The temperature drops from 24°C to 10°C in 12 hours. a. Find the average temperature change per hour. b. Predict the temperature in 2 more hours if the trend continues. c. Could you accurately predict the temperature in 24 hours?

1.3 Graphing Data 83. Figure 1-25 shows the masses of three substances for volumes between 0 and 60 cm3. a. What is the mass of 30 cm3 of each substance? b. If you had 100 g of each substance, what would be their volumes? c. In one or two sentences, describe the meaning of the slopes of the lines in this graph. d. What is the y-intercept of each line? What does it mean? Mass of Three Substances

Distance (cm)

5.0 10.0 15.0 20.0 25.0 30.0

24 49 75 99 120 145

a. Plot the values given in the table and draw the curve that best fits all points. b. Describe the resulting curve. c. Use the graph to write an equation relating the distance to the force. d. What is the constant in the equation? Find its units. e. Predict the distance traveled when a 22.0-N force is exerted on the object for 5 s.

85. The physics instructor from the previous problem changed the procedure. The mass was varied while the force was kept constant. Time and distance were measured, and the acceleration of each mass was calculated. The results of the experiment are shown in Table 1-6.

Acceleration of Different Masses

700 Mass (g)

Force (N)

Table 1-6

800 600

C

500 400

B

300 200 100 0

Table 1-5 Distance Traveled with Different Forces

A 10 20 30 40 50

Mass (kg)

Acceleration (m/s2)

1.0 2.0 3.0 4.0 5.0 6.0

12.0 5.9 4.1 3.0 2.5 2.0

Volume (cm3) ■

Figure 1-25

84. During a class demonstration, a physics instructor placed a mass on a horizontal table that was nearly frictionless. The instructor then applied various horizontal forces to the mass and measured the distance it traveled in 5 seconds for each force applied. The results of the experiment are shown in Table 1-5. physicspp.com/chapter_test

a. Plot the values given in the table and draw the curve that best fits all points. b. Describe the resulting curve. c. According to the graph, what is the relationship between mass and the acceleration produced by a constant force? d. Write the equation relating acceleration to mass given by the data in the graph. e. Find the units of the constant in the equation. f. Predict the acceleration of an 8.0-kg mass. Chapter 1 Assessment

27

86. During an experiment, a student measured the mass of 10.0 cm3 of alcohol. The student then measured the mass of 20.0 cm3 of alcohol. In this way, the data in Table 1-7 were collected.

Table 1-7 The Mass Values of Specific Volumes of Alcohol Volume

(cm3)

10.0 20.0 30.0 40.0 50.0

Mass (g) 7.9 15.8 23.7 31.6 39.6

a. Plot the values given in the table and draw the curve that best fits all the points. b. Describe the resulting curve. c. Use the graph to write an equation relating the volume to the mass of the alcohol. d. Find the units of the slope of the graph. What is the name given to this quantity? e. What is the mass of 32.5 cm3 of alcohol?

Mixed Review 87. Arrange the following numbers from most precise to least precise 0.0034 m 45.6 m

1234 m

88. Figure 1-26 shows the toroidal (doughnut-shaped) interior of the now-dismantled Tokamak Fusion Test Reactor. Explain why a width of 80 m would be an unreasonable value for the width of the toroid. What would be a reasonable value?

90. You are given the following measurements of a rectangular bar: length 2.347 m, thickness 3.452 cm, height 2.31 mm, mass 1659 g. Determine the volume, in cubic meters, and density, in g/cm3, of the beam. Express your results in proper form.

91. A drop of water contains 1.71021 molecules. If the water evaporated at the rate of one million molecules per second, how many years would it take for the drop to completely evaporate?

92. A 17.6-gram sample of metal is placed in a graduated cylinder containing 10.0 cm3 of water. If the water level rises to 12.20 cm3, what is the density of the metal?

Thinking Critically 93. Apply Concepts It has been said that fools can ask more questions than the wise can answer. In science, it is frequently the case that one wise person is needed to ask the right question rather than to answer it. Explain.

94. Apply Concepts Find the approximate mass of water in kilograms needed to fill a container that is 1.40 m long and 0.600 m wide to a depth of 34.0 cm. Report your result to one significant digit. (Use a reference source to find the density of water.)

95. Analyze and Conclude A container of gas with a pressure of 101 kPa has a volume of 324 cm3 and a mass of 4.00 g. If the pressure is increased to 404 kPa, what is the density of the gas? Pressure and volume are inversely proportional.

96. Design an Experiment How high can you throw a ball? What variables might affect the answer to this question?

97. Calculate If the Sun suddenly ceased to shine, how long would it take Earth to become dark? (You will have to look up the speed of light in a vacuum and the distance from the Sun to Earth.) How long would it take the surface of Jupiter to become dark?

Writing in Physics 98. Research and describe a topic in the history of

■

Figure 1-26

89. You are cracking a code and have discovered the following conversion factors: 1.23 longs 23.0 mediums, and 74.5 mediums 645 shorts. How many shorts are equal to one long?

28

physics. Explain how ideas about the topic changed over time. Be sure to include the contributions of scientists and to evaluate the impact of their contributions on scientific thought and the world outside the laboratory.

99. Explain how improved precision in measuring time would have led to more accurate predictions about how an object falls.

Chapter 1 A Physics Toolkit For more problems, go to Additional Problems, Appendix B.

Princeton Plasma Physics Laboratory

Lab A’s reading is more accurate than lab B’s. Lab A’s reading is less accurate than lab B’s. Lab A’s reading is more precise than lab B’s. Lab A’s reading is less precise than lab B’s. 2. Which of the following is equal to 86.2 cm? 8.62104 km 862 dm

8.62 m 0.862 mm

3. Jario has a problem to do involving time, distance, and velocity, but he has forgotten the formula. The question asks him for a measurement in seconds, and the numbers that are given have units of m/s and km. What could Jario do to get the answer in seconds? Multiply the km by the m/s, then multiply by 1000. Divide the km by the m/s, then multiply by 1000. Divide the km by the m/s, then divide by 1000. Multiply the km by the m/s, then divide by 1000.

m V

5. Which formula is equivalent to D ? m D

mD V D V m

V

V V Dm

Extended Answer 6. You want to calculate an acceleration, in units of m/s2, given a force, in N, and the mass, in g, on which the force acts. (1 N 1 kgm/s2) a.

Rewrite the equation F ma so a is in terms of m and F.

b.

What conversion factor will you need to multiply by to convert grams to kilograms?

c.

A force of 2.7 N acts on a 350-g mass. Write the equation you will use, including the conversion factor, to find the acceleration.

7. Find an equation for a line of best fit for the data shown below. Distance v. Time 12 10 Distance (m)

Multiple Choice 1. Two laboratories use radiocarbon dating to measure the age of two wooden spear handles found in the same grave. Lab A finds an age of 2250 40 years for the first object; lab B finds an age of 2215 50 years for the second object. Which of the following is true?

8 6 4

4. What is the slope of the graph? 0.25 m/s2 0.4 m/s2

2.5 m/s2 4.0 m/s2 Stopping Distance

2 0

1

2

3

4

5

6

7

Time (s)

Speed (m/s)

4 3

Skip Around if You Can 2 1 0

2

4

6

8

10

12

You may want to skip over difficult questions and come back to them later, after you’ve answered the easier questions. This will guarantee more points toward your final score. In fact, other questions may help you answer the ones you skipped. Just be sure you fill in the correct ovals on your answer sheet.

Time (s) physicspp.com/standardized_test

Chapter 1 Standardized Test Practice

29

What You’ll Learn • You will represent motion through the use of words, motion diagrams, and graphs. • You will use the terms position, distance, displacement, and time interval in a scientific manner to describe motion.

Why It’s Important Without ways to describe and analyze motion, travel by plane, train, or bus would be chaotic at best. Times and speeds determine the winners of races as well as transportation schedules. Running a Marathon As one runner passes another, the speed of the overtaking runner is greater than the speed of the other runner.

Think About This How can you represent the motion of two runners?

physicspp.com 30

AFP/Corbis

Which car is faster? Question In a race between two toy cars, can you explain which car is faster? Procedure

Analysis

1. Obtain two toy cars, either friction cars or windup cars. Place the cars on your lab table or other surface recommended by your teacher. 2. Decide on a starting line for the race. 3. Release both cars from the same starting line at the same time. Note that if you are using windup cars, you will need to wind them up before you release them. Be sure to pull the cars back before release if they are friction cars. 4. Observe Watch the two cars closely as they move and determine which car is moving faster. 5. Repeat steps 1–3, but this time collect one type of data to support your conclusion about which car is faster.

What data did you collect to show which car was moving faster? What other data could you collect to determine which car is faster? Critical Thinking Write an operational definition of average speed.

2.1 Picturing Motion

I

n the previous chapter, you learned about the scientific processes that will be useful in your study of physics. In this chapter, you will begin to use these tools to analyze motion. In subsequent chapters, you will apply them to all kinds of movement using sketches, diagrams, graphs, and equations. These concepts will help you to determine how fast and how far an object will move, whether the object is speeding up or slowing down, and whether it is standing still or moving at a constant speed. Perceiving motion is instinctive—your eyes naturally pay more attention to moving objects than to stationary ones. Movement is all around you— from fast trains and speedy skiers to slow breezes and lazy clouds. Movements travel in many directions, such as the straight-line path of a bowling ball in a lane’s gutter, the curved path of a tether ball, the spiral of a falling kite, and the swirls of water circling a drain.

Objectives • Draw motion diagrams to describe motion. • Develop a particle model to represent a moving object.

Vocabulary motion diagram particle model

Section 2.1 Picturing Motion

31

Horizons Companies

All Kinds of Motion What comes to your mind when you hear the word motion? A speeding automobile? A spinning ride at an amusement park? A baseball soaring over a fence for a home run? Or a child swinging back and forth in a regular rhythm? When an object is in motion, as shown in Figure 2-1, its position changes. Its position can change along the path of a straight line, a circle, an arc, or a back-and-forth vibration. Some of the types of motion described above appear to be more complicated than others. When beginning a new area of study, it is generally a good idea to begin with what appears to be the least complicated situation, learn as much as possible about it, and then gradually add more complexity to that simple model. In the case of motion, you will begin your study with movement along a straight line.

■ Figure 2-1 An object in motion changes its position as it moves. In this photo, the camera was focused on the rider, so the blurry background indicates that the rider’s position has changed.

Movement along a straight line Suppose that you are reading this textbook at home. At the beginning of Chapter 2, you glance over at your pet hamster and see that he is sitting in a corner of his cage. Sometime later, you look over again, and you see that he now is sitting by the food dish in the opposite corner of the cage. You can infer that he has moved from one place to another in the time in between your observations. Thus, a description of motion relates to place and time. You must be able to answer the questions of where and when an object is positioned to describe its motion. Next, you will look at some tools that are useful in determining when an object is at a particular place.

Motion Diagrams Consider an example of straight-line motion: a runner is jogging along a straight path. One way of representing the motion of the runner is to create a series of images showing the positions of the runner at equal time intervals. This can be done by photographing the runner in motion to obtain a series of images. Suppose you point a camera in a direction perpendicular to the direction of motion, and hold it still while the motion is occurring. Then you take a series of photographs of the runner at equal time intervals. Figure 2-2 shows what a series of consecutive images for a runner might look like. Notice that the runner is in a different position in each image, but everything in the background remains in the same position. This indicates that, relative to the ground, only the runner is in motion. What is another way of representing the runner’s motion?

■ Figure 2-2 If you relate the position of the runner to the background in each image over equal time intervals, you will conclude that she is in motion.

32

Chapter 2 Representing Motion

(t)Getty Images, (others)Hutchings Photography

Suppose that you stacked the images from Figure 2-2, one on top of the other. Figure 2-3 shows what such a stacked image might look like. You will see more than one image of the moving runner, but only a single image of the motionless objects in the background. A series of images showing the positions of a moving object at equal time intervals is called a motion diagram.

The Particle Model Keeping track of the motion of the runner is easier if you disregard the movement of the arms and legs, and instead concentrate on a single point at the center of her body. In effect, you can disregard the fact that she has some size and imagine that she is a very small object located precisely at that central point. A particle model is a simplified version of a motion diagram in which the object in motion is replaced by a series of single points. To use the particle model, the size of the object must be much less than the distance it moves. The internal motions of the object, such as the waving of the runner’s arms are ignored in the particle model. In the photographic motion diagram, you could identify one central point on the runner, such as a dot centered at her waistline, and take measurements of the position of the dot. The bottom part of Figure 2-3 shows the particle model for the runner’s motion. You can see that applying the particle model produces a simplified version of the motion diagram. In the next section, you will learn how to create and use a motion diagram that shows how far an object moved and how much time it took to move that far.

■ Figure

2-3 Stacking a series of images taken at regular time intervals and combining them into one image creates a motion diagram for the runner for one portion of her run. Reducing the runner’s motion to a series of single points results in a particle model of her motion.

2.1 Section Review 1. Motion Diagram of a Runner Use the particle model to draw a motion diagram for a bike rider riding at a constant pace. 2. Motion Diagram of a Bird Use the particle model to draw a simplified motion diagram corresponding to the motion diagram in Figure 2-4 for a flying bird. What point on the bird did you choose to represent it?

3. Motion Diagram of a Car Use the particle model to draw a simplified motion diagram corresponding to the motion diagram in Figure 2-5 for a car coming to a stop at a stop sign. What point on the car did you use to represent it?

■

■

Figure 2-5

4. Critical Thinking Use the particle model to draw motion diagrams for two runners in a race, when the first runner crosses the finish line as the other runner is three-fourths of the way to the finish line.

Figure 2-4 physicspp.com/self_check_quiz

Section 2.1 Picturing Motion

33

Hutchings Photography

2.2 Where and When?

Objectives • Define coordinate systems for motion problems. • Recognize that the chosen coordinate system affects the sign of objects’ positions. • Define displacement. • Determine a time interval. • Use a motion diagram to answer questions about an object’s position or displacement.

Vocabulary coordinate system origin position distance magnitude vectors scalars resultant time interval displacement

■ Figure 2-6 In these motion diagrams, the origin is at the left (a), and the positive values of distance extend horizontally to the right. The two arrows, drawn from the origin to points representing the runner, locate his position at two different times (b).

W

ould it be possible to take measurements of distance and time from a motion diagram, such as the motion diagram of the runner? Before taking the photographs, you could place a meterstick or a measuring tape on the ground along the path of the runner. The measuring tape would tell you where the runner was in each image. A stopwatch or clock within the view of the camera could tell the time. But where should you place the end of the measuring tape? When should you start the stopwatch?

Coordinate Systems When you decide where to place the measuring tape and when to start the stopwatch, you are defining a coordinate system, which tells you the location of the zero point of the variable you are studying and the direction in which the values of the variable increase. The origin is the point at which both variables have the value zero. In the example of the runner, the origin, represented by the zero end of the measuring tape, could be placed 6 m to the left of the tree. The motion is in a straight line; thus, your measuring tape should lie along that straight line. The straight line is an axis of the coordinate system. You probably would place the tape so that the meter scale increases to the right of the zero, but putting it in the opposite direction is equally correct. In Figure 2-6a, the origin of the coordinate system is on the left. You can indicate how far away the runner is from the origin at a particular time on the simplified motion diagram by drawing an arrow from the origin to the point representing the runner, as shown in Figure 2-6b. This arrow represents the runner’s position, the separation between an object and the origin. The length of the arrow indicates how far the object is from the origin, or the object’s distance from the origin. The arrow points from the origin to the location of the moving object at a particular time.

a

5

10

15

5

10

15

meters

20

25

30

20

25

30

d

b

34

Chapter 2 Representing Motion

meters

d

Is there such a thing as a negative position? Suppose you chose the coordinate system just described, placing the origin 4 m left of the tree with the d-axis extending in a positive direction to the right. A position 9 m to the left of the tree, 5 m left of the origin, would be a negative position, as shown in Figure 2-7. In the same way, you could discuss a time before the stopwatch was started.

5

5

10

meters

15

20

25

30

d

■ Figure 2-7 The arrow drawn on this motion diagram indicates a negative position.

Vectors and scalars Quantities that have both size, also called magnitude, and direction, are called vectors, and can be represented by arrows. Quantities that are just numbers without any direction, such as distance, time, or temperature, are called scalars. This textbook will use boldface letters to represent vector quantities and regular letters to represent scalars. You already know how to add scalars; for example, 0.6 0.2 0.8. How do you add vectors? Think about how you would solve the following problem. Your aunt asks you to get her some cold medicine at the store nearby. You walk 0.5 km east from your house to the store, buy the cold medicine, and then walk another 0.2 km east to your aunt’s house. How far from the origin are you at the end of your trip? The answer, of course, is 0.5 km east 0.2 km east 0.7 km east. You also could solve this problem graphically, using the following method. Using a ruler, measure and draw each vector. The length of a vector should be proportional to the magnitude of the quantity being represented, so you must decide on a scale for your drawing. For example, you might let 1 cm on paper represent 0.1 km. The important thing is to choose a scale that produces a diagram of reasonable size with a vector that is about 5–10 cm long. The vectors representing the two segments that made up your trip to your aunt’s house are shown in Figure 2-8, drawn to a scale of 1 cm, which represents 0.1 km. The vector that represents the total of these two, shown here with a dotted line, is 7 cm long. According to the established scale, you were 0.7 km from the origin at the end of your trip. The vector that represents the sum of the other two vectors is called the resultant. The resultant always points from the tail of the first vector to the tip of the last vector. , Aunt s house

Your house Store

5 cm

2 cm

■ Figure 2-8 Add two vectors by placing them tip to tail. The resultant points from the tail of the first vector to the tip of the last vector.

7 cm

Section 2.2 Where and When?

35

■ Figure 2-9 You can see that it took the runner 4.0 s to run from the tree to the lamppost. The initial position of the runner is used as a reference point. The vector from position 1 to position 2 indicates both the direction and amount of displacement during this time interval.

tf

ti

5

d

10

15

meters

20

25

30

d

Time Intervals and Displacements

• Displacement vectors are shown in green.

When analyzing the runner’s motion, you might want to know how long it took the runner to travel from the tree to the lamppost. This value is obtained by finding the difference in the stopwatch readings at each position. Assign the symbol ti to the time when the runner was at the tree and the symbol tf to the time when he was at the lamppost. The difference between two times is called a time interval. A common symbol for a time interval is t, where the Greek letter delta, , is used to represent a change in a quantity. The time interval is defined mathematically as follows. Time Interval

t tf ti

The time interval is equal to the final time minus the initial time.

■

Figure 2-10 Start with two vectors, A and B (a). To subtract vector B from vector A, first reverse vector B , then add them together to obtain the resultant, R (b).

a A

Although i and f are used to represent the initial and final times, they can be the initial and final times of any time interval you choose. In the example of the runner, the time it takes for him to go from the tree to the lamppost is tf ti 5.0 s 1.0 s 4.0 s. How did the runner’s position change when he ran from the tree to the lamppost, as shown in Figure 2-9? The symbol d may be used to represent position. In common speech, a position refers to a place; but in physics, a position is a vector with its tail at the origin of a coordinate system and its tip at the place. Figure 2-9 shows d, an arrow drawn from the runner’s position at the tree to his position at the lamppost. This vector represents his change in position, or displacement, during the time interval between ti and tf. The length of the arrow represents the distance the runner moved, while the direction the arrow points indicates the direction of the displacement. Displacement is mathematically defined as follows.

B

Displacement

Vectors A and B

d df di

Displacement is equal to the final position minus the initial position.

b

A B A (B) Resultant of A and (B)

36

Chapter 2 Representing Motion

Again, the initial and final positions are the beginning and end of any interval you choose. Also, while position can be considered a vector, it is common practice when doing calculations to drop the boldface, and use signs and magnitudes. This is because position usually is measured from the origin, and direction typically is included with the position indication.

■

df

a

di

d

df

b

d

di

Figure 2-11 The displacement of the runner during the 4.0-s time interval is found by subtracting df from di. In (a) the origin is at the left, and in (b) it is at the right. Regardless of your choice of coordinate system, d is the same.

How do you subtract vectors? Reverse the subtracted vector and add. This is because A B A (B). Figure 2-10a shows two vectors, A, 4 cm long pointing east, and B, 1 cm long also pointing east. Figure 2-10b shows B, which is 1 cm long pointing west. Figure 2-10b shows the resultant of A and B. It is 3 cm long pointing east. To determine the length and direction of the displacement vector, d df di, draw di , which is di reversed. Then draw df and copy di with its tail at df’s tip. Add df and di. In the example of the runner, his displacement is df di 25.0 m 5.0 m 20.0 m. He moved to the right of the tree. To completely describe an object’s displacement, you must indicate the distance it traveled and the direction it moved. Thus, displacement, a vector, is not identical to distance, a scalar; it is distance and direction. What would happen if you chose a different coordinate system; that is, if you measured the position of the runner from another location? Look at Figure 2-9, and suppose you change the right side of the d-axis to be zero. While the vectors drawn to represent each position change, the length and direction of the displacement vector does not, as shown in Figures 2-11a and b. The displacement, d, in the time interval from 1.0 s to 5.0 s does not change. Because displacement is the same in any coordinate system, you frequently will use displacement when studying the motion of an object. The displacement vector is always drawn with its flat end, or tail, at the earlier position, and its point, or tip, at the later position.

2.2 Section Review 5. Displacement The particle model for a car traveling on an interstate highway is shown below. The starting point is shown. Here

There

Make a copy of the particle model, and draw a vector to represent the displacement of the car from the starting time to the end of the third time interval.

6. Displacement The particle model for a boy walking to school is shown below. Home

School

Make a copy of the particle model, and draw vectors to represent the displacement between each pair of dots. physicspp.com/self_check_quiz

7. Position Two students compared the position vectors they each had drawn on a motion diagram to show the position of a moving object at the same time. They found that their vectors did not point in the same direction. Explain. 8. Critical Thinking A car travels straight along the street from the grocery store to the post office. To represent its motion you use a coordinate system with its origin at the grocery store and the direction the car is moving in as the positive direction. Your friend uses a coordinate system with its origin at the post office and the opposite direction as the positive direction. Would the two of you agree on the car’s position? Displacement? Distance? The time interval the trip took? Explain. Section 2.2 Where and When?

37

2.3 Position-Time Graphs

Objectives • Develop position-time graphs for moving objects. • Use a position-time graph to interpret an object’s position or displacement. • Make motion diagrams, pictorial representations, and position-time graphs that are equivalent representations describing an object’s motion.

W

hen analyzing motion, particularly when it is more complex than the examples considered so far, it often is useful to represent the motion of an object in a variety of ways. As you have seen, a motion diagram contains useful information about an object’s position at various times and can be helpful in determining the displacement of an object during time intervals. Graphs of the object’s position and time also contain this information. Review Figure 2-9, the motion diagram for the runner with a location to the left of the tree chosen as the origin. From this motion diagram, you can organize the times and corresponding positions of the runner, as in Table 2-1.

Vocabulary position-time graph instantaneous position

Using a Graph to Find Out Where and When The data from Table 2-1 can be presented by plotting the time data on a horizontal axis and the position data on a vertical axis, which is called a position-time graph. The graph of the runner’s motion is shown in Figure 2-12. To draw this graph, first plot the runner’s recorded positions. Then, draw a line that best fits the recorded points. Notice that this graph is not a picture of the path taken by the runner as he was moving—the graphed line is sloped, but the path that he ran was flat. The line represents the most likely positions of the runner at the times between the recorded data points. (Recall from Chapter 1 that this line often is referred to as a best-fit line.) This means that even though there is no data point to tell you exactly when the runner was 30.0 m beyond his starting point or where he was at t 4.5 s, you can use the graph to estimate his position. The following example problem shows how. Note that before estimating the runner’s position, the questions first are restated in the language of physics in terms of positions and times.

■ Figure 2-12 A position-time graph for the runner can be created by plotting his known position at each of several times. After these points are plotted, the line that best fits them is drawn. The best-fit line indicates the runner’s most likely positions at the times between the data points.

Table 2-1 Position v. Time

Position v. Time

38

Position d (m)

0.0 1.0 2.0 3.0 4.0 5.0 6.0

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Chapter 2 Representing Motion

30.0 Position (m)

Time t (s)

25.0 20.0 15.0 10.0 5.0 0.0

1.0

2.0

3.0

4.0

Time (s)

5.0

6.0

When did the runner whose motion is described in Figure 2-12 reach 30.0 m beyond the starting point? Where was he after 4.5 s?

Analyze the Problem • Restate the questions. Question 1: At what time was the position of the object equal to 30.0 m? Question 2: What was the position of the object at 4.5 s?

2

Solve for the Unknown

Position v. Time 30.0 Position (m)

1

Question 1 Examine the graph to find the intersection of the best-fit line with a horizontal line at the 30.0-m mark. Next, find where a vertical line from that point crosses the time axis. The value of t there is 6.0 s.

25.0 20.0 15.0 10.0 5.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

Time (s)

Question 2 Find the intersection of the graph with a vertical line at 4.5 s (halfway between 4.0 s and 5.0 s on this graph). Next, find where a horizontal line from that point crosses the position axis. The value of d is approximately 22.5 m.

Math Handbook Interpolating and Extrapolating page 849

The two intersections are shown on the graph above.

For problems 9–11, refer to Figure 2-13.

11. Answer the following questions about the car’s motion. Assume that the positive d-direction is east and the negative d-direction is west.

Position (m)

9. Describe the motion of the car shown by the graph. 10. Draw a motion diagram that corresponds to the graph.

50.0 0.0 1.0

50.0

12. Describe, in words, the motion of the two pedestrians shown by the lines in Figure 2-14. Assume that the positive direction is east on Broad Street and the origin is the intersection of Broad and High Streets.

Broad St.

3.0

■

c. Create a graph showing Odina’s motion.

Figure 2-14

East

A High St. B

a. 25.0 m from the cafeteria b. 25.0 m from the band room

7.0

5.0

Time (s)

Position (m)

13. Odina walked down the hall at school from the cafeteria to the band room, a distance of 100.0 m. A class of physics students recorded and graphed her position every 2.0 s, noting that she moved 2.6 m every 2.0 s. When was Odina in the following positions?

Figure 2-13

100.0

a. When was the car 25.0 m east of the origin? b. Where was the car at 1.0 s?

■

150.0

West

Time (s)

Section 2.3 Position-Time Graphs

39

Table 2-1

a

b

Position v. Time

Time t (s)

Position d (m)

0.0 1.0 2.0 3.0 4.0 5.0 6.0

0.0 5.0 10.0 15.0 20.0 25.0 30.0

■ Figure 2-15 The data table (a), position-time graph (b), and particle model (c) all represent the same moving object.

Position (m)

Position v. Time

30.0 25.0 20.0 15.0 10.0 5.0 0.0

1.0 2.0 3.0 4.0 5.0 6.0 Time (s)

c

Begin

End

How long did the runner spend at any location? Each position has been linked to a time, but how long did that time last? You could say “an instant,” but how long is that? If an instant lasts for any finite amount of time, then the runner would have stayed at the same position during that time, and he would not have been moving. However, as he was moving, an instant is not a finite period of time. This means that an instant of time lasts zero seconds. The symbol d represents the instantaneous position of the runner—the position at a particular instant. Equivalent representations As shown in Figure 2-15, you now have several different ways to describe motion: words, pictures (or pictorial representations), motion diagrams, data tables, and position-time graphs. All of these representations are equivalent. That is, they can all contain the same information about the runner’s motion. However, depending on what you want to find out about an object’s motion, some of the representations will be more useful than others. In the pages that follow, you will get some practice constructing these equivalent representations and learning which ones are the easiest to use in solving different kinds of problems. Considering the motion of multiple objects A position-time graph for two different runners in a race is shown in Example Problem 2. When and where does one runner pass the other? First, you need to restate this question in physics terms: At what time do the two objects have the same position? You can evaluate this question by identifying the point on the position-time graph at which the lines representing the two objects intersect.

Niram, Oliver, and Phil all enjoy exercising and often go to a path along the river for this purpose. Niram bicycles at a very consistent 40.25 km/h, Oliver runs south at a constant speed of 16.0 km/h, and Phil walks south at a brisk 6.5 km/h. Niram starts biking north at noon from the waterfalls. Oliver and Phil both start at 11:30 A.M. at the canoe dock, 20.0 km north of the falls. 1. Draw position-time graphs for each person. 2. At what time will the three exercise enthusiasts be within the smallest distance interval?

3. What is the length of that distance interval?

40

Chapter 2 Representing Motion

When and where does runner B pass runner A?

2

Analyze the Problem

200

• Restate the question. At what time do A and B have the same position?

150

Solve for the Unknown

100

Position (m)

1

In the figure at right, examine the graph to find the intersection of the line representing the motion of A with the line representing the motion of B.

A 50

B

0 15

These lines intersect at 45 s and at about 190 m.

50

B passes A about 190 m beyond the origin, 45 s after A has passed the origin.

25

35

45

55

Time (s)

100

Math Handbook Interpolating and Extrapolating page 849

For problems 14–17, refer to the figure in Example Problem 2.

14. What event occurred at t 0.0 s? 15. Which runner was ahead at t 48.0 s? 16. When runner A was at 0.0 m, where was runner B? 17. How far apart were runners A and B at t 20.0 s? 18. Juanita goes for a walk. Sometime later, her friend Heather starts to walk after her. Their motions are represented by the position-time graphs in Figure 2-16. a. How long had Juanita been walking when Heather started her walk? b. Will Heather catch up to Juanita? How can you tell? 6.0

ta an i

4.0

Ju

Position (km)

5.0 er

th

3.0

a He

2.0 1.0 0.0

0.5

1.0

1.5

2.0

Time (h) ■

Figure 2-16

Section 2.3 Position-Time Graphs

41

As you have seen, you can represent the motion of more than one object on a position-time graph. The intersection of two lines tells you when the two objects have the same position. Does this mean that they will collide? Not necessarily. For example, if the two objects are runners and if they are in different lanes, they will not collide. Later in this textbook, you will learn to represent motion in two dimensions. Is there anything else that you can learn from a position-time graph? Do you know what the slope of a line means? In the next section, you will use the slope of a line on a position-time graph to determine the velocity of an object. What about the area under a plotted line? In Chapter 3, you will draw other graphs and learn to interpret the areas under the plotted lines. In later chapters you will continue to refine your skills with creating and interpreting graphs.

2.3 Section Review 19. Position-Time Graph From the particle model in Figure 2-17 of a baby crawling across a kitchen floor, plot a position-time graph to represent his motion. The time interval between successive dots is 1 s.

20

40

60

80

100

120

140

160

Position (cm) ■

Figure 2-17

20. Motion Diagram Create a particle model from the position-time graph of a hockey puck gliding across a frozen pond in Figure 2-18.

22. Distance Use the position-time graph of the hockey puck to determine how far it moved between 0.0 s and 5.0 s. 23. Time Interval Use the position-time graph for the hockey puck to determine how much time it took for the puck to go from 40 m beyond the origin to 80 m beyond the origin. 24. Critical Thinking Look at the particle model and position-time graph shown in Figure 2-19. Do they describe the same motion? How do you know? Do not confuse the position coordinate system in the partical model with the horizontal axis in the position-time graph. The time intervals in the partical model are 2 s.

140 Position (m)

120 0

100

Position (m)

80 60 40 12

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

Time (s) Figure 2-18

Position (m)

20

■

10

8

4

For problems 21–23, refer to Figure 2-18.

21. Time Use the position-time graph of the hockey puck to determine when it was 10.0 m beyond the origin. 42

Chapter 2 Representing Motion

1

2

3

4

5

Time (s) ■

Figure 2-19

physicspp.com/self_check_quiz

2.4 How Fast?

Y

ou have learned how to use a motion diagram to show an object’s movement. How can you measure how fast it is moving? With devices such as a meterstick and a stopwatch, you can measure position and time. Can this information be used to describe the rate of motion?

• Define velocity. • Differentiate between speed and velocity. • Create pictorial, physical, and mathematical models of motion problems.

Velocity Suppose you recorded two joggers on one motion diagram, as shown in Figure 2-20a. From one frame to the next you can see that the position of the jogger in red shorts changes more than that of the one wearing blue. In other words, for a fixed time interval, the displacement, d, is greater for the jogger in red because she is moving faster. She covers a larger distance than the jogger in blue does in the same amount of time. Now, suppose that each jogger travels 100.0 m. The time interval, t, would be smaller for the jogger in red than for the one in blue.

1.0 m/s b

6.0 5.0 r

2.0 m/s

gg e

4.0

jo

3.0 m 2.0 m 3.0 s 2.0 s

3.0

d

a

d d tf ti

f i Blue slope

Re

6.0 m 2.0 m 3.0 s 1.0 s

■ Figure 2-20 The red jogger’s displacement is greater than the displacement of the blue jogger in each time interval because the jogger in red is moving faster than the jogger in blue (a). The position-time graph represents the motion of the red and blue joggers. The points used to calculate the slope of each line are shown (b).

Position (m)

d d tf ti

Vocabulary average velocity average speed instantaneous velocity

Average velocity From the example of the joggers, you can see that both the displacement, d, and time interval, t, might be needed to create the quantity that tells how fast an object is moving. How might they be combined? Compare the lines representing the red and blue joggers in the position-time graphs in Figure 2-20b. The slope of the red jogger’s line is steeper than the slope of the blue jogger’s line. A steeper slope indicates a greater change in displacement during each time interval. Recall from Chapter 1 that to find the slope, you first choose two points on the line. Next, you subtract the vertical coordinate (d in this case) of the first point from the vertical coordinate of the second point to obtain the rise of the line. After that, you subtract the horizontal coordinate (t in this case) of the first point from the horizontal coordinate of the second point to obtain the run. Finally, you divide the rise by the run to obtain the slope. The slopes of the two lines shown in Figure 2-20b are found as follows: f i Red slope

Objectives

2.0 1.0 0.0

r

ge

og ej

Blu 1.0

2.0

Section 2.4 How Fast?

3.0

43

Hutchings Photography

There are some important things to notice about this comparison. First, the slope of the faster runner is a greater number, so it is reasonable to assume that this number might be connected with the runner’s speed. Second, look at the units of the slope, meters per second. In other words, the slope tells how many meters the runner moved in 1 s. These units are similar to miles per hour, which also measure speed. Looking at how the slope is calculated, you can see that slope is the change in position, divided by the time interval during which that change took place, or (df di) / (tf ti), or d/t. When d gets larger, the slope gets larger; when t gets larger, the slope gets smaller. This agrees with the interpretation above of the movements of the red and blue joggers. The slope of a position-time graph for an object is the object’s average velocity and is represented by the ratio of the change of position to the time interval during which the change occurred.

Speed Records The world record for the men’s 100-m dash is 9.78 s, established in 2002 by Tim Montgomery. The world record for the women’s 100-m dash is 10.65 s, established in 1998 by Marion Jones. These sprinters often are referred to as the world’s fastest man and woman.

Average Velocity

d d

d f i v tf ti t

Average velocity is defined as the change in position, divided by the time during which the change occurred.

The symbol means that the left-hand side of the equation is defined by the right-hand side. It is a common misconception to say that the slope of a position-time graph gives the speed of the object. Consider the slope of the position-time graph shown in Figure 2-21. The slope of this position-time graph is 5.0 m/s. As you can see the slope indicates both the magnitude and direction. Recall that average velocity is a quantity that has both magnitude and direction. The slope of a position-time graph indicates the average velocity of the object and not its speed. Take another look at Figure 2-21. The slope of the graph is 5.0 m/s and thus, the object has a velocity of 5.0 m/s. The object starts out at a positive position and moves toward the origin. It moves in the negative direction at a rate of 5.0 m/s.

• Velocity vectors are red. • Displacement vectors are green.

■ Figure 2-21 The object whose motion is represented here is moving in the negative direction at a rate of 5.0 m/s.

20 15

Position (m)

10 5 0 5

1

2

3 Time (s)

10 15

44

Chapter 2 Representing Motion

Average speed The absolute value of the slope of a position-time graph tells you the average speed of the object; that is, how fast the object is moving. The sign of the slope tells you in what direction the object is moving. The combination of an object’s average speed, v, and the direction in which it is moving is the average velocity, v. Thus, for the object represented in Figure 2-21, the average velocity is 5.0 m/s, or 5.0 m/s in the negative direction. Its average speed is 5.0 m/s. Remember that if an object moves in the negative direction, then its displacement is negative. This means that the object’s velocity always will have the same sign as the object’s displacement. As you consider other types of motion to analyze in future chapters, sometimes the velocity will be the 4 5 important quantity to consider, while at other times, the speed will be the important quantity. Therefore, it is a good idea to make sure that you understand the differences between velocity and speed so that you will be sure to use the right one later.

The graph at the right describes the motion of a student riding his skateboard along a smooth, pedestrian-free sidewalk. What is his average velocity? What is his average speed?

Analyze and Sketch the Problem • Identify the coordinate system of the graph.

2

Solve for the Unknown

Position (m)

1

12.0

Unknown: v ?

9.0

6.0

3.0

v ?

Find the average velocity using two points on the line. d

v t

0.0

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 Time (s)

Use magnitudes with signs indicating directions.

d d t2 t1

1 2

12.0 m 6.0 m 8.0 s 4.0 s

Substitute d2 12.0 m, d1 6.0 m, t2 8.0 s, t1 4.0 s.

Math Handbook

v 1.5 m/s in the positive direction

Slope page 850

The average speed, v, is the absolute value of the average velocity, or 1.5 m/s. 3

Evaluate the Answer • Are the units correct? m/s are the units for both velocity and speed. • Do the signs make sense? The positive sign for the velocity agrees with the coordinate system. No direction is associated with speed.

25. The graph in Figure 2-22 describes the motion of a cruise ship during its voyage through calm waters. The positive d-direction is defined to be south.

Time (s) Position (m)

1

a. What is the ship’s average speed? b. What is its average velocity? 26. Describe, in words, the motion of the cruise ship in the previous problem.

3

4

1 2

■

Figure 2-22 20 Position (km)

27. The graph in Figure 2-23 represents the motion of a bicycle. Determine the bicycle’s average speed and average velocity, and describe its motion in words.

2

28. When Marilyn takes her pet dog for a walk, the dog walks at a very consistent pace of 0.55 m/s. Draw a motion diagram and position-time graph to represent Marilyn’s dog walking the 19.8-m distance from in front of her house to the nearest fire hydrant.

15 10 5 0

■

Figure 2-23

5

10 15 20 25 30 Time (min)

Section 2.4 How Fast?

45

Instantaneous Velocity

■ Figure 2-24 Average velocity vectors have the same direction as their corresponding displacement vectors. Their lengths are different, but proportional, and they have different units because they are obtained by dividing the displacement by the time interval.

Why is the quantity d/t called average velocity? Why isn’t it called velocity? Think about how a motion diagram is constructed. A motion diagram shows the position of a moving object at the beginning and end of a time interval. It does not, however, indicate what happened within that time interval. During the time interval, the speed of the object could have remained the same, increased, or decreased. The object may have stopped or even changed direction. All that can be determined from the motion diagram is an average velocity, which is found by dividing the total displacement by the time interval in which it took place. The speed and direction of an object at a particular instant is called the instantaneous velocity. In this textbook, the term velocity will refer to instantaneous velocity, represented by the symbol v.

Average Velocity on Motion Diagrams

Instantaneous Velocity Vectors 1. Attach a 1-m-long string to your hooked mass. 2. Hold the string in one hand with the mass suspended. 3. Carefully pull the mass to one side and release it. 4. Observe the motion, the speed, and direction of the mass for several swings. 5. Stop the mass from swinging. 6. Draw a diagram showing instantaneous velocity vectors at the following points: top of the swing, midpoint between top and bottom, bottom of the swing, midpoint between bottom and top, and back to the top of the swing. Analyze and Conclude 7. Where was the velocity greatest? 8. Where was the velocity least? 9. Explain how the average speed can be determined using your vector diagram.

46

Chapter 2 Representing Motion

How can you show average velocity on a motion diagram? Although the average velocity is in the same direction as displacement, the two quantities are not measured using the same units. Nevertheless, they are proportional—when displacement is greater during a given time interval, so is average velocity. A motion diagram is not a precise graph of average velocity, but you can indicate the direction and magnitude of the average velocity on it. Imagine two cars driving down the road at different speeds. A video camera records their motions at the rate of one frame every second. Imagine that each car has a paintbrush attached to it that automatically descends and paints a line on the ground for half a second every second. The faster car would paint a longer line on the ground. The vectors you draw on a motion diagram to represent the velocity are like the lines made by the paintbrushes on the ground below the cars. Red is used to indicate velocity vectors on motion diagrams. Figure 2-24 shows the particle models, complete with velocity vectors, for two cars: one moving to the right and the other moving more slowly to the left. Using equations Any time you graph a straight line, you can find an equation to describe it. There will be many cases for which it will be more efficient to use such an equation instead of a graph to solve problems. Take another look at the graph in Figure 2-21 on page 44 for the object moving with a constant velocity of 5.0 m/s. Recall from Chapter 1 that any straight line can be represented by the formula: y mx b where y is the quantity plotted on the vertical axis, m is the slope of the line, x is the quantity plotted on the horizontal axis, and b is the y-intercept of the line. For the graph in Figure 2-21, the quantity plotted on the vertical axis is position, and the variable used to represent position is d. The slope of the line is 5.0 m/s, which is the object’s average velocity, v. The quantity plotted on the horizontal axis is time, t. The y-intercept is 20.0 m. What does this 20.0 m represent? By inspecting the graph and thinking about how the object moves, you can figure out that the object was at a position of 20.0 m when t 0.0 s. This is called the initial position of the object, and is designated di. Table 2-2 summarizes how the general variables in the straight-line formula are changed to the specific variables you have been using to describe motion. It also shows the numerical values for the two constants in this equation.

Based on the information shown in Table 2-2, the equation y mx b becomes d vt di, or, by inserting the values of the constants, d (5.0 m/s)t 20.0 m. This equation describes the motion that is represented in Figure 2-21. You can check this by plugging a value of t into the equation and seeing that you obtain the same value of d as when you read it directly from the graph. To conduct an extra check to be sure the equation makes sense, take a look at the units. You cannot set two items with different units equal to each other in an equation. In this equation, the left-hand side is a position, so its units are meters. The first term on the right-hand side multiplies meters per second times seconds, so its units are also meters. The last term is in meters, too, so the units on this equation are valid. Equation of Motion for Average Velocity

Table 2-2 Comparison of Straight Lines with Position-Time Graphs General Variable

Specific Motion Variable

y m x b

d v t di

Value in Figure 2-21

5.0 m/s 20.0 m

d vt di

An object’s position is equal to the average velocity multiplied by time plus the initial position.

This equation gives you another way to represent the motion of an object. Note that once a coordinate system is chosen, the direction of d is specified by positive and negative values, and the boldface notation can be dispensed with, as in “d-axis.” However, the boldface notation for velocity will be retained to avoid confusing it with speed. Your toolbox of representations now includes words, motion diagrams, pictures, data tables, position-time graphs, and an equation. You should be able to use any one of these representations to generate at least the characteristics of the others. You will get more practice with this in the rest of this chapter and also in Chapter 3 as you apply these representations to help you solve problems.

2.4 Section Review For problems 29–31, refer to Figure 2-25.

29. Average Speed Rank the position-time graphs according to the average speed of the object, from greatest average speed to least average speed. Specifically indicate any ties.

Position (m)

B

D

A

Figure 2-25

31. Initial Position Rank the graphs according to the object’s initial position, from most positive position to most negative position. Specifically indicate any ties. Would your ranking be different if you had been asked to do the ranking according to initial distance from the origin? 32. Average Speed and Average Velocity Explain how average speed and average velocity are related to each other.

C ■

30. Average Velocity Rank the graphs according to average velocity, from greatest average velocity to least average velocity. Specifically indicate any ties.

Time (s) physicspp.com/self_check_quiz

33. Critical Thinking In solving a physics problem, why is it important to create pictorial and physical models before trying to solve an equation? Section 2.4 How Fast?

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In this activity you will construct motion diagrams for two toy cars. A motion diagram consists of a series of images showing the positions of a moving object at equal time intervals. Motion diagrams help describe the motion of an object. By looking at a motion diagram you can determine whether an object is speeding up, slowing down, or moving at constant speed.

QUESTION How do the motion diagrams of a fast toy car and a slow toy car differ?

Objectives

Procedure

■ Measure in SI the location of a moving object. ■ Recognize spatial relationships of moving

1. Mark a starting line on the lab table or the surface recommended by your teacher.

objects. ■ Describe the motion of a fast and slow object.

2. Place both toy cars at the starting line and release them at the same time. Be sure to wind them up before releasing them.

Safety Precautions

3. Observe both toy cars and determine which one is faster. 4. Place the slower toy car at the starting line. 5. Place a meterstick parallel to the path the toy car will take.

Materials video camera two toy windup cars meterstick foam board

6. Have one of the members of your group get ready to operate the video camera. 7. Release the slower toy car from the starting line. Be sure to wind up the toy car before releasing it. 8. Use the video camera to record the slower toy car’s motion parallel to the meterstick. 9. Set the recorder to play the tape frame by frame. Replay the video tape for 0.5 s, pressing the pause button every 0.1 s (3 frames). 10. Determine the toy car’s position for each time interval by reading the meterstick on the video tape and record it in the data table. 11. Repeat steps 5–10 with the faster car. 12. Place a piece of foam board at an angle of approximately 30° to form a ramp. 13. Place the meterstick on the ramp so that it is parallel to the path the toy car will take. 14. Place the slower toy car at the top of the ramp and repeat steps 6–10.

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Creating Motion Diagrams Alternate CBL instructions can be found on the Web site. physicspp.com

Data Table 1 Time (s)

Position of the Slower Toy Car (cm)

0.0 0.1 0.2 0.3 0.4 0.5

Data Table 2 Time (s)

Data Table 3 Position of the Faster Toy Car (cm)

Time (s)

0.0

0.0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.5

Analyze 1. Draw a motion diagram for the slower toy car using the data you collected. 2. Draw a motion diagram for the faster toy car using the data you collected. 3. Using the data you collected, draw a motion diagram for the slower toy car rolling down the ramp.

Position of the Slower Toy Car on the Ramp (cm)

4. What happens to the distance between points in the motion diagram in the previous question as the car slows down? 5. Draw a motion diagram for a car that starts moving slowly and then begins to speed up. 6. What happens to the distance between points in the motion diagram in the previous question as the car speeds up?

Real-World Physics Conclude and Apply How is the motion diagram of the faster toy car different from the motion diagram of the slower toy car?

Going Further

Suppose a car screeches to a halt to avoid an accident. If that car has antilock brakes that pump on and off automatically every fraction of a second, what might the tread marks on the road look like? Include a drawing along with your explanation of what the pattern of tread marks on the road might look like.

1. Draw a motion diagram for a car moving at a constant speed. 2. What appears to be the relationship between the distances between points in the motion diagram of a car moving at a constant speed? 3. Draw a motion diagram for a car that starts moving fast and then begins to slow down.

To find out more about representing motion, visit the Web site: physicspp.com

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Accurate Time What time is it really? You might use a clock to find out what time it is at any moment. A clock is a device that counts regularly recurring events in order to measure time. Suppose the clock in your classroom reads 9:00. Your watch, however, reads 8:55, and your friend’s watch reads 9:02. So what time is it, really? Which clock or watch is accurate? Many automated processes are controlled by clocks. For example, an automated bell that signals the end of a class period is controlled by a clock. Thus, if you wanted to be on time for a class you would have to synchronize your watch to the one controlling the bell. Other processes, such as GPS navigation, space travel, internet synchronization, transportation, and communication, rely on clocks with extreme precision and accuracy. A reliable standard clock that can measure when exactly one second has elapsed is needed.

The Standard Cesium Clock Atomic clocks, such as cesium clocks, address this need. Atomic clocks measure the number of times the atom used in the clock switches its energy state. Such oscillations in an atom’s energy occur very quickly and regularly. The National Institute of Standards and Technology (NIST) currently uses the oscillations of the cesium atom to determine the standard 1-s interval. One second is defined as the duration of 9,192,631,770 oscillations of the cesium atom. The cesium atom has a single electron in its outermost energy level. This outer electron spins and behaves like a miniature magnet. The cesium nucleus also spins and acts like a miniature magnet. The nucleus and electron may spin in such a manner that their north magnetic poles are aligned. The nucleus and electron also may spin in a way that causes opposite poles to be aligned. If the poles are 50

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National Institute of Standards and Technology

aligned, the cesium atom is in one energy state. If they are oppositely aligned, the atom is in another energy state. A microwave with a particular frequency can strike a cesium atom and cause the outside spinning electron to switch its magnetic pole orientation and change the atom’s energy state. As a result, the atom emits light. This occurs at cesium’s natural frequency of 9,192,631,770 cycles/s. This principle was used to design the cesium clock.

The cesium clock, NIST-F1, located at the NIST laboratories in Boulder, Colorado is among the most accurate clocks in the world.

How Does the Cesium Clock Work? The cesium clock consists of cesium atoms and a quartz crystal oscillator, which produces microwaves. When the oscillator’s microwave signal precisely equals cesium’s natural frequency, a large number of cesium atoms will change their energy state. Cesium’s natural frequency is equal to 9,192,631,770 microwave cycles. Thus, there are 9,192,631,770 cesium energy level changes in 1 s. Cesium clocks are so accurate that a modern cesium clock is off by less than 1 s in 20 million years.

Going Further 1. Research What processes require the precise measurement of time? 2. Analyze and Conclude Why is the precise measurement of time essential to space navigation?

2.1 Picturing Motion Vocabulary

Key Concepts

• motion diagram (p. 33) • particle model (p. 33)

• •

A motion diagram shows the position of an object at successive times. In the particle model, the object in the motion diagram is replaced by a series of single points.

2.2 Where and When? Vocabulary

Key Concepts

• • • • • • • • • •

•

coordinate system (p. 34) origin (p. 34) position (p. 34) distance (p. 34) magnitude (p. 35) vectors (p. 35) scalars (p. 35) resultant (p. 35) time interval (p. 36) displacement (p. 36)

•

You can define any coordinate system you wish in describing motion, but some are more useful than others. A time interval is the difference between two times. t tf ti

• •

A vector drawn from the origin of the coordinate system to the object indicates the object’s position. Change in position is displacement, which has both magnitude and direction. d df di

•

The length of the displacement vector represents how far the object was displaced, and the vector points in the direction of the displacement.

2.3 Position-Time Graphs Vocabulary

Key Concepts

• position-time graph (p. 38) • instantaneous position

• •

(p. 40)

Position-time graphs can be used to find the velocity and position of an object, as well as where and when two objects meet. Any motion can be described using words, motion diagrams, data tables, and graphs.

2.4 How Fast? Vocabulary

Key Concepts

• average velocity (p. 44) • average speed (p. 44) • instantaneous velocity

•

The slope of an object’s position-time graph is the average velocity of the object’s motion. ∆d

d d

f i v t t ∆t

(p. 46)

f

• • •

i

The average speed is the absolute value of the average velocity. An object’s velocity is how fast it is moving and in what direction it is moving. An object’s initial position, di, its constant average velocity, v, its position, d, and the time, t, since the object was at its initial position are related by a simple equation. d vt di

physicspp.com/vocabulary_puzzlemaker

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Concept Mapping following terms: words, equivalent representations, position-time graph.

data table

motion diagram

46. Figure 2-26 is a graph of two people running. a. Describe the position of runner A relative to runner B at the y-intercept. b. Which runner is faster? c. What occurs at point P and beyond?

er

B

n un

Position (m)

34. Complete the concept map below using the

R

ner

Run

A

Mastering Concepts Time (s)

35. What is the purpose of drawing a motion diagram?

■

(2.1)

36. Under what circumstances is it legitimate to treat an object as a point particle? (2.1)

37. The following quantities describe location or its change: position, distance, and displacement. Briefly describe the differences among them. (2.2)

47. The position-time graph in Figure 2-27 shows the motion of four cows walking from the pasture back to the barn. Rank the cows according to their average velocity, from slowest to fastest.

40. Walking Versus Running A walker and a runner

Be

Position (m)

graphs for two in-line skaters to determine if and when one in-line skater will pass the other one? (2.3)

Els

(2.2)

ss

ie

ie

38. How can you use a clock to find a time interval? 39. In-line Skating How can you use the position-time

da

olin

Mo

lly

Do

leave your front door at the same time. They move in the same direction at different constant velocities. Describe the position-time graphs of each. (2.4)

Time (s)

41. What does the slope of a position-time graph

■

measure? (2.4)

42. If you know the positions of a moving object at two points along its path, and you also know the time it took for the object to get from one point to the other, can you determine the particle’s instantaneous velocity? Its average velocity? Explain. (2.4)

running away from a dog. a. Describe how this graph would be different if the rabbit ran twice as fast. b. Describe how this graph would be different if the rabbit ran in the opposite direction. 3 Position (m)

each does not have the properties needed to describe the concept of velocity: d t, d t, d t, t/d.

Figure 2-27

48. Figure 2-28 is a position-time graph for a rabbit

Applying Concepts 43. Test the following combinations and explain why

Figure 2-26

2 1

44. Football When can a football be considered a point particle?

45. When can a football player be treated as a point particle?

52

1

2

Time (s)

Chapter 2 Representing Motion For more problems, go to Additional Problems, Appendix B.

3 ■

Figure 2-28

Mastering Problems

56. Figure 2-30 shows position-time graphs for Joszi

2.4 How Fast? 49. A bike travels at a constant speed of 4.0 m/s for 5.0 s. How far does it go?

50. Astronomy Light from the Sun reaches Earth in 8.3 min. The speed of light is 3.00108 m/s. How far is Earth from the Sun?

and Heike paddling canoes in a local river. a. At what time(s) are Joszi and Heike in the same place? b. How much time does Joszi spend on the river before he passes Heike? c. Where on the river does it appear that there might be a swift current? 18

51. A car is moving down a street at 55 km/h. A child

52. Nora jogs several times a week and always keeps track of how much time she runs each time she goes out. One day she forgets to take her stopwatch with her and wonders if there’s a way she can still have some idea of her time. As she passes a particular bank, she remembers that it is 4.3 km from her house. She knows from her previous training that she has a consistent pace of 4.0 m/s. How long has Nora been jogging when she reaches the bank?

53. Driving You and a friend each drive 50.0 km. You travel at 90.0 km/h; your friend travels at 95.0 km/h. How long will your friend have to wait for you at the end of the trip?

Mixed Review 54. Cycling A cyclist maintains a constant velocity of 5.0 m/s. At time t 0.0 s, the cyclist is 250 m from point A. a. Plot a position-time graph of the cyclist’s location from point A at 10.0-s intervals for 60.0 s. b. What is the cyclist’s position from point A at 60.0 s? c. What is the displacement from the starting position at 60.0 s?

16

Joszi

14 Position (km)

suddenly runs into the street. If it takes the driver 0.75 s to react and apply the brakes, how many meters will the car have moved before it begins to slow down?

Heike

12 10 8 6 4 2 0

0.5

1.0

1.5

2.0

2.5

Time (h) ■

Figure 2-30

57. Driving Both car A and car B leave school when a stopwatch reads zero. Car A travels at a constant 75 km/h, and car B travels at a constant 85 km/h. a. Draw a position-time graph showing the motion of both cars. How far are the two cars from school when the stopwatch reads 2.0 h? Calculate the distances and show them on your graph. b. Both cars passed a gas station 120 km from the school. When did each car pass the gas station? Calculate the times and show them on your graph.

58. Draw a position-time graph for two cars traveling to the beach, which is 50 km from school. At noon, Car A leaves a store that is 10 km closer to the beach than the school is and moves at 40 km/h. Car B starts from school at 12:30 P.M. and moves at 100 km/h. When does each car get to the beach?

59. Two cars travel along a straight road. When a 55. Figure 2-29 is a particle model for a chicken casually walking across the road. Time intervals are every 0.1 s. Draw the corresponding positiontime graph and equation to describe the chicken’s motion. This side

The other side

Time intervals are 0.1 s. ■

Figure 2-29 physicspp.com/chapter_test

stopwatch reads t 0.00 h, car A is at dA 48.0 km moving at a constant 36.0 km/h. Later, when the watch reads t 0.50 h, car B is at dB 0.00 km moving at 48.0 km/h. Answer the following questions, first, graphically by creating a positiontime graph, and second, algebraically by writing equations for the positions dA and dB as a function of the stopwatch time, t. a. What will the watch read when car B passes car A? b. At what position will car B pass car A? c. When the cars pass, how long will it have been since car A was at the reference point? Chapter 2 Assessment

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60. Figure 2-31 shows the position-time graph depicting Jim’s movement up and down the aisle at a store. The origin is at one end of the aisle. a. Write a story describing Jim’s movements at the store that would correspond to the motion represented by the graph. b. When does Jim have a position of 6.0 m? c. How much time passes between when Jim enters the aisle and when he gets to a position of 12.0 m? What is Jim’s average velocity between 37.0 s and 46.0 s? 14.0 12.0

Position (m)

10.0

4.0

red motorcycle is driven past your friend’s home, his father becomes angry because he thinks the motorcycle is going too fast for the posted 25 mph (40 km/h) speed limit. Describe a simple experiment you could do to determine whether or not the motorcycle is speeding the next time it is driven past your friend’s house.

Writing in Physics 10.0

20.0

30.0

40.0

50.0

60.0

Time (s)

Figure 2-31

Thinking Critically 61. Apply Calculators Members of a physics class stood 25 m apart and used stopwatches to measure the time at which a car traveling on the highway passed each person. Their data are shown in Table 2-3.

Table 2-3 Position v. Time Position (m) Time 0.0 1.3 2.7 3.6 5.1 5.9 7.0 8.6 10.3

0.0 25.0 50.0 75.0 100.0 125.0 150.0 175.0 200.0

Use a graphing calculator to fit a line to a positiontime graph of the data and to plot this line. Be sure to set the display range of the graph so that all the data fit on it. Find the slope of the line. What was the speed of the car?

54

63. Design an Experiment Every time a particular

position-time graph to be a horizontal line? A vertical line? If you answer yes to either situation, describe the associated motion in words.

6.0

2.0

■

want to average 90 km/h. You cover the first half of the distance at an average speed of only 48 km/h. What must your average speed be in the second half of the trip to meet your goal? Is this reasonable? Note that the velocities are based on half the distance, not half the time.

64. Interpret Graphs Is it possible for an object’s

8.0

0.00

62. Apply Concepts You plan a car trip for which you

65. Physicists have determined that the speed of light is 3.00108 m/s. How did they arrive at this number? Read about some of the series of experiments that were done to determine light’s speed. Describe how the experimental techniques improved to make the results of the experiments more accurate.

66. Some species of animals have good endurance, while others have the ability to move very quickly, but for only a short amount of time. Use reference sources to find two examples of each quality and describe how it is helpful to that animal.

Cumulative Review 67. Convert each of the following time measurements to its equivalent in seconds. (Chapter 1) a. 58 ns

c. 9270 ms

b. 0.046 Gs

d. 12.3 ks

68. State the number of significant digits in the following measurements. (Chapter 1) a. 3218 kg

c. 801 kg

b. 60.080 kg

d. 0.000534 kg

69. Using a calculator, Chris obtained the following results. Rewrite the answer to each operation using the correct number of significant digits. (Chapter 1) a. 5.32 mm 2.1 mm 7.4200000 mm b. 13.597 m 3.65 m 49.62905 m2 c. 83.2 kg 12.804 kg 70.3960000 kg

Chapter 2 Representing Motion For more problems, go to Additional Problems, Appendix B.

4. When is the person on the bicycle farthest away from the starting point? point A point B

The dots would form an evenly spaced pattern.

The dots would be close together to start, get farther apart, and become close together again as the airplane leveled off at cruising speed. 2. Which of the following statements about drawing vectors is false? A vector diagram is needed to solve all physics problems properly. The length of the vector should be proportional to the data. Vectors can be added by measuring the length of each vector and then adding them together.

section I section II

6. A squirrel descends an 8-m tree at a constant speed in 1.5 min. It remains still at the base of the tree for 2.3 min, and then walks toward an acorn on the ground for 0.7 min. A loud noise causes the squirrel to scamper back up the tree in 0.1 min to the exact position on the branch from which it started. Which of the following graphs would accurately represent the squirrel’s vertical displacement from the base of the tree?

Position

III II I

Position (m)

C

Time (min)

Time (min)

Vectors can be added in straight lines or in triangle formations. Use this graph for problems 3–5.

section III point IV

Position (m)

The dots would be close together to start with, and get farther apart as the plane accelerated.

5. Over what interval does the person on the bicycle travel the greatest distance?

Position (m)

The dots would be far apart at the beginning, but get closer together as the plane accelerated.

point C point D

Position (m)

Multiple Choice 1. Which of the following statements would be true about the particle model motion diagram for an airplane taking off from an airport?

IV D

Time (min)

Time (min)

B

Extended Answer 7. Find a rat’s total displacement at the exit if it takes the following path in a maze: start, 1.0 m north, 0.3 m east, 0.8 m south, 0.4 m east, finish.

A Time

3. The graph shows the motion of a person on a bicycle. When does the person have the greatest velocity? section I

point D

section III

point B

physicspp.com/standardized_test

Stock up on Supplies Bring all your test-taking tools: number two pencils, black and blue pens, erasers, correction fluid, a sharpener, a ruler, a calculator, and a protractor.

Chapter 2 Standardized Test Practice

55

What You’ll Learn • You will develop descriptions of accelerated motion. • You will use graphs and equations to solve problems involving moving objects. • You will describe the motion of objects in free fall.

Why It’s Important Objects do not always move at constant velocities. Understanding accelerated motion will help you better decribe the motion of many objects. Acceleration Cars, planes, subways, elevators, and other common forms of transportation often begin their journeys by speeding up quickly, and end by stopping rapidly.

Think About This The driver of a dragster on the starting line waits for the green light to signal the start of the race. At the signal, the driver will step on the gas pedal and try to speed up as quickly as possible. As the car speeds up, how will its position change?

physicspp.com 56 Rob Tringali/SportsChrome

Do all types of motion look the same when graphed? Question How does a graph showing constant speed compare to a graph of a vehicle speeding up? Procedure 1. Clamp a spark timer to the back edge of a lab table. 2. Cut a piece of timer tape approximately 50 cm in length, insert it into the timer, and tape it to vehicle 1. 3. Turn on the timer and release the vehicle. Label the tape with the vehicle number. 4. Raise one end of the lab table 8–10 cm by placing a couple of bricks under the back legs. CAUTION: Make sure the lab table remains stable. 5. Repeat steps 2–4 with vehicle 2, but hold the vehicle in place next to the timer and release it after the timer has been turned on. Catch the vehicle before it falls. 6. Construct and Organize Data Mark the first dark dot where the timer began as zero. Measure the distance to each dot from the zero dot for 10 intervals and record your data.

7. Make and Use Graphs Make a graph of total distance versus interval number. Place data for both vehicles on the same plot and label each graph. Analysis Which vehicle moved with constant speed? Which one sped up? Explain how you determined this by looking at the timer tape. Critical Thinking Describe the shape of each graph. How does the shape of the graph relate to the type of motion observed?

3.1 Acceleration

U

niform motion is one of the simplest kinds of motion. You learned in Chapter 2 that an object in uniform motion moves along a straight line with an unchanging velocity. From your own experiences, you know, however, that few objects move in this manner all of the time. In this chapter, you will expand your knowledge of motion by considering a slightly more complicated type of motion. You will be presented with situations in which the velocity of an object changes, while the object’s motion is still along a straight line. Examples of objects and situations you will encounter in this chapter include automobiles that are speeding up, drivers applying brakes, falling objects, and objects thrown straight upward. In Chapter 6, you will continue to add to your knowledge of motion by analyzing some common types of motion that are not confined to a straight line. These include motion along a circular path and the motion of thrown objects, such as baseballs.

Objectives • Define acceleration. • Relate velocity and acceleration to the motion of an object. • Create velocity-time graphs.

Vocabulary velocity-time graph acceleration average acceleration instantaneous acceleration

Section 3.1 Acceleration

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Changing Velocity

A Steel Ball Race If two steel balls are released at the same instant, will the steel balls get closer or farther apart as they roll down a ramp? 1. Assemble an inclined ramp from a piece of U-channel or two metersticks taped together. 2. Measure 40 cm from the top of the ramp and place a mark there. Place another mark 80 cm from the top. 3. Predict whether the steel balls will get closer or farther apart as they roll down the ramp. 4. At the same time, release one steel ball from the top of the ramp and the other steel ball from the 40-cm mark. 5. Next, release one steel ball from the top of the ramp. As soon as it reaches the 40-cm mark, release the other steel ball from the top of the ramp. Analyze and Conclude 6. Explain your observations in terms of velocities. 7. Do the steel balls have the same velocity as they roll down the ramp? Explain. 8. Do they have the same acceleration? Explain.

You can feel a difference between uniform and nonuniform motion. Uniform motion feels smooth. You could close your eyes and it would feel as though you were not moving at all. In contrast, when you move along a curve or up and down a roller coaster, you feel pushed or pulled. Consider the motion diagrams shown in Figure 3-1. How would you describe the motion of the person in each case? In one diagram, the person is motionless. In another, she is moving at a constant speed. In a third, she is speeding up, and in a fourth, she is slowing down. How do you know which one is which? What information do the motion diagrams contain that could be used to make these distinctions? The most important thing to notice in these motion diagrams is the distance between successive positions. You learned in Chapter 2 that motionless objects in the background of motion diagrams do not change positions. Therefore, because there is only one image of the person in Figure 3-1a, you can conclude that she is not moving; she is at rest. Figure 3-1b is like the constant-velocity motion diagrams in Chapter 2. The distances between images are the same, so the jogger is moving at a constant speed. The distance between successive positions changes in the two remaining diagrams. If the change in position gets larger, the jogger is speeding up, as shown in Figure 3-1c. If the change in position gets smaller, as in Figure 3-1d, the jogger is slowing down. What does a particle-model motion diagram look like for an object with changing velocity? Figure 3-2 shows the particle-model motion diagrams below the motion diagrams of the jogger speeding up and slowing down. There are two major indicators of the change in velocity in this form of the motion diagram. The change in the spacing of the dots and the differences in the lengths of the velocity vectors indicate the changes in velocity. If an object speeds up, each subsequent velocity vector is longer. If the object slows down, each vector is shorter than the previous one. Both types of motion diagrams give an idea of how an object’s velocity is changing.

Velocity-Time Graphs Just as it was useful to graph a changing position versus time, it also is useful to plot an object’s velocity versus time, which is called a velocitytime, or v-t graph. Table 3-1 on the next page shows the data for a car that starts at rest and speeds up along a straight stretch of road.

■ Figure 3-1 By noting the distance the jogger moves in equal time intervals, you can determine that the jogger is standing still (a), moving at a constant speed (b), speeding up (c), and slowing down (d).

58

Chapter 3 Accelerated Motion

a

b

c

d

■ Figure 3-2 The particle-model version of the motion diagram indicates the runner’s changing velocity not only by the change in spacing of the position dots, but also by the change in length of the velocity vectors.

The velocity-time graph obtained by plotting these data points is shown in Figure 3-3. The positive direction has been chosen to be the same as that of the motion of the car. Notice that this graph is a straight line, which means that the car was speeding up at a constant rate. The rate at which the car’s velocity is changing can be found by calculating the slope of the velocity-time graph. The graph shows that the slope is (10.0 m/s)/(2.00 s), or 5.00 m/s2. This means that every second, the velocity of the car increased by 5.00 m/s. Consider a pair of data points that are separated by 1 s, such as 4.00 s and 5.00 s. At 4.00 s, the car was moving at a velocity of 20.0 m/s. At 5.00 s, the car was traveling at 25.0 m/s. Thus, the car’s velocity increased by 5.00 m/s in 1.00 s. The rate at which an object’s velocity changes is called the acceleration of the object. When the velocity of an object changes at a constant rate, it has a constant acceleration.

Average and Instantaneous Acceleration The average acceleration of an object is the change in velocity during some measurable time interval divided by that time interval. Average acceleration is measured in m/s2. The change in velocity at an instant of time is called instantaneous acceleration. The instantaneous acceleration of an object can be found by drawing a tangent line on the velocity-time graph at the point of time in which you are interested. The slope of this line is equal to the instantaneous acceleration. Most of the situations considered in this textbook involve motion with acceleration in which the average and instantaneous accelerations are equal.

■ Figure 3-3 The slope of a velocity-time graph is the acceleration of the object represented.

Velocity v. Time

25.0

Table 3-1 20.0

Time (s)

Velocity (m/s)

0.00 1.00 2.00 3.00 4.00 5.00

0.00 5.00 10.0 15.0 20.0 25.0

Velocity (m/s)

Velocity v. Time

15.0

10.0 rise run

m

5.00

10.0 m/s 2.00 s

5.00 m/s2 0.00

1.00

2.00

3.00

4.00

5.00

Time (s) Section 3.1 Acceleration

59

■ Figure 3-4 Looking at two consecutive velocity vectors and finding the difference between them yields the average acceleration vector for that time interval.

a

vi

vf

vf

b vi vf

c

∆v

vi

a

Displaying Acceleration on a Motion Diagram • Acceleration vectors are violet. • Velocity vectors are red. • Displacement vectors are green.

For a motion diagram to give a full picture of an object’s movement, it also should contain information about acceleration. This can be done by including average acceleration vectors. These vectors will indicate how the velocity is changing. To determine the length and direction of an average acceleration vector, subtract two consecutive velocity vectors, as shown in Figures 3-4a and b. That is, v vf vi vf (vi). Then divide by the time interval, t. In Figures 3-4a and b, the time interval, t, is 1 s. This vector, (vf vi)/1 s, shown in violet in Figure 3-4c, is the average acceleration during that time interval. The velocities vi and vf refer to the velocities at the beginning and end of a chosen time interval.

Velocity and Acceleration How would you describe the sprinter’s velocity and acceleration as shown on the graph?

Analyze and Sketch the Problem • From the graph, note that the sprinter’s velocity starts at zero, increases rapidly for the first few seconds, and then, after reaching about 10.0 m/s, remains almost constant.

2

Known:

Unknown:

v varies

a?

12.0 Velocity (m/s)

1

Solve for the Unknown Draw a tangent to the curve at t 1.0 s and t 5.0 s. Solve for acceleration at 1.0 s: The slope of the line at 1.0 s is equal to rise a the acceleration at that time. run 11.0 m/s 2.8 m/s 2.4 s 0.00 s

3.4 m/s2 Solve for acceleration at 5.0 s: The slope of the line at 5.0 s is equal to rise a the acceleration at that time. run 10.3 m/s 10.0 m/s 10.0 s 0.00 s

6.0

0.00

5.00 Time (s)

Math Handbook Slope page 850

0.030 m/s2 The acceleration is not constant because it changes from 3.4 m/s2 to 0.03 m/s2 at 5.0 s. The acceleration is in the direction chosen to be positive because both values are positive. 3

Evaluate the Answer • Are the units correct? Acceleration is measured in m/s2.

60

Chapter 3 Accelerated Motion

10.0

Velocity (m/s)

1. A dog runs into a room and sees a cat at the other end of the room. The dog instantly stops running but slides along the wood floor until he stops, by slowing down with a constant acceleration. Sketch a motion diagram for this situation, and use the velocity vectors to find the acceleration vector. 2. Figure 3-5 is a v-t graph for Steven as he walks along the midway at the state fair. Sketch the corresponding motion diagram, complete with velocity vectors.

3. Refer to the v-t graph of the toy train in Figure 3-6 to answer the following questions. a. When is the train’s speed constant? b. During which time interval is the train’s acceleration positive? c. When is the train’s acceleration most negative?

1

2

3

4

5

6

7

8

9 10

Time (s) ■

Figure 3-5

12.0 Velocity (m/s)

4. Refer to Figure 3-6 to find the average acceleration of the train during the following time intervals. a. 0.0 s to 5.0 s b. 15.0 s to 20.0 s c. 0.0 s to 40.0 s 5. Plot a v-t graph representing the following motion. An elevator starts at rest from the ground floor of a three-story shopping mall. It accelerates upward for 2.0 s at a rate of 0.5 m/s2, continues up at a constant velocity of 1.0 m/s for 12.0 s, and then experiences a constant downward acceleration of 0.25 m/s2 for 4.0 s as it reaches the third floor.

10.0 8.0 6.0 4.0 2.0 0.0

10.0 20.0 30.0 40.0 Time (s) ■

Figure 3-6

Positive and Negative Acceleration Consider the four situations shown in Figure 3-7a. The first motion diagram shows an object moving in the positive direction and speeding up. The second motion diagram shows the object moving in the positive direction and slowing down. The third shows the object speeding up in the negative direction, and the fourth shows the object slowing down as it moves in the negative direction. Figure 3-7b shows the velocity vectors for the second time interval of each diagram, along with the corresponding acceleration vectors. Note ∆t is equal to 1 s. a

Begin

End

v2

b

v1

v a Begin

End

v

v2 v1

a

v2 End

Begin v

v1

a v

v2 End

Begin

v1 a

■ Figure 3-7 These four motion diagrams represent the four different possible ways to move along a straight line with constant acceleration (a). When the velocity vectors of the motion diagram and acceleration vectors point in the same direction, an object’s speed increases. When they point in opposite directions, the object slows down (b).

Section 3.1 Acceleration

61

East

Velocity (m/s)

A

C

E

West ■ Figure 3-8 Graphs A and E show motion with constant velocity in opposite directions. Graph B shows both positive velocity and positive acceleration. Graph C shows positive velocity and negative acceleration. Graph D shows motion with constant positive acceleration that slows down while velocity is negative and speeds up when velocity is positive.

In the first and third situations when the object is speeding up, the velocity and acceleration vectors point in the same direction in each case, as shown in Figure 3-7b. In the other two situations in which the acceleration vector is in the opposite direction from B the velocity vectors, the object is slowing down. In other words, when the object’s D acceleration is in the same direction as its velocity, the object’s speed increases. When Time (s) they are in opposite directions, the speed decreases. Both the direction of an object’s velocity and its direction of acceleration are needed to determine whether it is speeding up or slowing down. An object has a positive acceleration when the acceleration vector points in the positive direction and a negative acceleration, when the acceleration vector points in the negative direction. The sign of acceleration does not indicate whether the object is speeding up or slowing down.

Determining Acceleration from a v-t Graph Velocity and acceleration information also is contained in velocity-time graphs. Graphs A, B, C, D, and E, shown in Figure 3-8, represent the motions of five different runners. Assume that the positive direction has been chosen to be east. The slopes of Graphs A and E are zero. Thus, the accelerations are zero. Both Graphs A and E show motion at a constant velocity—Graph A to the east and Graph E to the west. Graph B shows motion with a positive velocity. The slope of this graph indicates a constant, positive acceleration. You also can infer from Graph B that the speed increased because it shows positive velocity and acceleration. Graph C has a negative slope. Graph C shows motion that begins with a positive velocity, slows down, and then stops. This means that the acceleration and velocity are in opposite directions. The point at which Graphs C and B cross shows that the runners’ velocities are equal at that point. It does not, however, give any information about the runners’ positions. Graph D indicates movement that starts out toward the west, slows down, and for an instant gets to zero velocity, and then moves east with increasing speed. The slope of Graph D is positive. Because the velocity and acceleration are in opposite directions, the speed decreases and equals zero at the time the graph crosses the axis. After that time, the velocity and acceleration are in the same direction and the speed increases. Calculating acceleration How can you describe acceleration mathematically? The following equation expresses average acceleration as the slope of the velocity-time graph. Average Acceleration

v

v v

f i a t t t f

i

Average acceleration is equal to the change in velocity, divided by the time it takes to make that change.

62

Chapter 3 Accelerated Motion

Suppose you run wind sprints back and forth across the gym. You first run at 4.0 m/s toward the wall. Then, 10.0 s later, you run at 4.0 m/s away from the wall. What is your average acceleration if the positive direction is toward the wall? v

v v

f i a t t t f

i

(4.0 m/s) (4.0 m/s) 8 .0 m/s 0.80 m/s2 10.0 s 10.0 s

The negative sign indicates that the direction of acceleration is away from the wall. The velocity changes when the direction of motion changes, because velocity includes the direction of motion. A change in velocity results in acceleration. Thus, acceleration also is associated with a change in the direction of motion.

Acceleration Describe the motion of a ball as it rolls up a slanted driveway. The ball starts at 2.50 m/s, slows down for 5.00 s, stops for an instant, and then rolls back down at an increasing speed. The positive direction is chosen to be up the driveway, and the origin is at the place where the motion begins. What is the sign of the ball’s acceleration as it rolls up the driveway? What is the magnitude of the ball’s acceleration as it rolls up the driveway? 1

Analyze and Sketch the Problem

x

x

Begin

End

• Sketch the situation. • Draw the coordinate system based on the motion diagram. Unknown:

vi 2.5 m/s vf 0.00 m/s at t 5.00 s

a?

Solve for the Unknown

0.00

3.00

5.00 Find the magnitude of the acceleration from the slope of Time (s) the graph. Solve for the change in velocity and the time taken to make that change. v vf vi 0.00 m/s 2.50 m/s Substitute vf 0.00 m/s at tf 5.00 s, vi 2.50 m/s at ti 0.00 s 2.50 m/s t tf ti 5.00 s 0.00 s Substitute tf 5.00 s, ti 0.00 s 5.00 s

10.0

Math Handbook

Solve for the acceleration. v a t 2.50 m/s 5.00 s

Same point

3.00 Velocity (m/s)

2

Known:

a

Substitute v 2.50 m/s, t 5.00 s

Operations with Significant Digits pages 835–836

0.500 m/s2 or 0.500 m/s2 down the driveway 3

Evaluate the Answer • Are the units correct? Acceleration is measured in m/s2. • Do the directions make sense? In the first 5.00 s, the direction of the acceleration is opposite to that of the velocity, and the ball slows down.

Section 3.1 Acceleration

63

6. A race car’s velocity increases from 4.0 m/s to 36 m/s over a 4.0-s time interval. What is its average acceleration? 7. The race car in the previous problem slows from 36 m/s to 15 m/s over 3.0 s. What is its average acceleration? 8. A car is coasting backwards downhill at a speed of 3.0 m/s when the driver gets the engine started. After 2.5 s, the car is moving uphill at 4.5 m/s. If uphill is chosen as the positive direction, what is the car’s average acceleration? 9. A bus is moving at 25 m/s when the driver steps on the brakes and brings the bus to a stop in 3.0 s. a. What is the average acceleration of the bus while braking? b. If the bus took twice as long to stop, how would the acceleration compare with what you found in part a? 10. Rohith has been jogging to the bus stop for 2.0 min at 3.5 m/s when he looks at his watch and sees that he has plenty of time before the bus arrives. Over the next 10.0 s, he slows his pace to a leisurely 0.75 m/s. What was his average acceleration during this 10.0 s? 11. If the rate of continental drift were to abruptly slow from 1.0 cm/y to 0.5 cm/y over the time interval of a year, what would be the average acceleration?

There are several parallels between acceleration and velocity. Both are rates of change: acceleration is the time rate of change of velocity, and velocity is the time rate of change of position. Both acceleration and velocity have average and instantaneous forms. You will learn later in this chapter that the area under a velocity-time graph is equal to the object’s displacement and that the area under an acceleration-time graph is equal to the object’s velocity.

3.1 Section Review 12. Velocity-Time Graph What information can you obtain from a velocity-time graph? 13. Position-Time and Velocity-Time Graphs Two joggers run at a constant velocity of 7.5 m/s toward the east. At time t 0, one is 15 m east of the origin and the other is 15 m west. a. What would be the difference(s) in the positiontime graphs of their motion? b. What would be the difference(s) in their velocitytime graphs? 14. Velocity Explain how you would use a velocitytime graph to find the time at which an object had a specified velocity. 15. Velocity-Time Graph Sketch a velocity-time graph for a car that goes east at 25 m/s for 100 s, then west at 25 m/s for another 100 s. 64

Chapter 3 Accelerated Motion

16. Average Velocity and Average Acceleration A canoeist paddles upstream at 2 m/s and then turns around and floats downstream at 4 m/s. The turnaround time is 8 s. a. What is the average velocity of the canoe? b. What is the average acceleration of the canoe? 17. Critical Thinking A police officer clocked a driver going 32 km/h over the speed limit just as the driver passed a slower car. Both drivers were issued speeding tickets. The judge agreed with the officer that both were guilty. The judgement was issued based on the assumption that the cars must have been going the same speed because they were observed next to each other. Are the judge and the police officer correct? Explain with a sketch, a motion diagram, and a position-time graph. physicspp.com/self_check_quiz

3.2 Motion with Constant Acceleration

Y

ou have learned that the definition of average velocity can be algebraically rearranged to show the new position after a period of time, given the initial position and the average velocity. The definition of average acceleration can be manipulated similarly to show the new velocity after a period of time, given the initial velocity and the average acceleration.

Velocity with Average Acceleration If you know an object’s average acceleration during a time interval, you can use it to determine how much the velocity changed during that time. The definition of average acceleration, v

Objectives • Interpret position-time graphs for motion with constant acceleration. • Determine mathematical relationships among position, velocity, acceleration, and time. • Apply graphical and mathematical relationships to solve problems related to constant acceleration.

a , can be rewritten as follows: t v at vf vi at The equation for final velocity with average acceleration can be written as follows. Final Velocity with Average Acceleration

vf vi at

The final velocity is equal to the initial velocity plus the product of the average acceleration and time interval.

In cases in which the acceleration is constant, the average acceleration, a, is the same as the instantaneous acceleration, a. This equation can be rearranged to find the time at which an object with constant acceleration has a given velocity. It also can be used to calculate the initial velocity of an object when both the velocity and the time at which it occurred are given.

18. A golf ball rolls up a hill toward a miniature-golf hole. Assume that the direction toward the hole is positive. a. If the golf ball starts with a speed of 2.0 m/s and slows at a constant rate of 0.50 m/s2, what is its velocity after 2.0 s? b. What is the golf ball’s velocity if the constant acceleration continues for 6.0 s? c. Describe the motion of the golf ball in words and with a motion diagram. 19. A bus that is traveling at 30.0 km/h speeds up at a constant rate of 3.5 m/s2. What velocity does it reach 6.8 s later? 20. If a car accelerates from rest at a constant 5.5 m/s2, how long will it take for the car to reach a velocity of 28 m/s? 21. A car slows from 22 m/s to 3.0 m/s at a constant rate of 2.1 m/s2. How many seconds are required before the car is traveling at 3.0 m/s?

Section 3.2 Motion with Constant Acceleration

65

Position-Time Graph for a Car

70.0

Table 3-2 60.0

Time (s)

Position (m)

0.00 1.00 2.00 3.00 4.00 5.00

0.00 2.50 10.0 22.5 40.0 62.5

Displacement (m)

Position-Time Data for a Car

60.0 m 20.0 m 5.00 s 3.00 s

m

50.0

20.0 m/s

40.0 30.0 20.0

m 10.0

10.0 m/s

0.00

1.00

■

Figure 3-9 The slope of a position-time graph of a car moving with a constant acceleration gets steeper as time goes on.

20.0 m 0.00 m 3.00 s 1.00 s

2.00

3.00

4.00

5.00

Time (s)

Position with Constant Acceleration You have learned that an object experiencing constant acceleration changes its velocity at a constant rate. How does the position of an object with constant acceleration change? The position data at different time intervals for a car with constant acceleration are shown in Table 3-2. The data from Table 3-2 are graphed in Figure 3-9. The graph shows that the car’s motion is not uniform: the displacements for equal time intervals on the graph get larger and larger. Notice that the slope of the line in Figure 3-9 gets steeper as time goes on. The slopes from the positiontime graph can be used to create a velocity-time graph. Note that the slopes shown in Figure 3-9 are the same as the velocities graphed in Figure 3-10a. A unique position-time graph cannot be created using a velocity-time graph because it does not contain any information about the object’s position. However, the velocity-time graph does contain information about the object’s displacement. Recall that for an object moving at a constant velocity, v v d/t, so d vt. On the graph in Figure 3-10b, v is the height of the plotted line above the t-axis, while t is the width of the shaded rectangle. The area of the rectangle, then, is vt, or d. Thus, the area under the v-t graph is equal to the object’s displacement.

a

b

25.0

Figure 3-10 The slopes of the p-t graph in Figure 3-9 are the values of the corresponding v-t graph (a). For any v-t graph, the displacement during a given time interval is the area under the graph (b).

66

Chapter 3 Accelerated Motion

15.0 10.0 5.00 0.00

20.0 m/s 15.0 m/s m 4.00 s 3.00 s

5.00 m/s2 1.00 2.00 3.00 4.00 5.00 Time (s)

Velocity (m/s)

■

Velocity (m/s)

20.0

10 9 8 7 6 5 4 3 2 1 0

v

∆t 1

2 Time (s)

3

4

Finding the Displacement from a v-t Graph The v-t graph below shows the motion of an airplane. Find the displacement of the airplane at t 1.0 s and at t 2.0 s. 1

Analyze and Sketch the Problem • The displacement is the area under the v-t graph. • The time intervals begin at t 0.0. Unknown:

v 75 m/s t 1.0 s t 2.0 s

d ?

80 Velocity (m/s)

2

Known:

82

Solve for the Unknown Solve for displacement during t 1.0 s. d vt (75 m/s)(1.0 s) Substitute v 75 m/s, t 1.0 s 75 m Solve for displacement during t 2.0 s. d vt (75 m/s)(2.0 s) Substitute v 75 m/s, t 2.0 s 150 m

3

78 76 74

∆t

72

∆t

0.0

1.0

2.0

3.0

Time (s)

Math Handbook Operations with Significant Digits pages 835–836

Evaluate the Answer

• Are the units correct? Displacement is measured in meters. • Do the signs make sense? The positive sign agrees with the graph. • Is the magnitude realistic? Moving a distance equal to about one football field is reasonable for an airplane.

25. A car is driven at a constant velocity of 25 m/s for 10.0 min. The car runs out of gas and the driver walks in the same direction at 1.5 m/s for 20.0 min to the nearest gas station. The driver takes 2.0 min to fill a gasoline can, then walks back to the car at 1.2 m/s and eventually drives home at 25 m/s in the direction opposite that of the original trip. a. Draw a v-t graph using seconds as your time unit. Calculate the distance the driver walked to the gas station to find the time it took him to walk back to the car. b. Draw a position-time graph for the situation using the areas under the velocity-time graph.

Figure 3-11

80 Velocity (m/s)

24. A position-time graph for a pony running in a field is shown in Figure 3-12. Draw the corresponding velocity-time graph using the same time scale.

■

82

78 76 74 72 70 0.0

1.0

2.0

3.0

Time (s) Displacement (m)

22. Use Figure 3-11 to determine the velocity of an airplane that is speeding up at each of the following times. a. 1.0 s b. 2.0 s c. 2.5 s 23. Use dimensional analysis to convert an airplane’s speed of 75 m/s to km/h.

y

Time (s)

■

Figure 3-12

x

Section 3.2 Motion with Constant Acceleration

67

Velocity (m/s)

vf vi 0

ti

tf Time (s)

■

Figure 3-13 The displacement of an object moving with constant acceleration can be found by computing the area under the v-t graph.

The area under the v-t graph is equal to the object’s displacement. Consider the v-t graph in Figure 3-13 for an object moving with constant acceleration that started with an initial velocity of vi. What is the object’s displacement? The area under the graph can be calculated by dividing it into a rectangle and a triangle. The area of the rectangle can be found by drectangle vit, and the area of the triangle can be found by dtriangle 12vt. Because average acceleration, a, is equal to v/t, v can be rewritten as at. Substituting v at into the equation for the triangle’s area yields dtriangle 12(at)t, or 12a(t)2. Solving for the total area under the graph results in the following: 1 2

d drectangle dtriangle vi(t) a(t)2 When the initial or final position of the object is known, the equation can be written as follows: 1 2

1 2

df di vi(t) a(t)2 or df di vi(t) a(t)2 If the initial time is ti 0, the equation then becomes the following. 1 2 An object’s position at a time after the initial time is equal to the sum of its initial position, the product of the initial velocity and the time, and half the product of the acceleration and the square of the time.

df di vitf atf 2

Position with Average Acceleration

An Alternative Expression Often, it is useful to relate position, velocity, and constant acceleration without including time. Rearrange the equation vf vi atf to solve for Drag Racing A dragster driver tries to obtain maximum acceleration over a 402-m (quarter-mile) course. The fastest time on record for the 402-m course is 4.480 s. The highest final speed on record is 147.63 m/s (330.23 mph).

v v a

f i . time: tf

Rewriting df di vitf 12 atf2 by substituting tf yields the following: v v a

v v 2 a

1 f f i i a df di vi 2

This equation can be solved for the velocity, vf , at any time, tf . Velocity with Constant Acceleration

vf2 vi2 2a(df di)

The square of the final velocity equals the sum of the square of the initial velocity and twice the product of the acceleration and the displacement since the initial time.

The three equations for motion with constant acceleration are summarized in Table 3-3. Note that in a multi-step problem, it is useful to add additional subscripts to identify which step is under consideration.

Table 3-3 Equations of Motion for Uniform Acceleration Equation

Variables

Initial Conditions

vf vi at f

t f, vf, a

vi

df di vit f at f2

t f, df, a

di , vi

vf2 vi2 2a(df di )

df, vf, a

di , vi

1 2

68

Chapter 3 Accelerated Motion

Displacement An automobile starts at rest and speeds up at 3.5 m/s2 after the traffic light turns green. How far will it have gone when it is traveling at 25 m/s? 1

Analyze and Sketch the Problem • Sketch the situation. • Establish coordinate axes. • Draw a motion diagram. Known:

End

x

Unknown:

di 0.00 m df ? vi 0.00 m/s vf 25 m/s a a 3.5 m/s2 2

Begin

v a

Begin

Solve for the Unknown

Math Handbook Order of Operations page 843

Solve for df . vf2 vi2 2a(df di) v2 v2 2a (25 m/s)2 (0.00 m/s)2 0.00 m 2(3.5 m/s2 )

End

f i df di

Substitute di 0.00 m, vf 25 m/s, vi 0.00 m/s

89 m 3

Evaluate the Answer • Are the units correct? Position is measured in meters. • Does the sign make sense? The positive sign agrees with both the pictorial and physical models. • Is the magnitude realistic? The displacement is almost the length of a football field. It seems large, but 25 m/s is fast (about 55 mph); therefore, the result is reasonable.

26. A skateboarder is moving at a constant velocity of 1.75 m/s when she starts up an incline that causes her to slow down with a constant acceleration of 0.20 m/s2. How much time passes from when she begins to slow down until she begins to move back down the incline? 27. A race car travels on a racetrack at 44 m/s and slows at a constant rate to a velocity of 22 m/s over 11 s. How far does it move during this time? 28. A car accelerates at a constant rate from 15 m/s to 25 m/s while it travels a distance of 125 m. How long does it take to achieve this speed? 29. A bike rider pedals with constant acceleration to reach a velocity of 7.5 m/s over a time of 4.5 s. During the period of acceleration, the bike’s displacement is 19 m. What was the initial velocity of the bike?

Section 3.2 Motion with Constant Acceleration

69

Two-Part Motion You are driving a car, traveling at a constant velocity of 25 m/s, when you see a child suddenly run onto the road. It takes 0.45 s for you to react and apply the brakes. As a result, the car slows with a steady acceleration of 8.5 m/s2 and comes to a stop. What is the total distance that the car moves before it stops? 1

Analyze and Sketch the Problem

Begin

• Sketch the situation. • Choose a coordinate system with the motion of the car in the positive direction. • Draw the motion diagram and label v and a.

2

Known:

Unknown:

vreacting 25 m/s treacting 0.45 s a abraking 8.5 m/s2 vi, braking 25 m/s vf, braking 0.00 m/s

dreacting ? dbraking ? dtotal ?

Reacting

End

Braking

x

0 Begin

v a

Solve for the Unknown Reacting: Solve for the distance the car travels at a constant speed. dreacting vreactingtreacting dreacting (25 m/s)(0.45 s) Substitute vreacting 25 m/s, treacting 0.45 s 11 m Braking: Solve for the distance the car moves while braking. vf, braking2 vreacting2 2abraking(dbraking) Math Handbook

Solve for dbraking. vf, braking2

Isolating a Variable page 845

vreacting2

dbraking 2abraking

(0.00 m/s) (25 m/s) 2 2

2(8.5 m/s )

2

Substitute vf, braking 0.00 m/s, vreacting 25 m/s, abraking 8.5 m/s2

37 m The total distance traveled is the sum of the reaction distance and the braking distance. Solve for dtotal. dtotal dreacting dbraking 11 m 37 m 48 m 3

Substitute dreacting 11 m, dbraking 37 m

Evaluate the Answer • Are the units correct? Distance is measured in meters. • Do the signs make sense? Both dreacting and dbraking are positive, as they should be. • Is the magnitude realistic? The braking distance is small because the magnitude of the acceleration is large.

70

Chapter 3 Accelerated Motion

End

30. A man runs at a velocity of 4.5 m/s for 15.0 min. When going up an increasingly steep hill, he slows down at a constant rate of 0.05 m/s2 for 90.0 s and comes to a stop. How far did he run? 31. Sekazi is learning to ride a bike without training wheels. His father pushes him with a constant acceleration of 0.50 m/s2 for 6.0 s, and then Sekazi continues at 3.0 m/s for another 6.0 s before falling. What is Sekazi’s displacement? Solve this problem by constructing a velocity-time graph for Sekazi’s motion and computing the area underneath the graphed line. 32. You start your bicycle ride at the top of a hill. You coast down the hill at a constant acceleration of 2.00 m/s2. When you get to the bottom of the hill, you are moving at 18.0 m/s, and you pedal to maintain that speed. If you continue at this speed for 1.00 min, how far will you have gone from the time you left the hilltop? 33. Sunee is training for an upcoming 5.0-km race. She starts out her training run by moving at a constant pace of 4.3 m/s for 19 min. Then she accelerates at a constant rate until she crosses the finish line, 19.4 s later. What is her acceleration during the last portion of the training run?

You have learned several different tools that you can apply when solving problems dealing with motion in one dimension: motion diagrams, graphs, and equations. As you gain more experience, it will become easier to decide which tools are most appropriate in solving a given problem. In the following section, you will practice using these tools to investigate the motion of falling objects.

3.2 Section Review 34. Acceleration A woman driving at a speed of 23 m/s sees a deer on the road ahead and applies the brakes when she is 210 m from the deer. If the deer does not move and the car stops right before it hits the deer, what is the acceleration provided by the car’s brakes? 35. Displacement If you were given initial and final velocities and the constant acceleration of an object, and you were asked to find the displacement, what equation would you use? 36. Distance An in-line skater first accelerates from 0.0 m/s to 5.0 m/s in 4.5 s, then continues at this constant speed for another 4.5 s. What is the total distance traveled by the in-line skater? 37. Final Velocity A plane travels a distance of 5.0102 m while being accelerated uniformly from rest at the rate of 5.0 m/s2. What final velocity does it attain? 38. Final Velocity An airplane accelerated uniformly from rest at the rate of 5.0 m/s2 for 14 s. What final velocity did it attain? physicspp.com/self_check_quiz

39. Distance An airplane starts from rest and accelerates at a constant 3.00 m/s2 for 30.0 s before leaving the ground. a. How far did it move? b. How fast was the airplane going when it took off? 40. Graphs A sprinter walks up to the starting blocks at a constant speed and positions herself for the start of the race. She waits until she hears the starting pistol go off, and then accelerates rapidly until she attains a constant velocity. She maintains this velocity until she crosses the finish line, and then she slows down to a walk, taking more time to slow down than she did to speed up at the beginning of the race. Sketch a velocity-time and a position-time graph to represent her motion. Draw them one above the other on the same time scale. Indicate on your p-t graph where the starting blocks and finish line are. 41. Critical Thinking Describe how you could calculate the acceleration of an automobile. Specify the measuring instruments and the procedures that you would use. Section 3.2 Motion with Constant Acceleration

71

3.3 Free Fall

Objectives • Define acceleration due to gravity. • Solve problems involving objects in free fall.

Vocabulary free fall acceleration due to gravity

D

rop a sheet of paper. Crumple it, and then drop it again. Drop a rock or a pebble. How do the three motions compare with each other? Do heavier objects fall faster than lighter ones? A light, spread-out object, such as a smooth sheet of paper or a feather, does not fall in the same manner as something more compact, such as a pebble. Why? As an object falls, it bumps into particles in the air. For an object such as a feather, these little collisions have a greater effect than they do on pebbles or rocks. To understand the behavior of falling objects, first consider the simplest case: an object such as a rock, for which the air does not have an appreciable effect on its motion. The term used to describe the motion of such objects is free fall, which is the motion of a body when air resistance is negligible and the action can be considered due to gravity alone.

Acceleration Due to Gravity

■ Figure 3-14 An egg accelerates at 9.80 m/s2 in free fall. If the upward direction is chosen as positive, then both the velocity and the acceleration of this egg in free fall are negative.

About 400 years ago, Galileo Galilei recognized that to make progress in the study of the motion of falling objects, the effects of the substance through which the object falls have to be ignored. At that time, Galileo had no means of taking position or velocity data for falling objects, so he rolled balls down inclined planes. By “diluting” gravity in this way, he could make careful measurements even with simple instruments. Galileo concluded that, neglecting the effect of the air, all objects in free fall had the same acceleration. It didn’t matter what they were made of, how much they weighed, what height they were dropped from, or whether they were dropped or thrown. The acceleration of falling objects, given a special symbol, g, is equal to 9.80 m/s2. It is now known that there are small variations in g at different places on Earth, and that 9.80 m/s2 is the average value. The acceleration due to gravity is the acceleration of an object in free fall that results from the influence of Earth’s gravity. Suppose you drop a rock. After 1 s, its velocity is 9.80 m/s downward, and 1 s after that, its velocity is 19.60 m/s downward. For each second that the rock is falling, its downward velocity increases by 9.80 m/s. Note that g is a positive number. When analyzing free fall, whether you treat the acceleration as positive or negative depends upon the coordinate system that you use. If your coordinate system defines upward to be the positive direction, then the acceleration due to gravity is equal to g; if you decide that downward is the positive direction, then the acceleration due to gravity is g. A strobe photo of a dropped egg is shown in Figure 3-14. The time interval between the images is 0.06 s. The displacement between each pair of images increases, so the speed is increasing. If the upward direction is chosen as positive, then the velocity is becoming more and more negative. Ball thrown upward Instead of a dropped egg, could this photo also illustrate a ball thrown upward? If upward is chosen to be the positive direction, then the ball leaves the hand with a positive velocity of, for example, 20.0 m/s. The acceleration is downward, so a is negative. That is, a g 9.80 m/s2. Because the velocity and acceleration are in opposite directions, the speed of the ball decreases, which is in agreement with the strobe photo.

72

Chapter 3 Accelerated Motion

Richard Megna/Fundamental Photographs

20.0

0.00

20.0

1

2

3

0.00

0.50 2.00

4

Time (s)

Displacement (m) 0

1

2 Time (s)

2.08

20.41

d

Displacement (m) 0

2.04 Time (s)

25

c

■ Figure 3-15 In a coordinate system in which the upward direction is positive, the velocity of the thrown ball decreases until it becomes zero at 2.04 s. Then it increases in the negative direction as the ball falls (a, b). The p-t graphs show the height of the ball at corresponding time intervals (c, d).

0.50

b Velocity (m/s)

Velocity (m/s)

a

3

4

20.40

20.39 2.00

2.04

2.08

Time (s)

After 1 s, the ball’s velocity is reduced by 9.80 m/s, so it now is traveling at 10.2 m/s. After 2 s, the velocity is 0.4 m/s, and the ball still is moving upward. What happens during the next second? The ball’s velocity is reduced by another 9.80 m/s, and is equal to 9.4 m/s. The ball now is moving downward. After 4 s, the velocity is 19.2 m/s, meaning that the ball is falling even faster. Figure 3-15a shows the velocity-time graph for the ball as it goes up and comes back down. At around 2 s, the velocity changes smoothly from positive to negative. Figure 3-15b shows a closer view of the v-t graph around that point. At an instant of time, near 2.04 s, the ball’s velocity is zero. Look at the position-time graphs in Figure 3-15c and d, which show how the ball’s height changes. How are the ball’s position and velocity related? The ball reaches its maximum height at the instant of time when its velocity is zero. At 2.04 s, the ball reaches its maximum height and its velocity is zero. What is the ball’s acceleration at that point? The slope of the line in the v-t graphs in Figure 3-15a and 3-15b is constant at 9.80 m/s2. Often, when people are asked about the acceleration of an object at the top of its flight, they do not take the time to fully analyze the situation, and respond that the acceleration at this point is zero. However, this is not the case. At the top of the flight, the ball’s velocity is 0 m/s. What would happen if its acceleration were also zero? Then the ball’s velocity would not be changing and would remain at 0 m/s. If this were the case, the ball would not gain any downward velocity and would simply hover in the air at the top of its flight. Because this is not the way objects tossed in the air behave on Earth, you know that the acceleration of an object at the top of its flight must not be zero. Further, because you know that the object will fall from that height, you know that the acceleration must be downward. Section 3.3 Free Fall

73

Free-fall rides Amusement parks use the concept of free fall to design rides that give the riders the sensation of free fall. These types of rides usually consist of three parts: the ride to the top, momentary suspension, and the plunge downward. Motors provide the force needed to move the cars to the top of the ride. When the cars are in free fall, the most massive rider and the least massive rider will have the same acceleration. Suppose the free-fall ride at an amusement park starts at rest and is in free fall for 1.5 s. What would be its velocity at the end of 1.5 s? Choose a coordinate system with a positive axis upward and the origin at the initial position of the car. Because the car starts at rest, vi would be equal to 0.00 m/s. To calculate the final velocity, use the equation for velocity with constant acceleration. vf vi atf 0.00 m/s (9.80 m/s2)(1.5 s) 15 m/s How far does the car fall? Use the equation for displacement when time and constant acceleration are known. 1 2

df di vitf atf2 1 2

0.00 m (0.00 m/s)(1.5 s) (9.80 m/s2)(1.5 s)2 11 m

42. A construction worker accidentally drops a brick from a high scaffold. a. What is the velocity of the brick after 4.0 s? b. How far does the brick fall during this time? 43. Suppose for the previous problem you choose your coordinate system so that the opposite direction is positive. a. What is the brick’s velocity after 4.0 s? b. How far does the brick fall during this time? 44. A student drops a ball from a window 3.5 m above the sidewalk. How fast is it moving when it hits the sidewalk? 45. A tennis ball is thrown straight up with an initial speed of 22.5 m/s. It is caught at the same distance above the ground. a. How high does the ball rise? b. How long does the ball remain in the air? Hint: The time it takes the ball to rise equals the time it takes to fall. 46. You decide to flip a coin to determine whether to do your physics or English homework first. The coin is flipped straight up. a. If the coin reaches a high point of 0.25 m above where you released it, what was its initial speed? b. If you catch it at the same height as you released it, how much time did it spend in the air?

74

Chapter 3 Accelerated Motion

You notice a water balloon fall past your classroom window. You estimate that it took the balloon about t seconds to fall the length of the window and that the window is about y meters high. Suppose the balloon started from rest. Approximately how high above the top of the window was it released? Your answer should be in terms of t, y, g, and numerical constants.

Remember to define the positive direction when establishing your coordinate system. As motion problems increase in complexity, it becomes increasingly important to keep all the signs consistent. This means that any displacement, velocity, or acceleration that is in the same direction as the one chosen to be positive will be positive. Thus, any displacement, velocity, or acceleration that is in the direction opposite to the one chosen to be positive should be indicated with a negative sign. Sometimes it might be appropriate to choose upward as positive. At other times, it might be easier to choose downward as positive. You can choose either direction you want, as long as you stay consistent with that convention throughout the solution of that particular problem. Suppose you solve one of the practice problems on the preceding page again, choosing the direction opposite to the one you previously designated as the positive direction for the coordinate system. You should arrive at the same answer, provided that you assigned signs to each of the quantities that were consistent with the coordinate system. It is important to be consistent with the coordinate system to avoid getting the signs mixed up.

3.3 Section Review 47. Maximum Height and Flight Time Acceleration due to gravity on Mars is about one-third that on Earth. Suppose you throw a ball upward with the same velocity on Mars as on Earth. a. How would the ball’s maximum height compare to that on Earth? b. How would its flight time compare? 48. Velocity and Acceleration Suppose you throw a ball straight up into the air. Describe the changes in the velocity of the ball. Describe the changes in the acceleration of the ball. 49. Final Velocity Your sister drops your house keys down to you from the second floor window. If you catch them 4.3 m from where your sister dropped them, what is the velocity of the keys when you catch them?

physicspp.com/self_check_quiz

50. Initial Velocity A student trying out for the football team kicks the football straight up in the air. The ball hits him on the way back down. If it took 3.0 s from the time when the student punted the ball until he gets hit by the ball, what was the football’s initial velocity? 51. Maximum Height When the student in the previous problem kicked the football, approximately how high did the football travel? 52. Critical Thinking When a ball is thrown vertically upward, it continues upward until it reaches a certain position, and then it falls downward. At that highest point, its velocity is instantaneously zero. Is the ball accelerating at the highest point? Devise an experiment to prove or disprove your answer.

Section 3.3 Free Fall

75

Acceleration Due to Gravity Alternate CBL instructions can be found on the Web site. physicspp.com

Small variations in the acceleration due to gravity, g, occur at different places on Earth. This is because g varies with distance from the center of Earth and is influenced by the subsurface geology. In addition, g varies with latitude due to Earth’s rotation. For motion with constant acceleration, the displacement is df di vi(tf ti) 1 1 a(tf ti)2. If di 0 and ti 0, then the displacement is df vitf atf2. 2 2 Dividing both sides of the equation by tf yields the following: df /tf vi 12 atf . The slope of a graph of df/tf versus tf, is equal to 12 a. The initial velocity, vi, is determined by the y-intercept. In this activity, you will be using a spark timer to collect free-fall data and use it to determine the acceleration due to gravity, g.

QUESTION How does the value of g vary from place to place?

Objectives

Procedure

■ Measure free-fall data. ■ Make and use graphs of velocity versus time. ■ Compare and contrast values of g for different

1. Attach the spark timer to the edge of the lab table with the C-clamp.

locations.

Safety Precautions

■ Keep clear of falling masses.

Materials spark timer timer tape 1-kg mass C-clamp stack of newspapers masking tape

2. If the timer needs to be calibrated, follow your teacher’s instructions or those provided with the timer. Determine the period of the timer and record it in your data table. 3. Place the stack of newspapers on the floor, directly below the timer so that the mass, when released, will not damage the floor. 4. Cut a piece of timer tape approximately 70 cm in length and slide it into the spark timer. 5. Attach the timer tape to the 1-kg mass with a small piece of masking tape. Hold the mass next to the spark timer, over the edge of the table so that it is above the newspaper stack. 6. Turn on the spark timer and release the mass. 7. Inspect the timer tape to make sure that there are dots marked on it and that there are no gaps in the dot sequence. If your timer tape is defective, repeat steps 4–6 with another piece of timer tape. 8. Have each member of your group perform the experiment and collect his or her own data. 9. Choose a dot near the beginning of the timer tape, a few centimeters from the point where the timer began to record dots, and label it 0. Label the dots after that 1, 2, 3, 4, 5, etc. until you get near the end where the mass is no longer in free fall. If the dots stop, or the distance between them begins to get smaller, the mass is no longer in free fall.

76 Horizons Companies

Data Table Time period (#/s) Interval

Distance (cm)

Time (s)

Speed (cm/s)

1 2 3 4 5 6 7 8

10. Measure the total distance to each numbered dot from the zero dot, to the nearest millimeter and record it in your data table. Using the timer period, record the total time associated with each distance measurement and record it in your data table.

Real-World Physics Why do designers of free-fall amusement-park rides design exit tracks that gradually curve toward the ground? Why is there a stretch of straight track?

Analyze 1. Use Numbers Calculate the values for speed and record them in the data table. 2. Make and Use Graphs Draw a graph of speed versus time. Draw the best-fit straight line for your data. 3. Calculate the slope of the line. Convert your result to m/s2.

Conclude and Apply 1. Recall that the slope is equal to 12a. What is the acceleration due to gravity? 2. Find the relative error for your experimental value of g by comparing it to the accepted value.

Communicate the average value of g to others. Go to physicspp.com/internet_lab and post the name of your school, city, state, elevation above sea level, and average value of g for your class. Obtain a map for your state and a map of the United States. Using the data posted on the Web site by other students, mark the values for g at the appropriate locations on the maps. Do you notice any variation in the acceleration due to gravity for different locations, regions and elevations?

Relative error Accepted value Experimental value 100 Accepted value

3. What was the mass’s velocity, vi , when you began measuring distance and time?

Going Further

To find out more about accelerated motion, visit the Web site: physicspp.com

What is the advantage of measuring several centimeters away from the beginning of the timer tape rather than from the very first dot? 77

Mirror

Time Dilation at High Velocities

Observer inside the spacecraft

t0

Can time pass differently in two reference frames? How can one of a pair of twins age more than the other?

Light clock Mirror

Light Clock Consider the following thought experiment using a light clock. A light clock is a vertical tube with a mirror at each end. A short pulse of light is introduced at one end and allowed to bounce back and forth within the tube. Time is measured by counting the number of bounces made by the pulse of light. The clock will be accurate because the speed of a pulse of light is always c, which is 3108 m/s, regardless of the velocity of the light source or the observer. Suppose this light clock is placed in a very fast spacecraft. When the spacecraft goes at slow speeds, the light beam bounces vertically in the tube. If the spacecraft is moving fast, the light beam still bounces vertically—at least as seen by the observer in the spacecraft. A stationary observer on Earth, however, sees the pulse of light move diagonally because of the movement of the spacecraft. Thus, to the stationary observer, the light beam moves a greater distance. Distance velocity time, so if the distance traveled by the light beam increases, the product (velocity time) also must increase. Because the speed of the light pulse, c, is the same for any observer, time must be increasing for the stationary observer. That is, the stationary observer sees the moving clock ticking slower than the same clock on Earth. Suppose the time per tick seen by the stationary observer on Earth is ts , the time seen by the observer on the spacecraft is to, the length of the light clock is cto , the velocity of the spacecraft is v, and the speed of light is c. For every tick, the spacecraft moves vts and the light pulse moves cto. This leads to the following equation: t

o ts

v 1 c 2

2

To the stationary observer, the closer v is to 78

Extreme Physics

D

D

Observer on Earth

c, the slower the clock ticks. To the observer on the spacecraft, however, the clock keeps perfect time.

Time Dilation This phenomenon is called time dilation and it applies to every process associated with time aboard the spacecraft. For example, biological aging will proceed more slowly in the spacecraft than on Earth. So if the observer on the spacecraft is one of a pair of twins, he or she would age more slowly than the other twin on Earth. This is called the twin paradox. Time dilation has resulted in a lot of speculation about space travel. If spacecraft were able to travel at speeds close to the speed of light, trips to distant stars would take only a few years for the astronaut.

Going Further 1. Calculate Find the time dilation ts/to for Earth’s orbit about the Sun if vEarth 10,889 km/s. 2. Calculate Derive the equation for ts above. 3. Discuss How is time dilation similar to or different from time travel?

3.1 Acceleration Vocabulary

Key Concepts

• velocity-time graph (p. 58) • acceleration (p.59) • average acceleration

• •

A velocity-time graph can be used to find the velocity and acceleration of an object. The average acceleration of an object is the slope of its velocity-time graph.

(p. 59)

v

v v

f i a t t t

• instantaneous acceleration (p. 59)

f

• • •

i

Average acceleration vectors on a motion diagram indicate the size and direction of the average acceleration during a time interval. When the acceleration and velocity are in the same direction, the object speeds up; when they are in opposite directions, the object slows down. Velocity-time graphs and motion diagrams can be used to determine the sign of an object’s acceleration.

3.2 Motion with Constant Acceleration Key Concepts

•

If an object’s average acceleration during a time interval is known, the change in velocity during that time can be found. vf vi at

• •

The area under an object’s velocity-time graph is its displacement. In motion with constant acceleration, there are relationships among the position, velocity, acceleration, and time. 1 2

df di vitf atf 2

•

The velocity of an object with constant acceleration can be found using the following equation. vf 2 vi2 2a(df di)

3.3 Free Fall Vocabulary

Key Concepts

• free fall (p. 72) • acceleration due to gravity (p. 72)

• •

The acceleration due to gravity on Earth, g, is 9.80 m/s2 downward. The sign associated with g in equations depends upon the choice of the coordinate system. Equations for motion with constant acceleration can be used to solve problems involving objects in free fall.

physicspp.com/vocabulary_puzzlemaker

79

Concept Mapping 53. Complete the following concept map using the following symbols or terms: d, velocity, m/s2, v, m, acceleration.

61. What quantity is represented by the area under a velocity-time graph? (3.2)

62. Write a summary of the equations for position, velocity, and time for an object experiencing motion with uniform acceleration. (3.2)

Quantities of motion

63. Explain why an aluminum ball and a steel ball of similar size and shape, dropped from the same height, reach the ground at the same time. (3.3)

position

64. Give some examples of falling objects for which air a

resistance cannot be ignored. (3.3)

65. Give some examples of falling objects for which air m/s

resistance can be ignored. (3.3)

Mastering Concepts

Applying Concepts

54. How are velocity and acceleration related? (3.1) 55. Give an example of each of the following. (3.1)

66. Does a car that is slowing down always have a

a. an object that is slowing down, but has a positive acceleration b. an object that is speeding up, but has a negative acceleration

56. Figure 3-16 shows the velocity-time graph for an automobile on a test track. Describe how the velocity changes with time. (3.1)

negative acceleration? Explain.

67. Croquet A croquet ball, after being hit by a mallet, slows down and stops. Do the velocity and acceleration of the ball have the same signs?

68. If an object has zero acceleration, does it mean its velocity is zero? Give an example.

69. If an object has zero velocity at some instant, is its acceleration zero? Give an example.

70. If you were given a table of velocities of an object at

25

various times, how would you find out whether the acceleration was constant?

20 15

71. The three notches in the graph in Figure 3-16 occur

5 0

5

10

15

20

25

30

35

Time (s) ■

Figure 3-16

57. What does the slope of the tangent to the curve on a velocity-time graph measure? (3.1)

58. Can a car traveling on an interstate highway have a negative velocity and a positive acceleration at the same time? Explain. Can the car’s velocity change signs while it is traveling with constant acceleration? Explain. (3.1)

59. Can the velocity of an object change when its acceleration is constant? If so, give an example. If not, explain. (3.1)

60. If an object’s velocity-time graph is a straight line parallel to the t-axis, what can you conclude about the object’s acceleration? (3.1)

80

where the driver changed gears. Describe the changes in velocity and acceleration of the car while in first gear. Is the acceleration just before a gear change larger or smaller than the acceleration just after the change? Explain your answer.

72. Use the graph in Figure 3-16 and determine the time interval during which the acceleration is largest and the time interval during which the acceleration is smallest.

73. Explain how you would walk to produce each of the position-time graphs in Figure 3-17. Displacement

10

Displacement

Velocity (m/s)

30

Time

Chapter 3 Accelerated Motion For more problems, go to Additional Problems, Appendix B.

C

A

D

E

B

F

G

H

Time ■

Figure 3-17

74. Draw a velocity-time graph for each of the graphs in

■

Displacement

Displacement Time

80. Find the uniform acceleration that causes a car’s velocity to change from 32 m/s to 96 m/s in an 8.0-s period.

Displacement

Figure 3-18.

Time

81. A car with a velocity of 22 m/s is accelerated uniformly at the rate of 1.6 m/s2 for 6.8 s. What is its final velocity?

82. Refer to Figure 3-19 to find the acceleration of the

Time

Figure 3-18

moving object at each of the following times. a. during the first 5.0 s of travel b. between 5.0 s and 10.0 s c. between 10.0 s and 15.0 s d. between 20.0 s and 25.0 s

reaches its maximum height. A second object falling from rest takes 7.0 s to reach the ground. Compare the displacements of the two objects during this time interval.

76. The Moon The value of g on the Moon is one-sixth of its value on Earth. a. Would a ball that is dropped by an astronaut hit the surface of the Moon with a greater, equal, or lesser speed than that of a ball dropped from the same height to Earth? b. Would it take the ball more, less, or equal time to fall?

77. Jupiter The planet Jupiter has about three times the gravitational acceleration of Earth. Suppose a ball is thrown vertically upward with the same initial velocity on Earth and on Jupiter. Neglect the effects of Jupiter’s atmospheric resistance and assume that gravity is the only force on the ball. a. How does the maximum height reached by the ball on Jupiter compare to the maximum height reached on Earth? b. If the ball on Jupiter were thrown with an initial velocity that is three times greater, how would this affect your answer to part a?

78. Rock A is dropped from a cliff and rock B is thrown upward from the same position. a. When they reach the ground at the bottom of the cliff, which rock has a greater velocity? b. Which has a greater acceleration? c. Which arrives first?

Mastering Problems 3.1 Acceleration 79. A car is driven for 2.0 h at 40.0 km/h, then for another 2.0 h at 60.0 km/h in the same direction. a. What is the car’s average velocity? b. What is the car’s average velocity if it is driven 1.0102 km at each of the two speeds? physicspp.com/chapter_test

Velocity (m/s)

75. An object shot straight up rises for 7.0 s before it

30.0 20.0 10.0 0.0

5.0

10.0

15.0

20.0

25.0

30.0

Time (s) ■

Figure 3-19

83. Plot a velocity-time graph using the information in Table 3-4, and answer the following questions. a. During what time interval is the object speeding up? Slowing down? b. At what time does the object reverse direction? c. How does the average acceleration of the object in the interval between 0.0 s and 2.0 s differ from the average acceleration in the interval between 7.0 s and 12.0 s?

Table 3-4 Velocity v. Time Time (s)

Velocity (m/s)

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.0 11.0 12.0

4.00 8.00 12.0 14.0 16.0 16.0 14.0 12.0 8.00 4.00 0.00 4.00 8.00

Chapter 3 Assessment

81

84. Determine the final velocity of a proton that has an initial velocity of 2.35105 m/s and then is accelerated uniformly in an electric field at the rate of 1.101012 m/s2 for 1.50107 s.

85. Sports Cars Marco is looking for a used sports car. He wants to buy the one with the greatest acceleration. Car A can go from 0 m/s to 17.9 m/s in 4.0 s; car B can accelerate from 0 m/s to 22.4 m/s in 3.5 s; and car C can go from 0 to 26.8 m/s in 6.0 s. Rank the three cars from greatest acceleration to least, specifically indicating any ties.

86. Supersonic Jet A supersonic jet flying at 145 m/s experiences uniform acceleration at the rate of 23.1 m/s2 for 20.0 s. a. What is its final velocity? b. The speed of sound in air is 331 m/s. What is the plane’s speed in terms of the speed of sound?

3.2 Motion with Constant Acceleration 87. Refer to Figure 3-19 to find the distance traveled during the following time intervals. a. t 0.0 s and t 5.0 s b. t 5.0 s and t 10.0 s c. t 10.0 s and t 15.0 s d. t 0.0 s and t 25.0 s

88. A dragster starting from rest accelerates at 49 m/s2. How fast is it going when it has traveled 325 m?

89. A car moves at 12 m/s and coasts up a hill with a uniform acceleration of 1.6 m/s2. a. What is its displacement after 6.0 s? b. What is its displacement after 9.0 s?

90. Race Car A race car can be slowed with a constant acceleration of 11 m/s2. a. If the car is going 55 m/s, how many meters will it travel before it stops? b. How many meters will it take to stop a car going twice as fast?

91. A car is traveling 20.0 m/s when the driver sees a child standing on the road. She takes 0.80 s to react, then steps on the brakes and slows at 7.0 m/s2. How far does the car go before it stops?

92. Airplane Determine the displacement of a plane that experiences uniform acceleration from 66 m/s to 88 m/s in 12 s.

93. How far does a plane fly in 15 s while its velocity is

95. Road Barrier The driver of a car going 90.0 km/h suddenly sees the lights of a barrier 40.0 m ahead. It takes the driver 0.75 s to apply the brakes, and the average acceleration during braking is 10.0 m/s2. a. Determine whether the car hits the barrier. b. What is the maximum speed at which the car could be moving and not hit the barrier 40.0 m ahead? Assume that the acceleration doesn’t change.

3.3 Free Fall 96. A student drops a penny from the top of a tower and decides that she will establish a coordinate system in which the direction of the penny’s motion is positive. What is the sign of the acceleration of the penny?

97. Suppose an astronaut drops a feather from 1.2 m above the surface of the Moon. If the acceleration due to gravity on the Moon is 1.62 m/s2 downward, how long does it take the feather to hit the Moon’s surface?

98. A stone that starts at rest is in free fall for 8.0 s. a. Calculate the stone’s velocity after 8.0 s. b. What is the stone’s displacement during this time?

99. A bag is dropped from a hovering helicopter. The bag has fallen for 2.0 s. What is the bag’s velocity? How far has the bag fallen?

100. You throw a ball downward from a window at a speed of 2.0 m/s. How fast will it be moving when it hits the sidewalk 2.5 m below?

101. If you throw the ball in the previous problem up instead of down, how fast will it be moving when it hits the sidewalk?

102. Beanbag You throw a beanbag in the air and catch it 2.2 s later. a. How high did it go? b. What was its initial velocity?

Mixed Review 103. A spaceship far from any star or planet experiences uniform acceleration from 65.0 m/s to 162.0 m/s in 10.0 s. How far does it move?

104. Figure 3-20 is a strobe photo of a horizontally moving ball. What information about the photo would you need and what measurements would you make to estimate the acceleration?

changing from 145 m/s to 75 m/s at a uniform rate of acceleration?

94. Police Car A speeding car is traveling at a constant speed of 30.0 m/s when it passes a stopped police car. The police car accelerates at 7.0 m/s2. How fast will it be going when it catches up with the speeding car?

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Chapter 3 Accelerated Motion For more problems, go to Additional Problems, Appendix B.

Skip Peticolas/Fundamental Photographs

■

Figure 3-20

105. Bicycle A bicycle accelerates from 0.0 m/s to 4.0 m/s in 4.0 s. What distance does it travel?

106. A weather balloon is floating at a constant height above Earth when it releases a pack of instruments. a. If the pack hits the ground with a velocity of 73.5 m/s, how far did the pack fall? b. How long did it take for the pack to fall?

107. Baseball A baseball pitcher throws a fastball at a speed of 44 m/s. The acceleration occurs as the pitcher holds the ball in his hand and moves it through an almost straight-line distance of 3.5 m. Calculate the acceleration, assuming that it is constant and uniform. Compare this acceleration to the acceleration due to gravity.

108. The total distance a steel ball rolls down an incline at various times is given in Table 3-5. a. Draw a position-time graph of the motion of the ball. When setting up the axes, use five divisions for each 10 m of travel on the d-axis. Use five divisions for 1 s of time on the t-axis. b. Calculate the distance the ball has rolled at the end of 2.2 s.

Table 3-5 Distance v. Time Time (s)

Distance (m)

0.0 1.0 2.0 3.0 4.0 5.0

0.0 2.0 8.0 18.0 32.0 50.0

109. Engineers are developing new types of guns that might someday be used to launch satellites as if they were bullets. One such gun can give a small object a velocity of 3.5 km/s while moving it through a distance of only 2.0 cm. a. What acceleration does the gun give this object? b. Over what time interval does the acceleration take place?

111. The velocity of a car changes over an 8.0-s time period, as shown in Table 3-6. a. Plot the velocity-time graph of the motion. b. Determine the displacement of the car during the first 2.0 s. c. What displacement does the car have during the first 4.0 s? d. What is the displacement of the car during the entire 8.0 s? e. Find the slope of the line between t 0.0 s and t 4.0 s. What does this slope represent? f. Find the slope of the line between t 5.0 s and t 7.0 s. What does this slope indicate?

Table 3-6 Velocity v. Time Time (s)

Velocity (m/s)

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

0.0 4.0 8.0 12.0 16.0 20.0 20.0 20.0 20.0

112. A truck is stopped at a stoplight. When the light turns green, the truck accelerates at 2.5 m/s2. At the same instant, a car passes the truck going 15 m/s. Where and when does the truck catch up with the car?

113. Safety Barriers Highway safety engineers build soft barriers, such as the one shown in Figure 3-21, so that cars hitting them will slow down at a safe rate. A person wearing a safety belt can withstand an acceleration of 3.0102 m/s2. How thick should barriers be to safely stop a car that hits a barrier at 110 km/h?

110. Sleds Rocket-powered sleds are used to test the responses of humans to acceleration. Starting from rest, one sled can reach a speed of 444 m/s in 1.80 s and can be brought to a stop again in 2.15 s. a. Calculate the acceleration of the sled when starting, and compare it to the magnitude of the acceleration due to gravity, 9.80 m/s2. b. Find the acceleration of the sled as it is braking and compare it to the magnitude of the acceleration due to gravity. ■

physicspp.com/chapter_test

Figure 3-21

Chapter 3 Assessment

83

Joel Bennett/Peter Arnold, Inc.

114. Karate The position-time and velocity-time

Velocity (m/s)

Fist Displacement (cm)

graphs of George’s fist breaking a wooden board during karate practice are shown in Figure 3-22. a. Use the velocity-time graph to describe the motion of George’s fist during the first 10 ms. b. Estimate the slope of the velocity-time graph to determine the acceleration of his fist when it suddenly stops. c. Express the acceleration as a multiple of the gravitational acceleration, g 9.80 m/s2. d. Determine the area under the velocity-time curve to find the displacement of the fist in the first 6 ms. Compare this with the positiontime graph. 10.0 5.0 0.0 5.0 0.0 5.0 10.0 15.0

5.0 10.0 15.0 20.0 25.0 30.0 Time (ms)

■

Figure 3-22

118. Analyze and Conclude An express train, traveling at 36.0 m/s, is accidentally sidetracked onto a local train track. The express engineer spots a local train exactly 1.00102 m ahead on the same track and traveling in the same direction. The local engineer is unaware of the situation. The express engineer jams on the brakes and slows the express train at a constant rate of 3.00 m/s2. If the speed of the local train is 11.0 m/s, will the express train be able to stop in time, or will there be a collision? To solve this problem, take the position of the express train when the engineer first sights the local train as a point of origin. Next, keeping in mind that the local train has exactly a 1.00102 m lead, calculate how far each train is from the origin at the end of the 12.0 s it would take the express train to stop (accelerate at 3.00 m/s2 from 36 m/s to 0 m/s). a. On the basis of your calculations, would you conclude that a collision will occur? b. The calculations that you made do not allow for the possibility that a collision might take place before the end of the 12 s required for the express train to come to a halt. To check this, take the position of the express train when the engineer first sights the local train as the point of origin and calculate the position of each train at the end of each second after the sighting. Make a table showing the distance of each train from the origin at the end of each second. Plot these positions on the same graph and draw two lines. Use your graph to check your answer to part a.

115. Cargo A helicopter is rising at 5.0 m/s when a bag of its cargo is dropped. The bag falls for 2.0 s. a. What is the bag’s velocity? b. How far has the bag fallen? c. How far below the helicopter is the bag?

Writing in Physics 119. Research and describe Galileo’s contributions to physics.

120. Research the maximum acceleration a human

Thinking Critically 116. Apply CBLs Design a lab to measure the distance an accelerated object moves over time. Use equal time intervals so that you can plot velocity over time as well as distance. A pulley at the edge of a table with a mass attached is a good way to achieve uniform acceleration. Suggested materials include a motion detector, CBL, lab cart, string, pulley, C-clamp, and masses. Generate distancetime and velocity-time graphs using different masses on the pulley. How does the change in mass affect your graphs?

117. Analyze and Conclude Which has the greater acceleration: a car that increases its speed from 50 km/h to 60 km/h, or a bike that goes from 0 km/h to 10 km/h in the same time? Explain.

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body can withstand without blacking out. Discuss how this impacts the design of three common entertainment or transportation devices.

Cumulative Review 121. Solve the following problems. Express your answers in scientific notation. (Chapter 1) a. 6.2104 m 5.7103 m b. 8.7108 km 3.4107 m c. (9.21105 cm)(1.83108 cm) d. (2.63106 m)/(4.08106 s)

122. The equation below describes the motion of an object. Create the corresponding position-time graph and motion diagram. Then write a physics problem that could be solved using that equation. Be creative. d (35.0 m/s) t 5.0 m (Chapter 2)

Chapter 3 Accelerated Motion For more problems, go to Additional Problems, Appendix B.

Multiple Choice Use the following information to answer the first two questions. A ball rolls down a hill with a constant acceleration of 2.0 m/s2. The ball starts at rest and travels for 4.0 s before it stops.

8. The graph shows the motion of a farmer’s truck. What is the truck’s total displacement? Assume that north is the positive direction. 150 m south 125 m north

300 m north 600 m south

1. How far did the ball travel before it stopped? 16 m 20 m

2. What was the ball’s velocity just before it stopped? 2.0 m/s 8.0 m/s

12 m/s 16 m/s

3. A driver of a car enters a new 110-km/h speed zone on the highway. The driver begins to accelerate immediately and reaches 110 km/h after driving 500 m. If the original speed was 80 km/h, what was the driver’s rate of acceleration? 0.44 m/s2 0.60 m/s2

25.0

Velocity (m/s)

8.0 m 12 m

5. A rock climber’s shoe loosens a rock, and her climbing buddy at the bottom of the cliff notices that the rock takes 3.20 s to fall to the ground. How high up the cliff is the rock climber? 15.0 m 31.0 m

Time (s) 5.00

15.0

25.0

35.0

45.0

15.0

by calculating the slope of the tangent on a distance-time graph by calculating the area under the graph on a distance-time graph by calculating the area under the graph on a velocity-time graph by calculating the slope of the tangent on a velocity-time graph

Extended Answer 10. Graph the following data, and then show calculations for acceleration and displacement after 12.0 s on the graph. Time (s)

50.0 m 1.00102 m

0.00

6. A car traveling at 91.0 km/h approaches the turnoff for a restaurant 30.0 m ahead. If the driver slams on the brakes with an acceleration of 6.40 m/s2, what will be her stopping distance? 14.0 m 29.0 m

5.00

9. How can the instantaneous acceleration of an object with varying acceleration be calculated?

8.4 m/s2 9.80 m/s2

8.7 s 17 s

5.00 0.00

25.0

4. A flowerpot falls off the balcony of a penthouse suite 85 m above the street. How long does it take to hit the ground? 4.2 s 8.3 s

15.0

Velocity (m/s) 8.10

6.00

36.9

9.00

51.3

12.00

65.7

50.0 m 100.0 m

7. What is the correct formula manipulation to find acceleration when using the equation vf2 vi2 2ad? (vf2 vi2)/d (vf2 vi2)/2d

(vf vi)2/2d (vf2 vi2)/2d

physicspp.com/standardized_test

Tables If a test question involves a table, skim the table before reading the question. Read the title, column heads, and row heads. Then read the question and interpret the information in the table.

Chapter 3 Standardized Test Practice

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What You’ll Learn • You will use Newton’s laws to solve problems. • You will determine the magnitude and direction of the net force that causes a change in an object’s motion. • You will classify forces according to the agents that cause them.

Why It’s Important Forces act on you and everything around you at all times. Sports A soccer ball is headed by a player. Before play began, the ball was motionless. During play, the ball started, stopped, and changed directions many times.

Think About This What causes a soccer ball, or any other object, to stop, start, or change direction?

physicspp.com 86 Joe McBride/CORBIS

Which force is stronger? Question What forces can act on an object that is suspended by a string? Procedure 1. Tie a piece of heavy cord around the middle of a book. Tie one piece of lightweight string to the center of the cord on the top of the book. Tie another piece to the bottom. 2. While someone holds the end of the top lightweight string so that the book is suspended in the air, pull very slowly, but firmly, on the end of the bottom lightweight string. Record your observations. CAUTION: Keep feet clear of falling objects. 3. Replace the broken string and repeat step 2, but this time pull very fast and very hard on the bottom string. Record your observations. Analysis Which string broke in step 2? Why? Which string broke in step 3? Why? Critical Thinking Draw a diagram of the experimental set-up. Use arrows to show the forces acting on the book.

4.1 Force and Motion

I

magine that a train is speeding down a railroad track at 80 km/h when suddenly the engineer sees a truck stalled at a railroad crossing ahead. The engineer applies the brakes to try to stop the train before it crashes into the truck. Because the brakes cause an acceleration in the direction opposite the train’s velocity, the train will slow down. Imagine that, in this case, the engineer is able to stop the train just before it crashes into the truck. But what if instead of moving at 80 km/h the train had been moving at 100 km/h? What would have to happen for the train to avoid hitting the truck? The acceleration provided by the train’s brakes would have to be greater because the engineer still has the same distance in which to stop the train. Similarly, if the train was going 80 km/h but had been much closer to the truck when the engineer started to apply the brake, the acceleration also would need to be greater because the train would need to stop in less time.

Objectives • Define force. • Apply Newton’s second law to solve problems. • Explain the meaning of Newton’s first law.

Vocabulary force free-body diagram net force Newton’s second law Newton’s first law inertia equilibrium

Section 4.1 Force and Motion

87

Horizons Companies

Force and Motion

System

Ftable on book

Fhand on book

FEarth’s

What happened to make the train slow down? A force is a push or pull exerted on an object. Forces can cause objects to speed up, slow down, or change direction as they move. When an engineer applies the brakes, the brakes exert a force on the wheels and cause the train to slow down. Based on the definitions of velocity and acceleration, this can be restated as follows: a force exerted on an object causes that object’s velocity to change; that is, a force causes an acceleration. Consider a textbook resting on a table. How can you cause it to move? Two possibilites are that you can push on it or you can pull on it. The push or pull is a force that you exert on the textbook. If you push harder on an object, you have a greater effect on its motion. The direction in which the force is exerted also matters—if you push the book to the right, the book moves in a different direction from the direction it would move if you pushed it to the left. The symbol F is a vector and represents the size and direction of a force, while F represents only the magnitude. When considering how a force affects motion, it is important to identify the object of interest. This object is called the system. Everything around the object that exerts forces on it is called the external world. In the case of the book in Figure 4-1, the book is the system. Your hand and gravity are parts of the external world that can interact with the book by pushing or pulling on it and potentially causing its motion to change.

mass on book

Contact Forces and Field Forces ■ Figure 4-1 The book is the system. The table, the hand, and Earth’s mass (through gravity) all exert forces on the book.

88

Again, think about the different ways in which you could move a textbook. You could touch it directly and push or pull it, or you could tie a string around it and pull on the string. These are examples of contact forces. A contact force exists when an object from the external world touches a system and thereby exerts a force on it. If you are holding this physics textbook right now, your hands are exerting a contact force on it. If you place the book on a table, you are no longer exerting a force on the book. The table, however, is exerting a force because the table and the book are in contact. There are other ways in which you could change the motion of the textbook. You could drop it, and as you learned in Chapter 3, it would accelerate as it falls to the ground. The gravitational force of Earth acting on the book causes this acceleration. This force affects the book whether or not Earth is actually touching it. This is an example of a field force. Field forces are exerted without contact. Can you think of other kinds of field forces? If you have ever experimented with magnets, you know that they exert forces without touching. You will investigate magnetism and other similar forces in more detail in future chapters. For now, the only field force that you need to consider is the gravitational force. Forces result from interactions; thus, each force has a specific and identifiable cause called the agent. You should be able to name the agent exerting each force, as well as the system upon which the force is exerted. For example, when you push your textbook, your hand (the agent) exerts a force on the textbook (the system). If there are not both an agent and a system, a force does not exist. What about the gravitational force? If you allow your textbook to fall, the agent is the mass of Earth exerting a field force on the book.

Chapter 4 Forces in One Dimension

a

y

Contact with external world

b

y

Contact with external world System

System

Frope on ball

Fhand on ball

FEarth’s mass

FEarth’s mass

on ball

■ Figure 4-2 To make a physical model of the forces acting on an object, apply the particle model and draw an arrow to represent each force. Label each force, including its agent.

on ball

Free-body diagrams Just as pictorial models and motion diagrams are useful in solving problems about motion, similar representations will help you to analyze how forces affect motion. The first step in solving any problem is to create a pictorial model. For example, to represent the forces on a ball tied to a string or held in your hand, sketch the situations, as shown in Figures 4-2a and 4-2b. Circle the system and identify every place where the system touches the external world. It is at these places that contact forces are exerted. Identify the contact forces. Then identify any field forces on the system. This gives you the pictorial model. To make a physical representation of the forces acting on the ball in Figures 4-2a and 4-2b, apply the particle model and represent the object with a dot. Represent each force with a blue arrow that points in the direction that the force is applied. Try to make the length of each arrow proportional to the size of the force. Often, you will draw these diagrams before you know the magnitudes of all the forces. In such cases, make your best estimate. Always draw the force arrows pointing away from the particle, even when the force is a push. Make sure that you label each force. Use the symbol F with a subscript label to identify both the agent and the object on which the force is exerted. Finally, choose a direction to be positive and indicate this off to the side of your diagram. Usually, you select the positive direction to be in the direction of the greatest amount of force. This typically makes the problem easiest to solve by reducing the number of negative values in your calculations. This type of physical model, which represents the forces acting on a system, is called a free-body diagram.

For each of the following situations, specify the system and draw a motion diagram and a free-body diagram. Label all forces with their agents, and indicate the direction of the acceleration and of the net force. Draw vectors of appropriate lengths.

1. A flowerpot falls freely from a windowsill. (Ignore any forces due to air resistance.) 2. A sky diver falls downward through the air at constant velocity. (The air exerts an upward force on the person.) 3. A cable pulls a crate at a constant speed across a horizontal surface. The surface provides a force that resists the crate’s motion. 4. A rope lifts a bucket at a constant speed. (Ignore air resistance.) 5. A rope lowers a bucket at a constant speed. (Ignore air resistance.)

Section 4.1 Force and Motion

89

Force and Acceleration

■ Figure 4-3 Because the rubber band is stretched a constant amount, it applies a constant force on the cart, which is designed to be low-friction (a). The cart’s motion can be graphed and shown to be a linear relationship (b).

How does an object move when one or more forces are exerted on it? One way to find out is by doing experiments. As before, begin by considering a simple situation. Once you fully understand that situation, then you can add more complications to it. In this case, begin with one controlled force exerted horizontally on an object. The horizontal direction is a good place to start because gravity does not act horizontally. Also, to reduce complications resulting from the object rubbing against the surface, do the experiments on a very smooth surface, such as ice or a very well-polished table, and use an object with wheels that spin easily. In other words, you are trying to reduce the resistance to motion in the situation. To determine how force, acceleration, and velocity are related, you need to be able to exert a constant and controlled force on an object. How can you exert such a controlled force? A stretched rubber band exerts a pulling force; the farther you stretch it, the greater the force with which it pulls back. If you always stretch the rubber band the same amount, you always exert the same force. Figure 4-3a shows a rubber band, stretched a constant 1 cm, pulling a low-resistance cart. If you perform this experiment and determine the cart’s velocity for some period of time, you could construct a graph like the one shown in Figure 4-3b. Does this graph look different from what you expected? What do you notice about the velocity? The constant increase in the velocity is a result of the constant acceleration the stretched rubber band gives the cart. How does this acceleration depend upon the force? To find out, repeat the experiment, this time with the rubber band stretched to a constant 2 cm, and then repeat it again with the rubber band stretched longer and longer each time. For each experiment, plot a velocity-time graph like the one in Figure 4-3b, calculate the acceleration, and then plot the accelerations and forces for all the trials to make a force-acceleration graph, as shown in Figure 4-4a. What is the relationship between the force and acceleration? It’s a linear relationship where the greater the force is, the greater the resulting acceleration. As you did in Chapters 2 and 3, you can apply the straight-line equation y mx b to this graph.

Cart Pulled by Stretched Rubber Band (1 cm)

b

Velocity (m/s)

a

1.50 1.00 0.50 0

0.50 1.00 1.50 2.00 2.50 3.00 3.50 Time (s) a v

90 Matt Meadows

Chapter 4 Forces in One Dimension

Acceleration of Cart

Acceleration v. Force

b

1.2

Acceleration (m/s2)

Acceleration (m/s2)

a

0.8

0.4

1 cart

1

pe

m

slo

1

e 2m slop

2 carts 3 carts

1

slope 3m

0.0 0

1

2

3

Application of force (in cm of stretch)

4

0 Force (N) ■ Figure 4-4 The graph shows that as the force increases, so does the acceleration (a). You can see that the slope of the force-acceleration graph depends upon the number of carts (b).

What is the physical meaning of this slope? Perhaps it describes something about the object that is accelerating. What happens if the object changes? Suppose that a second, identical cart is placed on top of the first, and then a third cart is added. The rubber band would be pulling two carts, and then three. A plot of the force versus acceleration for one, two, and three carts is shown in Figure 4-4b. The graph shows that if the same force is applied in each situation, the acceleration of two carts is 12 the acceleration of one cart, and the acceleration of three carts is 13 the acceleration of one cart. This means that as the number of carts is increased, a greater force is needed to produce the same acceleration. In this example, you would have to stretch the rubber band farther to get a greater amount of force. The slopes of the lines in Figure 4-4b depend upon the number of carts; that is, the slope depends on the total mass of the carts. If the slope, k in this case, is defined as the reciprocal of the mass (k 1/m), then a F/m, or F ma. What information is contained in the equation a F/m? It tells you that a force applied to an object causes that object to experience a change in motion—the force causes the object to accelerate. It also tells you that for the same object, if you double the force, you will double the object’s acceleration. Lastly, if you apply the same force to several different objects, the one with the most mass will have the smallest acceleration and the one with the least mass will have the greatest acceleration. What are the proper units for measuring force? Because F ma, one unit of force causes a 1-kg mass to accelerate at 1 m/s2, so one force unit has the dimensions 1 kgm/s2. The unit 1 kgm/s2 is called the newton, represented Table 4-1 by N. One newton of force applied to a 1-kg Common Forces object will cause it to have an acceleration Description of 1 m/s2. Do these units make sense? Force of gravity on a coin (nickel) Think about a sky diver who is falling through the air. The properties affecting his Force of gravity on 1 lb (0.45 kg) of sugar motion are his mass and the acceleration Force of gravity on a 150-lb (70-kg) person due to the gravitational force, so these units Force of an accelerating car do make sense. Table 4-1 shows the magniForce of a rocket motor tude of some common forces.

F (N) 0.05 4.5 686 3000 5,000,000

Section 4.1 Force and Motion

91

Combining Forces What happens if you and a friend each push a table and exert 100 N of force on it? When you and your friend push together, you give the table a greater acceleration than when you push against each other. In fact, when you push together, you give the table twice the acceleration that it would have if just one of you applied 100 N of force. When you push on the table in opposite directions with the same amount of force, as in Figure 4-5a, the table does not move at all. Figure 4-5b and c show free-body diagrams for these two situations. Figure 4-5d shows a third free-body diagram in which your friend pushes on the table twice as hard as you in the opposite direction. Below each free-body diagram is a vector representing the total result of the two forces. When the force vectors are in the same direction, they can be replaced by one vector with a length equal to their combined length. When the forces are in opposite directions, the resulting vector is the length of the difference between the two vectors. Another term for the vector sum of all the forces on an object is the net force. You also can analyze the situation mathematically. Assume that you are pushing the table in the positive direction with a 100 N force in the above cases. In the first case, your friend is pushing with a negative force of 100 N. Adding them together gives a total force of 0 N, which means there is no acceleration. In the second case, your friend’s force is 100 N, so the total force is 200 N in the positive direction and the table accelerates in the positive direction. In the third case, your friend’s force is 200 N, so the total force is 100 N and the table will accelerate in the negative direction. ■ Figure 4-5 Pushing on the table with equal force in opposite directions (a) results in no net force on the table, as shown by the vector addition in the freebody diagram (b). However, there is a net force applied in (c) and (d), as shown by the free-body diagrams.

a x

F1

b

c

F2 100 N

92 Aaron Haupt

Chapter 4 Forces in One Dimension

F1 100 N

F2

F1 100 N

F2 100 N

d F2 200 N

F1 100 N

Fnet 0 N

Fnet 200 N

Fnet 100 N

Equal forces Opposite directions

Equal forces Same direction

Unequal forces Opposite directions

Newton’s Second Law You could conduct a series of experiments in which you and your friend vary the net force exerted on the table and measure the acceleration in each case. You would find that the acceleration of the table is proportional to the net force exerted on it and inversely proportional to its mass. In other words, if the net force of you and your friend acting on the table is 100 N, the table will experience the same acceleration as it would if only you were acting on it with a force of 100 N. Taking this into account, the mathematical relationship among force, mass, and acceleration can be rewritten in terms of the net force. The observation that the acceleration of an object is proportional to the net force and inversely proportional to the mass of the object being accelerated is Newton’s second law, which is represented by the following equation. Newton’s Second Law

F m

a net

The acceleration of an object is equal to the sum of the forces acting on the object, divided by the mass of the object.

Notice that Newton’s second law can be rearranged to the form F ma, which you learned about previously. If the table that you and your friend were pushing was 15.0 kg and the two of you each pushed with a force of 50.0 N in the same direction, what would be the acceleration of the table? To find out, calculate the net force, 50.0 N 50.0 N 100.0 N, and apply Newton’s second law by dividing the net force of 100.0 N by the mass of the table, 15.0 kg, to get an acceleration of 6.67 m/s2. Here is a useful strategy for finding how the motion of an object depends on the forces exerted on it. First, identify all the forces acting on the object. Draw a free-body diagram showing the direction and relative strength of each force acting on the system. Then, add the forces to find the net force. Next, use Newton’s second law to calculate the acceleration. Finally, if necessary, use kinematics to find the velocity or position of the object. When you learned about kinematics in Chapters 2 and 3, you studied the motion of objects without regard for the causes of motion. You now know that an unbalanced force, a net force, is the cause of a change in velocity (an acceleration).

6. Two horizontal forces, 225 N and 165 N, are exerted on a canoe. If these forces are applied in the same direction, find the net horizontal force on the canoe. 7. If the same two forces as in the previous problem are exerted on the canoe in opposite directions, what is the net horizontal force on the canoe? Be sure to indicate the direction of the net force. 8. Three confused sleigh dogs are trying to pull a sled across the Alaskan snow. Alutia pulls east with a force of 35 N, Seward also pulls east but with a force of 42 N, and big Kodiak pulls west with a force of 53 N. What is the net force on the sled?

Section 4.1 Force and Motion

93

Newton’s First Law What is the motion of an object with no net force acting on it? A stationary object with no net force acting on it will stay at its position. Consider a moving object, such as a ball rolling on a surface. How long will the ball continue to roll? It will depend on the quality of the surface. If the ball is rolled on a thick carpet that offers much resistance, it will come to rest quickly. If it is rolled on a hard, smooth surface that offers little resistance, such as a bowling alley, the ball will roll for a long time with little change in velocity. Galileo did many experiments, and he concluded that in the ideal case of zero resistance, horizontal motion would never stop. Galileo was the first to recognize that the general principles of motion could be found by extrapolating experimental results to the ideal case, in which there is no resistance to slow down an object’s motion. In the absence of a net force, the motion (or lack of motion) of both the moving ball and the stationary object continues as it was. Newton recognized this and generalized Galileo’s results in a single statement. This statement, “an object that is at rest will remain at rest, and an object that is moving will continue to move in a straight line with constant speed, if and only if the net force acting on that object is zero,” is called Newton’s first law.

Table 4-2 Some Types of Forces Force

94

Symbol

Definition

Friction

Ff

The contact force that acts to oppose sliding motion between surfaces

Parallel to the surface and opposite the direction of sliding

Normal

FN

The contact force exerted by a surface on an object

Perpendicular to and away from the surface

Spring

Fsp

A restoring force; that is, the push or pull a spring exerts on an object

Opposite the displacement of the object at the end of the spring

Tension

FT

The pull exerted by a string, rope, or cable when attached to a body and pulled taut

Away from the object and parallel to the string, rope, or cable at the point of attachment

Thrust

Fthrust

A general term for the forces that move objects such as rockets, planes, cars, and people

In the same direction as the acceleration of the object, barring any resistive forces

Weight

Fg

A field force due to gravitational attraction between two objects, generally Earth and an object

Straight down toward the center of Earth

Chapter 4 Forces in One Dimension

Direction

Inertia Newton’s first law is sometimes called the law of inertia. Is inertia a force? No. Inertia is the tendency of an object to resist change. If an object is at rest, it tends to remain at rest. If it is moving at a constant velocity, it tends to continue moving at that velocity. Forces are results of interactions between two objects; they are not properties of single objects, so inertia cannot be a force. Remember that because velocity includes both the speed and direction of motion, a net force is required to change either the speed or direction of an object’s motion. Equilibrium According to Newton’s first law, a net force is something that causes the velocity of an object to change. If the net force on an object is zero, then the object is in equilibrium. An object is in equilibrium if it is at rest or if it is moving at a constant velocity. Note that being at rest is simply a special case of the state of constant velocity, v 0. Newton’s first law identifies a net force as something that disturbs a state of equilibrium. Thus, if there is no net force acting on the object, then the object does not experience a change in speed or direction and is in equilibrium. By understanding and applying Newton’s first and second laws, you can often figure out something about the relative sizes of forces, even in situations in which you do not have numbers to work with. Before looking at an example of this, review Table 4-2, which lists some of the common types of forces. You will be dealing with many of these throughout your study of physics. When analyzing forces and motion, it is important to keep in mind that the world is dominated by resistance. Newton’s ideal, resistance-free world is not easy to visualize. If you analyze a situation and find that the result is different from a similar experience that you have had, ask yourself if this is because of the presence of resistance. In addition, many terms used in physics have everyday meanings that are different from those understood in physics. When talking or writing about physics issues, be careful to use these terms in their precise, scientific way.

Shuttle Engine Thrust The Space Shuttle Main Engines (SSMEs) each are rated to provide 1.6 million N of thrust. Powered by the combustion of hydrogen and oxygen, the SSMEs are throttled anywhere from 65 percent to 109 percent of their rated thrust.

4.1 Section Review 9. Force Identify each of the following as either a, b, or c: weight, mass, inertia, the push of a hand, thrust, resistance, air resistance, spring force, and acceleration. a. a contact force b. a field force c. not a force 10. Inertia Can you feel the inertia of a pencil? Of a book? If you can, describe how. 11. Free-Body Diagram Draw a free-body diagram of a bag of sugar being lifted by your hand at a constant speed. Specifically identify the system. Label all forces with their agents and make the arrows the correct lengths. physicspp.com/self_check_quiz

12. Direction of Velocity If you push a book in the forward direction, does this mean its velocity has to be forward? 13. Free-Body Diagram Draw a free-body diagram of a water bucket being lifted by a rope at a decreasing speed. Specifically identify the system. Label all forces with their agents and make the arrows the correct lengths. 14. Critical Thinking A force of 1 N is the only force exerted on a block, and the acceleration of the block is measured. When the same force is the only force exerted on a second block, the acceleration is three times as large. What can you conclude about the masses of the two blocks? Section 4.1 Force and Motion

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4.2 Using Newton’s Laws

Objectives • Describe how the weight and the mass of an object are related. • Differentiate between actual weight and apparent weight.

Using Newton’s Second Law

Vocabulary apparent weight weightlessness drag force terminal velocity

■ Figure 4-6 The net force on the ball is the weight force, Fg.

System v

Known

a

ag m

Unknown Fg

N

ewton’s second law describes the connection between the cause of a change in an object’s velocity and the resulting displacement. This law identifies the relationship between the net force exerted on an object and the object’s acceleration.

Fg

Fnet ma Fnet Fg and a g so Fg mg

What is the weight force, Fg , exerted on an object of mass m? Newton’s second law can help answer this question. Consider the pictorial and physical models in Figure 4-6, which show a free-falling ball in midair. With what objects is it interacting? Because it is touching nothing and air resistance can be neglected, the only force acting on it is Fg . You know from Chapter 3 that the ball’s acceleration is g. Newton’s second law then becomes Fg mg. Both the force and the acceleration are downward. The magnitude of an object’s weight is equal to its mass times the acceleration it would have if it were falling freely. It is important to keep in mind that even when an object is not experiencing free-fall, the gravitational force of Earth is still acting on the object. This result is true on Earth, as well as on any other planet, although the magnitude of g will be different on other planets. Because the value of g is much less on the Moon than on Earth, astronauts who landed on the Moon weighed much less while on the Moon, even though their mass had not changed. Scales A bathroom scale contains springs. When you stand on the scale, the scale exerts an upward force on you because you are in contact with it. Because you are not accelerating, the net force acting on you must be zero. Therefore, the spring force, Fsp, pushing up on you must be the same magnitude as your weight, Fg , pulling down on you, as shown in the pictorial and physical models in Figure 4-7. The reading on the scale is determined by the amount of force the springs inside it exert on you. A spring scale, therefore, measures weight, not mass. If you were on a different planet, the compression of the spring would be different, and consequently, the scale’s reading would be different. Remember that the proper unit for expressing mass is kilograms and because weight is a force, the proper unit used to express weight is the newton. a

■ Figure 4-7 The upward force of the spring in the scale is equal to your weight when you step on the bathroom scale (a). The free-body diagram in (b) shows that the system is in equilibrium because the force of the spring is equal to your weight.

96 Aaron Haupt

Chapter 4 Forces in One Dimension

b

y

Fsp

System Fg

Fighting Over a Toy Anudja is holding a stuffed dog, with a mass of 0.30 kg, when Sarah decides that she wants it and tries to pull it away from Anudja. If Sarah pulls horizontally on the dog with a force of 10.0 N and Anudja pulls with a horizontal force of 11.0 N, what is the horizontal acceleration of the dog? 1

Analyze and Sketch the Problem • Sketch the situation. • Identify the dog as the system and the direction in which Anudja pulls as positive. • Draw the free-body diagram. Label the forces. Known: m 0.30 kg FAnudja on dog 11.0 N FSarah on dog 10.0 N

2

Unknown: a?

Solve for the Unknown

F1

Fnet FAnudja on dog (FSarah on dog)

F2

Use Newton’s second law. F m FAnudja on dog (FSarah on dog) Substitute Fnet FAnudja on dog (FSarah on dog) m 11.0 N 10.0 N Substitute FAnudja on dog 11.0 N, FSarah on dog 10.0 N, m 0.30 kg 0.30 kg

a net

3.3 m/s2 a 3.3 m/s2 toward Anudja 3

Math Handbook Operations with Significant Digits pages 835–836

Evaluate the Answer

• Are the units correct? m/s2 is the correct unit for acceleration. • Does the sign make sense? The acceleration is in the positive direction, which is expected, because Anudja is pulling in the positive direction with a greater force than Sarah is pulling in the negative direction. • Is the magnitude realistic? It is a reasonable acceleration for a light, stuffed toy.

15. You place a watermelon on a spring scale at the supermarket. If the mass of the watermelon is 4.0 kg, what is the reading on the scale? 16. Kamaria is learning how to ice-skate. She wants her mother to pull her along so that she has an acceleration of 0.80 m/s2. If Kamaria’s mass is 27.2 kg, with what force does her mother need to pull her? (Neglect any resistance between the ice and Kamaria’s skates.) 17. Taru and Reiko simultaneously grab a 0.75-kg piece of rope and begin tugging on it in opposite directions. If Taru pulls with a force of 16.0 N and the rope accelerates away from her at 1.25 m/s2, with what force is Reiko pulling? 18. In Figure 4-8, the block has a mass of 1.2 kg and the sphere has a mass of 3.0 kg. What are the readings on the two scales? (Neglect the masses of the scales.)

■

Figure 4-8

Section 4.2 Using Newton’s Laws

97

System y

Apparent weight What is weight? Because the weight force is defined as Fg mg, Fg changes when g varies. On or near the surface of Earth, g is approximately constant, so an object’s Fnet weight does not change appreciably as it moves around near Earth’s surface. If a bathroom scale provides the only upward force on you, then it reads your weight. What would it read if you stood with one foot on the scale and one foot on the floor? Fg What if a friend pushed down on your shoulders or up on your elbows? Then there would be other contact forces on you, and the scale would not read your weight. What happens if you stand on a scale in an elevator? As long as the elevator is in equilibrium, the scale will read your weight. What would the scale read if the elevator accelerates upward? Figure 4-9 shows the pictorial and physical models for this situation. You are the system, and upward is the positive direction. Because the acceleration of the system is upward, the upward force of the scale must be greater than the downward force of your weight. Therefore, the scale reading is greater than your weight. If you ride in an elevator like this, you would feel heavier because the floor would press harder on your feet. On the other hand, if the acceleration is downward, then you would feel lighter, and the scale would have a lower reading. The force exerted by the scale is called the apparent weight. An object’s apparent weight is the force an object experiences as a result of all the forces acting on it, giving the object an acceleration. Imagine that the cable holding the elevator breaks. What would the scale read then? The scale and you would both accelerate with a g. According to this formula, the scale would read zero and your apparent weight would be zero. That is, you would be weightless. However, weightlessness does not mean that an object’s weight is actually zero; rather, it means that there are no contact forces pushing up on the object, and the object’s apparent weight is zero. Fscale

Fscale

v

Fg ■ Figure 4-9 If you stand on a scale in an elevator accelerating upward, the scale must exert an upward force greater than the downward force of your weight.

a

Force and Motion When solving force and motion problems, use the following strategies. 1. Read the problem carefully, and sketch a pictorial model. 2. Circle the system and choose a coordinate system. 3. Determine which quantities are known and which are unknown. 4. Create a physical model by drawing a motion diagram showing the direction of the acceleration, and create a free-body diagram showing the net force. 5. Use Newton’s laws to link acceleration and net force. 6. Rearrange the equation to solve for the unknown quantity. 7. Substitute known quantities with their units into the equation and solve. 8. Check your results to see if they are reasonable.

98

Chapter 4 Forces in One Dimension

Real and Apparent Weight Your mass is 75.0 kg, and you are standing on a bathroom scale in an elevator. Starting from rest, the elevator accelerates upward at 2.00 m/s2 for 2.00 s and then continues at a constant speed. Is the scale reading during acceleration greater than, equal to, or less than the scale reading when the elevator is at rest? 1

Analyze and Sketch the Problem • Sketch the situation. • Choose a coordinate system with the positive direction as upward. • Draw the motion diagram. Label v and a. • Draw the free-body diagram. The net force is in the same direction as the acceleration, so the upward force is greater than the downward force.

2

Known:

Unknown:

m 75.0 kg a 2.00 m/s2 t 2.00 s g 9.80 N

Fscale ?

Fscale

System y

Fscale

Fnet

v

a Fg

Fg

Solve for the Unknown Fnet ma Fnet Fscale (Fg)

Fg is negative because it is in the negative direction defined by the coordinate system.

Solve for Fscale. Fscale Fnet Fg Elevator at rest: Fscale Fnet Fg Fg mg (75.0 kg)(9.80 m/s2) 735 N Acceleration of the elevator:

The elevator is not accelerating. Thus, Fnet 0.00 N. Substitute Fnet 0.00 N Substitute Fg mg Math Handbook Substitute m 75.0 kg, g 9.80 m/s2 Operations with Significant Digits pages 835–836

Fscale Fnet Fg ma mg m(a g) (75.0 kg)(2.00 m/s2 9.80 m/s2) 885 N

Substitute Fnet ma, Fg mg Substitute m 75.0 kg, a 2.00 m/s2, g 9.80 m/s2

The scale reading when the elevator is accelerating (885 N) is larger than the scale reading at rest (735 N). 3

Evaluate the Answer • Are the units correct? kgm/s2 is the force unit, N. • Does the sign make sense? The positive sign agrees with the coordinate system. • Is the magnitude realistic? Fscale is larger than it would be at rest when Fscale would be 735 N, so the magnitude is reasonable.

Section 4.2 Using Newton’s Laws

99

19. On Earth, a scale shows that you weigh 585 N. a. What is your mass? b. What would the scale read on the Moon (g 1.60 m/s2)? 20. Use the results from Example Problem 2 to answer questions about a scale in an elevator on Earth. What force would be exerted by the scale on a person in the following situations? a. The elevator moves at constant speed. b. It slows at 2.00 m/s2 while moving upward. c. It speeds up at 2.00 m/s2 while moving downward. d. It moves downward at constant speed. e. It slows to a stop at a constant magnitude of acceleration.

Drag Force and Terminal Velocity It is true that the particles in the air around an object exert forces on it. Air actually exerts a huge force, but in most cases, it exerts a balanced force on all sides, and therefore it has no net effect. Can you think of any experiences that help to prove that air exerts a force? When you stick a suction cup on a smooth wall or table, you remove air from the “inside” of it. The suction cup is difficult to remove because of the net force of the air on the “outside.” So far, you have neglected the force of air on an object moving through the air. In actuality, when an object moves through any fluid, such as air or water, the fluid exerts a drag force on the moving object in the direction opposite to its motion. A drag force is the force exerted by a fluid on the object moving through the fluid. This force is dependent on the motion of the object, the properties of the object, and the properties of the fluid that the object is moving through. For example, as the speed of the object increases, so does the magnitude of the drag force. The size and shape of the object also affects the drag force. The drag force is also affected by the properties of the fluid, such as its viscosity and temperature.

An air-track glider passes through a photoelectric gate at an initial speed of 0.25 m/s. As it passes through the gate, a constant force of 0.40 N is applied to the glider in the same direction as its motion. The glider has a mass of 0.50 kg. 1. What is the acceleration of the glider? 2. It takes the glider 1.3 s to pass through a second gate. What is the distance between the two gates?

3. The 0.40-N force is applied by means of a string attached to the glider. The other end of the string passes over a resistance-free pulley and is attached to a hanging mass, m. How big is m?

4. Derive an expression for the tension, T, in the string as a function of the mass, M, of the glider, the mass, m, of the hanging mass, and g.

100

Chapter 4 Forces in One Dimension

If you drop a table-tennis ball, as in Figure 4-10, it has very little velocity at the start, and thus only a small drag force. The downward force of gravity is much stronger than the upward drag force, so Fd v there is a downward acceleration. As the ball’s Fd velocity increases, so does the drag force. Soon, the Fg drag force equals the force of gravity. When this Fg happens, there is no net force, and so there is no acceleration. The constant velocity that is reached when the drag force equals the force of gravity is called the terminal velocity. When light objects with large surface areas are falling, the drag force has a substantial effect on their motion, and they quickly reach terminal velocity. Heavier, more-compact objects are not affected as much by the drag force. For example, the terminal velocity of a table-tennis ball in air is 9 m/s, that of a basketball is 20 m/s, and that of a baseball is 42 m/s. Competitive skiers increase their terminal velocities by decreasing the drag force on them. They hold their bodies in an egg shape and wear smooth clothing and streamlined helmets. Sky divers can increase or decrease their terminal velocity by changing their body orientation and shape. A horizontal, spread-eagle shape produces the slowest terminal velocity, about 60 m/s. Because a parachute changes the shape of the sky diver when it opens, a sky diver becomes part of a very large object with a correspondingly large drag force and a terminal velocity of about 5 m/s.

v Fd

v

Fg

Fd

v

Fg

■

Figure 4-10 The drag force on an object increases as its velocity increases. When the drag force increases to the point that it equals the force of gravity, the object will no longer be accelerated.

4.2 Section Review 21. Lunar Gravity Compare the force holding a 10.0-kg rock on Earth and on the Moon. The acceleration due to gravity on the Moon is 1.62 m/s2. 22. Real and Apparent Weight You take a ride in a fast elevator to the top of a tall building and ride back down while standing on a bathroom scale. During which parts of the ride will your apparent and real weights be the same? During which parts will your apparent weight be less than your real weight? More than your real weight? Sketch freebody diagrams to support your answers. 23. Acceleration Tecle, with a mass of 65.0 kg, is standing by the boards at the side of an iceskating rink. He pushes off the boards with a force of 9.0 N. What is his resulting acceleration? 24. Motion of an Elevator You are riding in an elevator holding a spring scale with a 1-kg mass suspended from it. You look at the scale and see that it reads 9.3 N. What, if anything, can you conclude about the elevator’s motion at that time? physicspp.com/self_check_quiz

25. Mass Marcos is playing tug-of-war with his cat using a stuffed toy. At one instant during the game, Marcos pulls on the toy with a force of 22 N, the cat pulls in the opposite direction with a force of 19.5 N, and the toy experiences an acceleration of 6.25 m/s2. What is the mass of the toy? 26. Acceleration A sky diver falls at a constant speed in the spread-eagle position. After he opens his parachute, is the sky diver accelerating? If so, in which direction? Explain your answer using Newton’s laws. 27. Critical Thinking You have a job at a meat warehouse loading inventory onto trucks for shipment to grocery stores. Each truck has a weight limit of 10,000 N of cargo. You push each crate of meat along a low-resistance roller belt to a scale and weigh it before moving it onto the truck. However, right after you weigh a 1000-N crate, the scale breaks. Describe a way in which you could apply Newton’s laws to figure out the approximate masses of the remaining crates. Section 4.2 Using Newton’s Laws

101

4.3 Interaction Forces

Objectives • Define Newton’s third law. • Explain the tension in ropes and strings in terms of Newton’s third law. • Define the normal force. • Determine the value of the normal force by applying Newton’s second law.

Vocabulary interaction pair Newton’s third law tension normal force

FA on B ■

FB on A

Figure 4-11 When you exert a force on your friend to push him forward, he exerts an equal and opposite force on you, which causes you to move backwards.

Y

ou have learned that when an agent exerts a net force upon an object, the object undergoes acceleration. You know that this force can be either a field force or a contact force. But what causes the force? If you experiment with two magnets, you can feel each magnet pushing or pulling the other. Similarly, if you pull on a lever, you can feel the lever pulling back against you. Which is the agent and which is the object?

Identifying Interaction Forces Imagine that you and a friend are each wearing in-line skates (with all the proper safety gear), and your friend is standing right in front of you, with his back to you. You push your friend so that he starts rolling forward. What happens to you? You move backwards. Why? Recall that a force is the result of an interaction between two objects. When you push your friend forward, you come into contact with him and exert a force that moves him forward. However, because he is also in contact with you, he also exerts a force on you, and this results in a change in your motion. Forces always come in pairs. Consider you (Student A) as one system and your friend (Student B) as another. What horizontal forces act on each of the two systems? Figure 4-11 shows the free-body diagram for the systems. Looking at this diagram, you can see that each system experiences a force exerted by the other. The two forces, FA on B and FB on A, are the forces of interaction between the two of you. Notice the symmetry in the subscripts: A on B and B on A. What do you notice about the directions of these forces? What do you expect to be true about their relative magnitudes? The forces FA on B and FB on A are an interaction pair. An interaction pair is two forces that are in opposite directions and have equal magnitude. Sometimes, this also is called an action-reaction pair of forces. This might suggest that one causes the other; however, this is not true. For example, the force of you pushing your friend doesn’t cause your friend to exert a force on you. The two forces either exist together or not at all. They both result from the contact between the two of you.

Newton’s Third Law The force of you on your friend is equal in magnitude and opposite in direction to the force of your friend on you. This is summarized in Newton’s third law, which states that all forces come in pairs. The two forces in a pair act on different objects and are equal in strength and opposite in direction. Newton’s Third Law

FA on B FB on A

The force of A on B is equal in magnitude and opposite in direction of the force of B on A.

Consider the situation of you holding a book in your hand. Draw one free-body diagram each for you and for the book. Are there any interaction 102

Chapter 4 Forces in One Dimension

pairs? When identifying interaction pairs, keep in mind that they always will occur in two different free-body diagrams, and they always will have the symmetry of subscripts noted on the previous page. In this case, there is one interaction pair, Fbook on hand and Fhand on book. Notice also that each object has a weight. If the weight force is due to an interaction between the object and Earth’s mass, then shouldn’t each of these objects also exert a force on Earth? If this is the case, shouldn’t Earth be accelerating? Consider a soccer ball sitting on a table. The table, in turn, is sitting on Earth, as shown in Figure 4-12. First, analyze the forces acting on the ball. The table exerts an upward force on the ball, and the mass of Earth exerts a downward gravitational force on the ball. Even though these forces are in the opposite direction on the same object, they are not an interaction pair. They are simply two forces acting on the same object, not the interaction between two objects. Consider the ball and the table together. In addition to the upward force exerted by the table on the ball, the ball exerts a downward force on the table. This is one pair of forces. The ball and Earth also have an interaction pair. Thus, the interaction pairs related to the soccer ball are Fball on table Ftable on ball and Fball on Earth FEarth on ball. It is important to keep in mind that an interaction pair must consist of two forces of equal magnitude pointing in opposite directions. These opposing forces must act on two different objects that can exert a force against each other. The acceleration caused by the force of an object interacting with Earth is usually a very small number. Under most circumstances, the number is so small that for problems involving falling or stationary objects, Earth can be treated as part of the external world rather than as a second system. Consider Example Problem 3 using the following problem-solving strategies.

■ Figure 4-12 A soccer ball on a table on Earth is part of two interaction pairs—the interaction between the ball and table and the interaction between the ball and Earth. (Not to scale)

Tug-of-War Challenge In a tug-of-war, predict how the force you exert on your end of the rope compares to the force your opponent exerts if you pull and your opponent just holds the rope.

Interaction Pairs Use these strategies to solve problems in which there is an interaction between objects in two different systems. 1. Separate the system or systems from the external world. 2. Draw a pictorial model with coordinate systems for each system and a physical model that includes free-body diagrams for each system. 3. Connect interaction pairs by dashed lines.

1. Predict how the forces compare if the rope moves in your direction. 2. Test your prediction. CAUTION: Do not suddenly let go of the rope. Analyze and Conclude 3. Compare the force on your end of the rope to the force on your opponent’s end of the rope. What happened when you started to move your opponent’s direction?

4. To calculate your answer, use Newton’s second law to relate the net force and acceleration for each system. 5. Use Newton’s third law to equate the magnitudes of the interaction pairs and give the relative direction of each force. 6. Solve the problem and check the units, signs, and magnitudes for reasonableness.

Section 4.3 Interaction Forces

103

Earth’s Acceleration When a softball with a mass of 0.18 kg is dropped, its acceleration toward Earth is equal to g, the acceleration due to gravity. What is the force on Earth due to the ball, and what is Earth’s resulting acceleration? Earth’s mass is 6.01024 kg. 1

Analyze and Sketch the Problem • Draw free-body diagrams for the two systems: the ball and Earth. • Connect the interaction pair by a dashed line.

2

Known:

Unknown:

mball 0.18 kg mEarth 6.01024 kg g 9.80 m/s2

FEarth on ball ? aEarth ?

Softball

y FEarth on ball Fball on Earth

Earth

Solve for the Unknown

Use Newton’s second law to find the weight of the ball. FEarth on ball mballa mball(g) Substitute a g (0.18 kg)(9.80 m/s2) Substitute m ball 0.18 kg, g 9.80 m/s2 1.8 N Use Newton’s third law to find Fball on Earth. Fball on Earth FEarth on ball (1.8 N) Substitute FEarth on ball 1.8 N 1.8 N Use Newton’s second law to find aEarth. F mEarth 1.8 N 6.01024 kg

Math Handbook Operations with Scientific Notation pages 842–843

et aEarth n

Substitute Fnet 1.8 N, m Earth 6.01024 kg

2.91025 m/s2 toward the softball 3

Evaluate the Answer • Are the units correct? Dimensional analysis verifies force in N and acceleration in m/s2. • Do the signs make sense? Force and acceleration should be positive. • Is the magnitude realistic? Because of Earth’s large mass, the acceleration should be small.

28. You lift a relatively light bowling ball with your hand, accelerating it upward. What are the forces on the ball? What forces does the ball exert? What objects are these forces exerted on? 29. A brick falls from a construction scaffold. Identify any forces acting on the brick. Also identify any forces that the brick exerts and the objects on which these forces are exerted. (Air resistance may be ignored.) 30. You toss a ball up in the air. Draw a free-body diagram for the ball while it is still moving upward. Identify any forces acting on the ball. ■ Figure 4-13 Also identify any forces that the ball exerts and the objects on which these forces are exerted. 31. A suitcase sits on a stationary airport luggage cart, as in Figure 4-13. Draw a free-body diagram for each object and specifically indicate any interaction pairs between the two.

104

Chapter 4 Forces in One Dimension

Forces of Ropes and Strings Tension is simply a specific name for the force exerted by a string or rope. A simplification within this textbook is the assumption that all strings and ropes are massless. To understand tension in more detail, consider the situation in Figure 4-14, where a bucket hangs from a rope attached to the ceiling. The rope is about to break in the middle. If the rope breaks, the bucket will fall; thus, before it breaks, there must be forces holding the rope together. The force that the top part of the rope exerts on the bottom part is Ftop on bottom. Newton’s third law states that this force must be part of an interaction pair. The other member of the pair is the force that the bottom part exerts on the top, Fbottom on top. These forces, equal in magnitude but opposite in direction, also are shown in Figure 4-14. Think about this situation in another way. Before the rope breaks, the bucket is in equilibrium. This means that the force of its weight downward must be equal in magnitude but opposite in direction to the tension in the rope upward. Similarly, if you look at the point in the rope just above the bucket, it also is in equilibrium. Therefore, the tension of the rope below it pulling down must be equal to the tension of the rope above it pulling up. You can move up the rope, considering any point in the rope, and see that the tension forces are pulling equally in both directions. Because the very bottom of the rope has a tension equal to the weight of the bucket, the tension everywhere in the rope is equal to the weight of the bucket. Thus, the tension in the rope is the weight of all objects below it. Because the rope is assumed to be massless, the tension everywhere in the rope is equal to the bucket’s weight. Tension forces also are at work in a tug-of-war, like the one shown in Figure 4-15. If team A, on the left, is exerting a force of 500 N and the rope does not move, then team B, on the right, also must be pulling with a force of 500 N. What is the tension in the rope in this case? If each team pulls with 500 N of force, is the tension 1000 N? To decide, think of the rope as divided into two halves. The left-hand end is not moving, so the net force on it is zero. Thus, FA on rope Fright on left 500 N. Similarly, FB on rope Fleft on right 500 N. But the two tensions, Fright on left and Fleft on right, are an interaction pair, so they are equal and opposite. Thus, the tension in the rope equals the force with which each team pulls, or 500 N. To verify this, you could cut the rope in half, tie the ends to a spring scale, and ask the two teams each to pull with 500 N of force. You would see that the scale reads 500 N.

FT (bottom on top) FT (top on bottom)

Fg

■ Figure 4-14 The tension in the rope is equal to the weight of all the objects hanging from it.

■ Figure 4-15 In a tug-of-war, the teams exert equal and opposite forces on each other via the tension in the rope, as long as neither side moves.

FA 500 N FB 500 N Fnet 0 N

Section 4.3 Interaction Forces

105 Tim Fuller

Lifting a Bucket A 50.0-kg bucket is being lifted by a rope. The rope will not break if the tension is 525 N or less. The bucket started at rest, and after being lifted 3.0 m, it is moving at 3.0 m/s. If the acceleration is constant, is the rope in danger of breaking? 1

Analyze and Sketch the Problem • • • •

Draw the situation and identify the forces on the system. Establish a coordinate system with the positive axis upward. Draw a motion diagram including v and a. Draw the free-body diagram, labeling the forces. Known: m 50.0 kg vi 0.0 m/s

2

y

v FT (rope on

Unknown:

vf 3.0 m/s d 3.0 m

bucket)

FT ?

a Fnet

Solve for the Unknown Fnet is the sum of the positive force of the rope pulling up, FT, System and the negative weight force, Fg, pulling down as defined by the coordinate system. Fnet FT (Fg) FT Fnet Fg ma mg Substitute Fnet ma, Fg mg m(a g) vi, vf, and d are known. Use this motion equation to solve for a. vf2 vi2 2ad v2 v2 2d vf2 2d

Fg (Earth’s mass on bucket)

f i a

Math Handbook Substitute vi 0.0 m/s2

FT m(a g)

(

v2 2d

)

Isolating a Variable page 845

v2 2d

m f g

Substitute a f 2

( (3.0 m/s)

)

(50.0 kg) 9.80 m/s2 Substitute m 50.0 kg, vf 3.0 m/s, d 3.0 m, g 9.80 m/s2 2(3.0 m) 570 N The rope is in danger of breaking because the tension exceeds 525 N. 3

Evaluate the Answer • Are the units correct? Dimensional analysis verifies kgm/s2, which is N. • Does the sign make sense? The upward force should be positive. • Is the magnitude realistic? The magnitude is a little larger than 490 N, which is the weight of the bucket. Fg mg (50.0 kg)(9.80 m/s2) 490 N

32. You are helping to repair a roof by loading equipment into a bucket that workers hoist to the rooftop. If the rope is guaranteed not to break as long as the tension does not exceed 450 N and you fill the bucket until it has a mass of 42 kg, what is the greatest acceleration that the workers can give the bucket as they pull it to the roof? 33. Diego and Mika are trying to fix a tire on Diego’s car, but they are having trouble getting the tire loose. When they pull together, Mika with a force of 23 N and Diego with a force of 31 N, they just barely get the tire to budge. What is the magnitude of the strength of the force between the tire and the wheel?

106

Chapter 4 Forces in One Dimension

a

b

c

FN

50.0 N 50.0 N

FN

FN

mg

mg

The Normal Force Any time two objects are in contact, they each exert a force on each other. Think about a box sitting on a table. There is a downward force due to the gravitational attraction of Earth. There also is an upward force that the table exerts on the box. This force must exist, because the box is in equilibrium. The normal force is the perpendicular contact force exerted by a surface on another object. The normal force always is perpendicular to the plane of contact between two objects, but is it always equal to the weight of an object as in Figure 4-16a? What if you tied a string to the box and pulled up on it a little bit, but not enough to move the box, as shown in Figure 4-16b? When you apply Newton’s second law to the box, you see that FN Fstring on box Fg ma 0, which rearranges to FN Fg Fstring on box. You can see that in this case, the normal force exerted by the table on the box is less than the box’s weight, Fg. Similarly, if you pushed down on the box on the table as shown in Figure 4-16c, the normal force would be more than the box’s weight. Finding the normal force will be important in the next chapter, when you begin dealing with resistance.

mg ■ Figure 4-16 The normal force on an object is not always equal to its weight. In (a) the normal force is equal to the object’s weight. In (b) the normal force is less than the object’s weight. In (c) the normal force is greater than the object’s weight.

4.3 Section Review 34. Force Hold a book motionless in your hand in the air. Identify each force and its interaction pair on the book. 35. Force Lower the book from problem 34 at increasing speed. Do any of the forces or their interaction-pair partners change? Explain. 36. Tension A block hangs from the ceiling by a massless rope. A second block is attached to the first block and hangs below it on another piece of massless rope. If each of the two blocks has a mass of 5.0 kg, what is the tension in each rope? physicspp.com/self_check_quiz

37. Tension If the bottom block in problem 36 has a mass of 3.0 kg and the tension in the top rope is 63.0 N, calculate the tension in the bottom rope and the mass of the top block. 38. Normal Force Poloma hands a 13-kg box to 61-kg Stephanie, who stands on a platform. What is the normal force exerted by the platform on Stephanie? 39. Critical Thinking A curtain prevents two tug-ofwar teams from seeing each other. One team ties its end of the rope to a tree. If the other team pulls with a 500-N force, what is the tension? Explain. Section 4.3 Interaction Forces

107

Forces in an Elevator Alternate CBL instructions can be found on the Web site. physicspp.com

Have you ever been in a fast-moving elevator? Was the ride comfortable? How about an amusement ride that quickly moves upward or one that free-falls? What forces are acting on you during your ride? In this experiment, you will investigate the forces that affect you during vertical motion when gravity is involved with a bathroom scale. Many bathroom scales measure weight in pounds mass (lbm) or pounds force (lbf) rather than newtons. In the experiment, you will need to convert weights measured on common household bathroom scales to SI units.

QUESTION What one-dimensional forces act on an object that is moving in a vertical direction in relation to the ground?

Objectives

Safety Precautions

■ Measure Examine forces that act on objects

that move vertically. ■ Compare and Contrast Differentiate between

■ Use caution when working around elevator

actual weight and apparent weight. ■ Analyze and Conclude Share and compare data of the acceleration of elevators.

■ Do not interfere with normal elevator traffic. ■ Watch that the mass on the spring scale

doors.

does not fall and hit someone’s feet or toes.

Materials elevator bathroom scale spring scale mass

Procedure 1. Securely attach a mass to the hook on a spring scale. Record the force of the mass in the data table. 2. Accelerate the mass upward, then move it upward at a constant velocity, and then slow the mass down. Record the greatest amount of force on the scale, the amount of force at constant velocity, and the lowest scale reading. 3. Get your teacher’s permission and proceed to an elevator on the ground floor. Before entering the elevator, measure your weight on a bathroom scale. Record this weight in the data table.

108 Laura Sifferlin

Data Table Force (step 1) Highest Reading (step 2) Reading at Constant Velocity (step 2) Lowest Reading (step 2) Your Weight (step 3) Highest Reading (step 4) Reading at Constant Velocity (step 5) Lowest Reading (step 6)

4. Place the scale in the elevator. Step on the scale and record the mass at rest. Select the highest floor that the elevator goes up to. Once the elevator starts, during its upward acceleration, record the highest reading on the scale in the data table. 5. When the velocity of the elevator becomes constant, record the reading on the scale in the data table. 6. As the elevator starts to decelerate, watch for the lowest reading on the scale and record it in the data table.

Analyze 1. Explain In step 2, why did the mass appear to gain weight when being accelerated upward? Provide a mathematical equation to summarize this concept.

Conclude and Apply How can you develop an experiment to find the acceleration of an amusement park ride that either drops rapidly or climbs rapidly?

Going Further How can a bathroom scale measure both pounds mass (lbm) and pounds force (lbf) at the same time?

Real-World Physics Forces on pilots in high-performance jet airplanes are measured in g’s or g-force. What does it mean if a pilot is pulling 6 g’s in a power climb?

2. Explain Why did the mass appear to lose weight when being decelerated at the end of its movement during step 3? Provide a mathematical equation to summarize this concept.

Communicate You can visit physicspp.com/ internet_lab to post the acceleration of your

3. Measure in SI Most bathroom scales read in pounds mass (lbm). Convert your reading in step 4 in pounds mass to kilograms. (1 kg 2.21 lbm) (Note: skip this step if your scale measures in kilograms.)

elevator and compare it to other elevators around the country, maybe even the world. Post a description of your elevator’s ride so that a comparison of acceleration versus ride comfort can be evaluated.

4. Measure in SI Some bathroom scales read in pounds force (lbf). Convert all of the readings you made in steps 4–6 to newtons. (1 N 0.225 lbf) 5. Analyze Calculate the acceleration of the elevator at the beginning of your elevator trip using the equation Fscale ma mg. 6. Use Numbers What is the acceleration of the elevator at the end of your trip?

To find out more about forces and acceleration, visit the Web site: physicspp.com

109

Bathroom Scale The portable weighing scale was patented in 1896 by John H. Hunter. People used coin-operated scales, usually located in stores, to weigh themselves until the advent of the home bathroom scale in 1946. How does a bathroom scale work?

1 There are two long and two

short levers that are attached to each other. Brackets in the lid of the scale sit on top of the levers to help evenly distribute your weight on the levers.

3 As the calibrating plate is pushed down

by weight on the scale, the crank pivots. This, in turn, moves the rack and rotates the pinion. As a result, the dial on the scale rotates.

Dial Main spring

Crank Dial spring Pinion

Rack

Calibrating plate

Fg

Lever

Thinking Critically 4 When the spring

force, Fsp, from the main spring being stretched is equal to Fg, the crank, rack, and pinion no longer move, and your weight is shown on the dial.

110

How It Works

2 The long levers rest on top of

a calibrating plate that has the main spring attached to it. When you step on the scale, your weight, Fg, is exerted on the levers, which, in turn, exert a force on the calibrating plate and cause the main spring to stretch.

1. Hypothesize Most springs in bathroom scales cannot exert a force larger than 20 lbs (89 N). How is it possible that you don’t break the scale every time you step on it? (Hint: Think about exerting a large force near the pivot of a see-saw.) 2. Solve If the largest reading on most scales is 240 lbs (1068 N) and the spring can exert a maximum of 20 lbs (89 N), what ratio does the lever use?

4.1 Force and Motion Vocabulary

Key Concepts

• • • •

• • • •

force (p. 88) free-body diagram (p. 89) net force (p. 92) Newton’s second law (p. 93)

• Newton’s first law (p. 94) • inertia (p. 95) • equilibrium (p. 95)

• •

An object that experiences a push or a pull has a force exerted on it. Forces have both direction and magnitude. Forces may be divided into contact and field forces. In a free-body diagram, always draw the force vectors leading away from the object, even if the force is a push. The forces acting upon an object can be added using vector addition to find the net force. Newton’s second law states that the acceleration of a system equals the net force acting on it, divided by its mass. F m

et a n

• •

Newton’s first law states that an object that is at rest will remain at rest, and an object that is moving will continue to move in a straight line with constant speed, if and only if the net force acting on that object is zero. An object with no net force acting on it is in equilibrium.

4.2 Using Newton’s Laws Vocabulary

Key Concepts

• • • •

•

apparent weight (p. 98) weightlessness (p. 98) drag force (p. 100) terminal velocity (p. 101)

• • • •

The weight of an object depends upon the acceleration due to gravity and the mass of the object. An object’s apparent weight is the force an object experiences as a result of the contact forces acting on it, giving the object an acceleration. An object with no apparent weight experiences weightlessness. The effect of drag on an object’s motion is determined by the object’s weight, size, and shape. If a falling object reaches a velocity such that the drag force is equal to the object’s weight, it maintains that velocity, called the terminal velocity.

4.3 Interaction Forces Vocabulary

Key Concepts

• • • •

• •

interaction pair (p. 102) Newton’s third law (p. 102) tension (p. 105) normal force (p. 107)

All forces result from interactions between objects. Newton’s third law states that the two forces that make up an interaction pair of forces are equal in magnitude, but opposite in direction and act on different objects. FA on B FB on A

• • •

In an interaction pair, FA on B does not cause FB on A. The two forces either exist together or not at all. Tension is the specific name for the force exerted by a rope or string. The normal force is a support force resulting from the contact of two objects. It is always perpendicular to the plane of contact between the two objects.

physicspp.com/vocabulary_puzzlemaker

111

Concept Mapping 40. Complete the following concept map using the following term and symbols: normal, FT, Fg. force

tension

48. Ramon pushes on a bed that has been pushed against a wall, as in Figure 4-17. Draw a free-body diagram for the bed and identify all the forces acting on it. Make a separate list of all the forces that the bed applies to other objects. (4.3)

gravity

FN

Mastering Concepts 41. A physics book is motionless on the top of a table. If you give it a hard push with your hand, it slides across the table and slowly comes to a stop. Use Newton’s laws to answer the following questions. (4.1) a. Why does the book remain motionless before the force of your hand is applied? b. Why does the book begin to move when your hand pushes hard enough on it? c. Under what conditions would the book remain in motion at a constant speed?

■

Figure 4-17

49. Figure 4-18 shows a block in four different situations. Rank them according to the magnitude of the normal force between the block and the surface, greatest to least. Specifically indicate any ties. (4.3)

42. Cycling Why do you have to push harder on the pedals of a single-speed bicycle to start it moving than to keep it moving at a constant velocity? (4.1)

43. Suppose that the acceleration of an object is zero. Does this mean that there are no forces acting on it? Give an example supporting your answer. (4.2)

44. Basketball When a basketball player dribbles a ball, it falls to the floor and bounces up. Is a force required to make it bounce? Why? If a force is needed, what is the agent involved? (4.2)

45. Before a sky diver opens her parachute, she may be falling at a velocity higher than the terminal velocity that she will have after the parachute opens. (4.2) a. Describe what happens to her velocity as she opens the parachute. b. Describe the sky diver’s velocity from when her parachute has been open for a time until she is about to land.

46. If your textbook is in equilibrium, what can you say about the forces acting on it? (4.2)

47. A rock is dropped from a bridge into a valley. Earth pulls on the rock and accelerates it downward. According to Newton’s third law, the rock must also be pulling on Earth, yet Earth does not seem to accelerate. Explain. (4.3)

112

■

Figure 4-18

50. Explain why the tension in a massless rope is constant throughout it. (4.3)

51. A bird sits on top of a statue of Einstein. Draw free-body diagrams for the bird and the statue. Specifically indicate any interaction pairs between the two diagrams. (4.3)

52. Baseball A slugger swings his bat and hits a baseball pitched to him. Draw free-body diagrams for the baseball and the bat at the moment of contact. Specifically indicate any interaction pairs between the two diagrams. (4.3)

Applying Concepts 53. Whiplash If you are in a car that is struck from behind, you can receive a serious neck injury called whiplash. a. Using Newton’s laws, explain what happens to cause such an injury. b. How does a headrest reduce whiplash?

Chapter 4 Forces in One Dimension For more problems, go to Additional Problems, Appendix B.

54. Space Should astronauts choose pencils with hard or soft lead for making notes in space? Explain.

55. When you look at the label of the product in Figure 4-19 to get an idea of how much the box contains, does it tell you its mass, weight, or both? Would you need to make any changes to this label to make it correct for consumption on the Moon?

62. Breaking the Wishbone After Thanksgiving, Kevin and Gamal use the turkey’s wishbone to make a wish. If Kevin pulls on it with a force 0.17 N larger than the force Gamal pulls with in the opposite direction, and the wishbone has a mass of 13 g, what is the wishbone’s initial acceleration?

4.2 Using Newton’s Laws 63. What is your weight in newtons? 64. Motorcycle Your new motorcycle weighs 2450 N. What is its mass in kilograms?

65. Three objects are dropped simultaneously from the

■

Figure 4-19

56. From the top of a tall building, you drop two tabletennis balls, one filled with air and the other with water. Both experience air resistance as they fall. Which ball reaches terminal velocity first? Do both hit the ground at the same time?

57. It can be said that 1 kg equals 2.2 lb. What does this statement mean? What would be the proper way of making the comparison?

58. You toss a ball straight up into the air. a. Draw a free-body diagram for the ball at three points during its motion: on the way up, at the very top, and on the way down. Specifically identify the forces acting on the ball and their agents. b. What is the velocity of the ball at the very top of the motion? c. What is the acceleration of the ball at this same point?

Mastering Problems 4.1 Force and Motion 59. What is the net force acting on a 1.0-kg ball in free-fall?

top of a tall building: a shot put, an air-filled balloon, and a basketball. a. Rank the objects in the order in which they will reach terminal velocity, from first to last. b. Rank the objects according to the order in which they will reach the ground, from first to last. c. What is the relationship between your answers to parts a and b?

66. What is the weight in pounds of a 100.0-N wooden shipping case?

67. You place a 7.50-kg television on a spring scale. If the scale reads 78.4 N, what is the acceleration due to gravity at that location?

68. Drag Racing A 873-kg (1930-lb) dragster, starting from rest, attains a speed of 26.3 m/s (58.9 mph) in 0.59 s. a. Find the average acceleration of the dragster during this time interval. b. What is the magnitude of the average net force on the dragster during this time? c. Assume that the driver has a mass of 68 kg. What horizontal force does the seat exert on the driver?

69. Assume that a scale is in an elevator on Earth. What force would the scale exert on a 53-kg person standing on it during the following situations? a. The elevator moves up at a constant speed. b. It slows at 2.0 m/s2 while moving upward. c. It speeds up at 2.0 m/s2 while moving downward. d. It moves downward at a constant speed. e. It slows to a stop while moving downward with a constant acceleration.

70. A grocery sack can withstand a maximum of 230 N

60. Skating Joyce and Efua are skating. Joyce pushes Efua, whose mass is 40.0-kg, with a force of 5.0 N. What is Efua’s resulting acceleration?

61. A car of mass 2300 kg slows down at a rate of 3.0 m/s2 when approaching a stop sign. What is the magnitude of the net force causing it to slow down? physicspp.com/chapter_test

before it rips. Will a bag holding 15 kg of groceries that is lifted from the checkout counter at an acceleration of 7.0 m/s2 hold?

71. A 0.50-kg guinea pig is lifted up from the ground. What is the smallest force needed to lift it? Describe its resulting motion. Chapter 4 Assessment

113 Aaron Haupt

72. Astronomy On the surface of Mercury, the gravitational acceleration is 0.38 times its value on Earth. a. What would a 6.0-kg mass weigh on Mercury? b. If the gravitational acceleration on the surface of Pluto is 0.08 times that of Mercury, what would a 7.0-kg mass weigh on Pluto?

73. A 65-kg diver jumps off of a 10.0-m tower. a. Find the diver’s velocity when he hits the water. b. The diver comes to a stop 2.0 m below the surface. Find the net force exerted by the water.

74. Car Racing A race car has a mass of 710 kg. It starts from rest and travels 40.0 m in 3.0 s. The car is uniformly accelerated during the entire time. What net force is exerted on it?

4.3 Interaction Forces 75. A 6.0-kg block rests on top of a 7.0-kg block, which rests on a horizontal table. a. What is the force (magnitude and direction) exerted by the 7.0-kg block on the 6.0-kg block? b. What is the force (magnitude and direction) exerted by the 6.0-kg block on the 7.0-kg block?

Mixed Review 81. The dragster in problem 68 completed a 402.3-m (0.2500-mi) run in 4.936 s. If the car had a constant acceleration, what was its acceleration and final velocity?

82. Jet A 2.75106-N catapult jet plane is ready for takeoff. If the jet’s engines supply a constant thrust of 6.35106 N, how much runway will it need to reach its minimum takeoff speed of 285 km/h?

83. The dragster in problem 68 crossed the finish line going 126.6 m/s. Does the assumption of constant acceleration hold true? What other piece of evidence could you use to determine if the acceleration was constant?

84. Suppose a 65-kg boy and a 45-kg girl use a massless rope in a tug-of-war on an icy, resistance-free surface as in Figure 4-21. If the acceleration of the girl toward the boy is 3.0 m/s2, find the magnitude of the acceleration of the boy toward the girl.

76. Rain A raindrop, with mass 2.45 mg, falls to the ground. As it is falling, what magnitude of force does it exert on Earth?

77. A 90.0-kg man and a 55-kg man have a tug-of-war.

?

3.0 m/s2

The 90.0-kg man pulls on the rope such that the 55-kg man accelerates at 0.025 m/s2. What force does the rope exert on the 90.0-kg man? ■

78. Male lions and human sprinters can both accelerate at about 10.0 m/s2. If a typical lion weighs 170 kg and a typical sprinter weighs 75 kg, what is the difference in the force exerted on the ground during a race between these two species?

79. A 4500-kg helicopter accelerates upward at 2.0 m/s2. What lift force is exerted by the air on the propellers?

80. Three blocks are stacked on top of one another, as in Figure 4-20. The top block has a mass of 4.6 kg, the middle one has a mass of 1.2 kg, and the bottom one has a mass of 3.7 kg. Identify and calculate any normal forces between the objects.

Figure 4-21

85. Space Station Pratish weighs 588 N and is weightless in a space station. If she pushes off the wall with a vertical acceleration of 3.00 m/s2, determine the force exerted by the wall during her push off.

86. Baseball As a baseball is being caught, its speed goes from 30.0 m/s to 0.0 m/s in about 0.0050 s. The mass of the baseball is 0.145 kg. a. What is the baseball’s acceleration? b. What are the magnitude and direction of the force acting on it? c. What are the magnitude and direction of the force acting on the player who caught it?

87. Air Hockey An air-hockey table works by pumping

■

114

Figure 4-20

air through thousands of tiny holes in a table to support light pucks. This allows the pucks to move around on cushions of air with very little resistance. One of these pucks has a mass of 0.25 kg and is pushed along by a 12.0-N force for 9.0 s. a. What is the puck’s acceleration? b. What is the puck’s final velocity?

Chapter 4 Forces in One Dimension For more problems, go to Additional Problems, Appendix B.

88. A student stands on a bathroom scale in an elevator at rest on the 64th floor of a building. The scale reads 836 N. a. As the elevator moves up, the scale reading increases to 936 N. Find the acceleration of the elevator. b. As the elevator approaches the 74th floor, the scale reading drops to 782 N. What is the acceleration of the elevator? c. Using your results from parts a and b, explain which change in velocity, starting or stopping, takes the longer time.

■

Figure 4-23

89. Weather Balloon The instruments attached to a weather balloon in Figure 4-22 have a mass of 5.0 kg. The balloon is released and exerts an upward force of 98 N on the instruments. a. What is the acceleration of the balloon and instruments? b. After the balloon has accelerated for 10.0 s, the instruments are released. What is the velocity of the instruments at the moment of their release? c. What net force acts on the instruments after their release? d. When does the direction of the instruments’ velocity first become downward?

92. Two blocks, one of mass 5.0 kg and the other of mass 3.0 kg, are tied together with a massless rope as in Figure 4-24. This rope is strung over a massless, resistance-free pulley. The blocks are released from rest. Find the following. a. the tension in the rope b. the acceleration of the blocks Hint: you will need to solve two simultaneous equations.

3.0 kg 98 N

5.0 kg ■

Figure 4-24

5.0 kg ■

Figure 4-22

90. When a horizontal force of 4.5 N acts on a block on a resistance-free surface, it produces an acceleration of 2.5 m/s2. Suppose a second 4.0-kg block is dropped onto the first. What is the magnitude of the acceleration of the combination if the same force continues to act? Assume that the second block does not slide on the first block.

91. Two blocks, masses 4.3 kg and 5.4 kg, are pushed across a frictionless surface by a horizontal force of 22.5 N, as shown in Figure 4-23. a. What is the acceleration of the blocks? b. What is the force of the 4.3-kg block on the 5.4-kg block? c. What is the force of the 5.4-kg block on the 4.3-kg block? physicspp.com/chapter_test

Thinking Critically 93. Formulate Models A 2.0-kg mass, mA, and a 3.0-kg

mass, mB, are connected to a lightweight cord that passes over a frictionless pulley. The pulley only changes the direction of the force exerted by the rope. The hanging masses are free to move. Choose coordinate systems for the two masses with the positive direction being up for mA and down for mB. a. Create a pictorial model. b. Create a physical model with motion and freebody diagrams. c. What is the acceleration of the smaller mass?

94. Use Models Suppose that the masses in problem 93 are now 1.00 kg and 4.00 kg. Find the acceleration of the larger mass. Chapter 4 Assessment

115

95. Infer The force exerted on a 0.145-kg baseball by a bat changes from 0.0 N to 1.0104 N in 0.0010 s, then drops back to zero in the same amount of time. The baseball was going toward the bat at 25 m/s. a. Draw a graph of force versus time. What is the average force exerted on the ball by the bat? b. What is the acceleration of the ball? c. What is the final velocity of the ball, assuming that it reverses direction?

96. Observe and Infer Three blocks that are connected by massless strings are pulled along a frictionless surface by a horizontal force, as shown in Figure 4-25. a. What is the acceleration of each block? b. What are the tension forces in each of the strings? Hint: Draw a separate free-body diagram for each block.

2.0 kg m1 ■

4.0 kg m2

FT2

write a one-page summary. Do you think his three laws of motion were his greatest accomplishments? Explain why or why not.

101. Review, analyze, and critique Newton’s first law. Can we prove this law? Explain. Be sure to consider the role of resistance.

102. Physicists classify all forces into four fundamental categories: gravitational, electromagnetic, strong nuclear, and weak nuclear. Investigate these four forces and describe the situations in which they are found.

Cumulative Review

6.0 kg m3

100. Research Newton’s contributions to physics and

F 36.0 N

Figure 4-25

97. Critique Using the Example Problems in this chapter as models, write a solution to the following problem. A block of mass 3.46 kg is suspended from two vertical ropes attached to the ceiling. What is the tension in each rope?

98. Think Critically Because of your physics knowledge, you are serving as a scientific consultant for a new science-fiction TV series about space exploration. In episode 3, the heroine, Misty Moonglow, has been asked to be the first person to ride in a new interplanetary transport for use in our solar system. She wants to be sure that the transport actually takes her to the planet she is supposed to be going to, so she needs to take a testing device along with her to measure the force of gravity when she arrives. The script writers don’t want her to just drop an object, because it will be hard to depict different accelerations of falling objects on TV. They think they’d like something involving a scale. It is your job to design a quick experiment Misty can conduct involving a scale to determine which planet in our solar system she has arrived on. Describe the experiment and include what the results would be for Pluto (g 0.30 m/s2 ), which is where she is supposed to go, and Mercury (g 3.70 m/s2), which is where she actually ends up.

103. Cross-Country Skiing Your friend is training for a cross-country skiing race, and you and some other friends have agreed to provide him with food and water along his training route. It is a bitterly cold day, so none of you wants to wait outside longer than you have to. Taro, whose house is the stop before yours, calls you at 8:25 A.M. to tell you that the skier just passed his house and is planning to move at an average speed of 8.0 km/h. If it is 5.2 km from Taro’s house to yours, when should you expect the skier to pass your house? (Chapter 2)

104. Figure 4-26 is a position-time graph of the motion of two cars on a road. (Chapter 3) a. At what time(s) does one car pass the other? b. Which car is moving faster at 7.0 s? c. At what time(s) do the cars have the same velocity? d. Over what time interval is car B speeding up all the time? e. Over what time interval is car B slowing down all the time?

Distance (m)

FT1

Writing in Physics

Position of Two Cars

12

B 6 A 0

2

4

116

8

Time (s) ■

99. Apply Concepts Develop a CBL lab, using a motion detector, that graphs the distance a freefalling object moves over equal intervals of time. Also graph velocity versus time. Compare and contrast your graphs. Using your velocity graph, determine the acceleration. Does it equal g?

6

Figure 4-26

105. Refer to Figure 4-26 to find the instantaneous speed for the following: (Chapter 3) a. car B at 2.0 s b. car B at 9.0 s c. car A at 2.0 s

Chapter 4 Forces in One Dimension For more problems, go to Additional Problems, Appendix B.

Multiple Choice 1. What is the acceleration of the car described by the graph below? 1.0 m/s2

0.40 m/s2

2.5 m/s2

Velocity (m/s)

0.20 m/s2

16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0

6. A 45-kg child sits on a 3.2-kg tire swing. What is the tension in the rope that hangs from a tree branch? 4.5102 N 4.7102 N

310 N 4.4102 N

7. The tree branch in problem 6 sags and the child’s feet rest on the ground. If the tension in the rope is reduced to 220 N, what is the value of the normal force being exerted on the child’s feet? 2.2102 N 2.5102 N 2.0

4.0

6.0

Time (s)

4.3102 N 6.9102 N

8. According the graph below, what is the force being exerted on the 16-kg cart? 4N 8N

2. What distance will the car described by the above graph have traveled after 4.0 s? 80 m

40 m

90 m

3. If the car in the above graph maintains a constant acceleration, what will its velocity be after 10 s? 10 km/h

90 km/h

25 km/h

120 km/h

4. In a tug-of-war, 13 children, with an average mass of 30 kg each, pull westward on a rope with an average force of 150 N per child. Five parents, with an average mass of 60 kg each, pull eastward on the other end of the rope with an average force of 475 N per adult. Assuming that the whole mass accelerates together as a single entity, what is the acceleration of the system? 0.62 m/s2 E

3.4 m/s2 E

2.8 m/s2 W

6.3 m/s2 W

5. What is the weight of a 225-kg space probe on the Moon? The acceleration of gravity on the Moon is 1.62 m/s2. 139 N

1.35103 N

364 N

2.21103 N physicspp.com/standardized_test

8.0 Velocity (m/s)

13 m

16 N 32 N

6.0 4.0 2.0 0.0

1.0 2.0 3.0 4.0 Time (s)

Extended Answer 9. Draw a free-body diagram of a dog sitting on a scale in an elevator. Using words and mathematical formulas, describe what happens to the apparent weight of the dog when: the elevator accelerates upward, the elevator travels at a constant speed downward, and the elevator falls freely downward.

Maximize Your Score If possible, find out how your standardized test will be scored. In order to do your best, you need to know if there is a penalty for guessing, and if so, what the penalty is. If there is no random-guessing penalty at all, you should always fill in an answer, even if you have not read the question.

Chapter 4 Standardized Test Practice

117

What You’ll Learn • You will represent vector quantities both graphically and algebraically. • You will use Newton’s laws to analyze motion when friction is involved. • You will use Newton’s laws and your knowledge of vectors to analyze motion in two dimensions.

Why It’s Important Most objects experience forces in more than one dimension. A car being towed, for example, experiences upward and forward forces from the tow truck and the downward force of gravity. Rock Climbing How do rock climbers keep from falling? This climber has more than one support point, and there are multiple forces acting on her in multiple directions.

Think About This A rock climber approaches a portion of the rock face that forces her to hang with her back to the ground. How will she use her equipment to apply the laws of physics in her favor and overcome this obstacle?

physicspp.com 118 CORBIS

Can 2 N 2 N 2 N? Question Under what conditions can two different forces equal one other force? Procedure

Analysis

1. Measure Use a spring scale to measure and record the weight of a 200-g object. 2. Obtain another spring scale, and attach one end of a 35-cm-long piece of string to the hooks on the bottom of each spring scale. 3. Tie one end of a 15-cm-long piece of string to the 200-g object. Loop the other end over the 35-cm-long piece of string and tie the end to the 200-g object. CAUTION: Avoid falling masses. 4. Hold the spring scales parallel to each other so that the string between them forms a 120° angle. Move the string with the hanging object until both scales have the same reading. Record the readings on each scale. 5. Collect and Organize Data Slowly pull the string more and more horizontal while it is still supporting the 200-g object. Describe your observations.

Does the sum of the forces measured by the two spring scales equal the weight of the hanging object? Is the sum greater than the weight? Less than the weight? Critical Thinking Draw an equilateral triangle, with one side vertical, on a sheet of paper. If the two sides of the triangle are 2.0 N, explain the size of the third side. How is it possible that 2 N 2 N 2 N?

5.1 Vectors

H

ow do rock climbers keep from falling in situations like the one shown on the preceding page? Notice that the climber has more than one support point and that there are multiple forces acting on her. She tightly grips crevices in the rock and has her feet planted on the rock face, so there are two contact forces acting on her. Gravity is pulling on her as well, so there are three total forces acting on the climber. One aspect of this situation that is different from the ones that you have studied in earlier chapters is that the forces exerted by the rock face on the climber are not horizontal or vertical forces. You know from previous chapters that you can pick your coordinate system and orient it in the way that is most useful to analyzing the situation. But what happens when the forces are not at right angles to each other? How can you set up a coordinate system and find for a net force when you are dealing with more than one dimension?

Objectives • Evaluate the sum of two or more vectors in two dimensions graphically. • Determine the components of vectors. • Solve for the sum of two or more vectors algebraically by adding the components of the vectors.

Vocabulary components vector resolution

Section 5.1 Vectors

119

Horizons Companies

Vectors Revisited 40 N 40 N

80 N ■ Figure 5-1 The sum of the two 40-N forces is shown by the resultant vector below them.

■ Figure 5-2 Add vectors by placing them tip-to-tail and drawing the resultant from the tail of the first vector to the tip of the last vector.

a

Consider an example with force vectors. Recall the case in Chapter 4 in which you and a friend both pushed on a table together. Suppose that you each exerted 40 N of force to the right. Figure 5-1 represents these vectors in a free-body diagram with the resultant vector, the net force, shown below it. The net force vector is 80 N, which is what you probably expected. But how was this net force vector obtained?

Vectors in Multiple Dimensions The process for adding vectors works even when the vectors do not point along the same straight line. If you are solving one of these two-dimensional problems graphically, you will need to use a protractor, both to draw the vectors at the correct angles and also to measure the direction and magnitude of the resultant vector. You can add vectors by placing them tip-to-tail and then drawing the resultant of the vector by connecting the tail of the first vector to the tip of the second vector, as shown in Figure 5-2. Figure 5-2a shows the two forces in the free-body diagram. In Figure 5-2b, one of the vectors has been moved so that its tail is at the same place as the tip of the other vector. Notice that its length and direction have not changed. Because the length and direction are the only important characteristics of the vector, the vector is unchanged by this movement. This is always true: if you move a vector so that its length and direction are unchanged, the vector is unchanged. Now, as in Figure 5-2c, you can draw the resultant vector pointing from the tail of the first vector to the tip of the last vector and measure it to obtain its magnitude. Use a protractor to measure the direction of the resultant vector. Sometimes you will need to use trigonometry to determine the length or direction of resultant vectors. Remember that the length of the hypotenuse of a right triangle can be found by using the Pythagorean theorem. If you were adding together two vectors at right angles, vector A pointing north and vector B pointing east, you could use the Pythagorean theorem to find the magnitude of the resultant, R. Pythagorean Theorem R2 A2 B2 If vector A is at a right angle to vector B, then the sum of the squares of the magnitudes is equal to the square of the magnitude of the resultant vector.

b

If the two vectors to be added are at an angle other than 90°, then you can use the law of cosines or the law of sines. Law of Cosines R2 A2 B2 2AB cos The square of the magnitude of the resultant vector is equal to the sum of the magnitudes of the squares of the two vectors, minus two times the product of the magnitudes of the vectors, multiplied by the cosine of the angle between them.

c

R A B sin sin a sin b The magnitude of the resultant, divided by the sine of the angle between two vectors, is equal to the magnitude of one of the vectors divided by the angle between that component vector and the resultant vector.

Law of Sines

120

Chapter 5 Forces in Two Dimensions

Finding the Magnitude of the Sum of Two Vectors Find the magnitude of the sum of a 15-km displacement and a 25-km displacement when the angle between them is 90° and when the angle between them is 135°. 1

Analyze and Sketch the Problem • Sketch the two displacement vectors, A and B, and the angle between them. Known: A 25 km 1 90° B 15 km 2 135°

2

Unknown:

B 2 A

R?

R

Solve for the Unknown When the angle is 90°, use the Pythagorean theorem to find the magnitude of the resultant vector. R 2 A2 B 2 R A2 B2 (25 km )2 (1 5 km)2 Substitute A 25 km, B 15 km 29 km

Math Handbook Square and Cube Roots pages 839–840

When the angle does not equal 90°, use the law of cosines to find the magnitude of the resultant vector. R 2 A2 B 2 2AB(cos 2) 2 B 2 2AB(cos R A 2)

(25 km )2 (1 5 km)2 2(25 km)( 15 km )(cos 135°) 37 km

3

Substitute A 25 km, B 15 km, 2 135°

Evaluate the Answer • Are the units correct? Each answer is a length measured in kilometers. • Do the signs make sense? The sums are positive. • Are the magnitudes realistic? The magnitudes are in the same range as the two combined vectors, but longer. This is because each resultant is the side opposite an obtuse angle. The second answer is larger than the first, which agrees with the graphical representation.

1. A car is driven 125.0 km due west, then 65.0 km due south. What is the magnitude of its displacement? Solve this problem both graphically and mathematically, and check your answers against each other. 2. Two shoppers walk from the door of the mall to their car, which is 250.0 m down a lane of cars, and then turn 90° to the right and walk an additional 60.0 m. What is the magnitude of the displacement of the shoppers’ car from the mall door? Solve this problem both graphically and mathematically, and check your answers against each other. 3. A hiker walks 4.5 km in one direction, then makes a 45° turn to the right and walks another 6.4 km. What is the magnitude of her displacement? 4. An ant is crawling on the sidewalk. At one moment, it is moving south a distance of 5.0 mm. It then turns southwest and crawls 4.0 mm. What is the magnitude of the ant’s displacement?

Section 5.1 Vectors

121

Components of Vectors a

Choosing a coordinate system, such as the one in Figure 5-3a, is similar to laying a grid drawn on a sheet of transparent plastic on top of a vector problem. You have to choose where to put the center of the grid (the origin) and establish the directions in which the axes point. Notice that in the coordinate system shown in Figure 5-3a, the x-axis is drawn through the origin with an arrow pointing in the positive direction. The positive y-axis is located 90° counterclockwise from the positive x-axis and crosses the x-axis at the origin. How do you choose the direction of the x-axis? There is never a single correct answer, but some choices make the problem easier to solve than others. When the motion you are describing is confined to the surface of Earth, it is often convenient to have the x-axis point east and the y-axis point north. When the motion involves an object moving through the air, the positive x-axis is often chosen to be horizontal and the positive y-axis vertical (upward). If the motion is on a hill, it’s convenient to place the positive x-axis in the direction of the motion and the y-axis perpendicular to the x-axis.

y

x

Origin

b y

A Ay

Component vectors Defining a coordinate system allows you to describe a vector in a different way. Vector A shown in Figure 5-3b, for example, could be described as going 5 units in the positive x-direction and 4 units in the positive y-direction. You can represent this information in the form of two vectors like the ones labeled Ax and Ay in the diagram. Notice that Ax is parallel to the x-axis, and Ay is parallel to the y-axis. Further, you can see that if you add Ax and Ay, the resultant is the original vector, A. A vector can be broken into its components, which are a vector parallel to the x-axis and another parallel to the y-axis. This can always be done and the following vector equation is always true.

x

Ax

■ Figure 5-3 A coordinate system has an origin and two perpendicular axes (a). The direction of a vector, A, is measured counterclockwise from the x-axis (b).

A Ax Ay

■ Figure 5-4 The sign of a component depends upon which of the four quadrants the component is in.

Second quadrant Ax 0 Ay 0

y

First quadrant Ax 0 Ay 0

This process of breaking a vector into its components is sometimes called vector resolution. Notice that the original vector is the hypotenuse of a right triangle. This means that the magnitude of the original vector will always be larger than the magnitudes of either component vector. Another reason for choosing a coordinate system is that the direction of any vector can be specified relative to those coordinates. The direction of a vector is defined as the angle that the vector makes with the x-axis, measured counterclockwise. In Figure 5-3b, the angle, , tells the direction of the vector, A. All algebraic calculations involve only the positive components of vectors, not the vectors themselves. In addition to measuring the lengths of the component vectors graphically, you can find the components by using trigonometry. The components are calculated using the equations below, where the angle, , is measured counterclockwise from the positive x-axis.

x

Ax 0 Ay 0 Third quadrant

122

Ax 0 Ay 0 Fourth quadrant

adjacent side hypotenuse

A A

opposite side hypotenuse

Ay

cos x ; therefore, Ax A cos sin ; therefore, Ay A sin A

When the angle that a vector makes with the x-axis is larger than 90°, the sign of one or more components is negative, as shown in Figure 5-4.

Chapter 5 Forces in Two Dimensions

Algebraic Addition of Vectors

y

y

You might be wondering why you need to resolve vectors into their components. The answer C Cy is that doing this often makes adding vectors B By together much easier mathematically. Two or more vectors (A, B, C, etc.) may be added by first resolvA Ay ing each vector into its x- and y-components. The x-components are added to form the x-component Ax Cx Bx of the resultant: Rx Ax Bx Cx. Similarly, the y-components are added to form the y-component of the resultant: Ry Ay By Cy. This process is illustrated graphically in Figure 5-5. Because Rx and Ry are at a right angle (90°), the magnitude of the resultant vector can be calculated using the Pythagorean theorem, R2 Rx2 Ry2. To find the angle or direction of the resultant, recall that the tangent of the angle that the vector makes with the x-axis is given by the following.

R

x

Ry

x

Rx

■ Figure 5-5 Rx is the sum of the x-components of A, B, and C. Ry is the sum of the y-components. The vector sum of Rx and Ry is the vector sum of A, B, and C.

( RR )

Angle of the Resultant Vector tan1 y

x

The angle of the resultant vector is equal to the inverse tangent of the quotient of the y-component divided by the x-component of the resultant vector.

You can find the angle by using the tan1 key on your calculator. Note that when tan 0, most calculators give the angle between 0° and 90°, and when tan 0, the angle is reported to be between 0° and 90°.

Vector Addition Use the following technique to solve problems for which you need to add or subtract vectors.

Math Review

1. Choose a coordinate system.

sin

2. Resolve the vectors into their x-components using Ax A cos , and their y-components using Ay A sin , where is the angle measured counterclockwise from the positive x-axis. 3. Add or subtract the component vectors in the x-direction. 4. Add or subtract the component vectors in the y-direction. 5. Use the Pythagorean theorem, R Rx2 Ry2, to find the magnitude of the resultant vector.

y

opposite side hypotenuse Ry R

adjacent side hypotenuse Rx R

cos

R

Ry

x y

R Rx

opposite side adjacent side Ry Rx

tan

x

y

R Ry Rx

x

(R )

6. Use tan1 to find the angle of the resultant Rx vector. y

Section 5.1 Vectors

123

Finding Your Way Home A GPS receiver indicates that your home is 15.0 km and 40.0° north of west, but the only path through the woods leads directly north. If you follow the path 5.0 km before it opens into a field, how far, and in what direction, would you have to walk to reach your home? 1

Analyze and Sketch the Problem

y

• Draw the resultant vector, R, from your original location to your home. • Draw A, the known vector, and draw B, the unknown vector.

2

Known:

Unknown:

A 5.0 km, due north R 15.0 km, 40.0° north of west 140.0°

B?

B A

R 40.0°

x

Solve for the Unknown Find the components of R. Rx R cos (15.0 km) cos 140.0° Substitute R 15.0 km, 140.0° 11.5 km Ry R sin (15.0 km) sin 140.0° Substitute R 15.0 km, 140.0° 9.64 km Because A is due north, Ax 0.0 km and Ay 5.0 km.

180.0° 40.0° 140.0°

Math Handbook Inverses of Sine, Cosine, and Tangent page 856

Use the components of R and A to find the components of B. Bx Rx Ax 11.5 km 0.0 km Substitute Rx 11.5 km, Ax 0.0 km 11.5 km The negative sign means that this component points west. By Ry Ay 9.64 km 5.0 km Substitute Ry 9.64 km, Ay 5.0 km 4.6 km This component points north. Use the components of vector B to find the magnitude of vector B. B Bx2 By2 2 (4 (11.5 km) .6 km)2

Substitute By 4.6 km, Bx 11.5 km

12 km Use the tangent to find the direction of vector B. By

tan1 Bx

tan1

4.6 km 11.5 km

22° or 158°

Substitute By 4.6 km, Bx 11.5 km Tangent of an angle is negative in quadrants II and IV, so two answers are possible.

Locate the tail of vector B at the origin of a coordinate system and draw the components Bx and By . The direction is in the third quadrant, at 158°, or 22° north of west. Thus, B 12 km at 22° north of west. 3

Evaluate the Answer • Are the units correct? Kilometers and degrees are correct. • Do the signs make sense? They agree with the diagram. • Is the magnitude realistic? The length of B should be longer than Rx because the angle between A and B is greater than 90°.

124

Chapter 5 Forces in Two Dimensions

Solve problems 5–10 algebraically. You may also choose to solve some of them graphically to check your answers.

5. Sudhir walks 0.40 km in a direction 60.0° west of north, then goes 0.50 km due west. What is his displacement? 6. Afua and Chrissy are going to sleep overnight in their tree house and are using some ropes to pull up a box containing their pillows and blankets, which have a total mass of 3.20 kg. The girls stand on different branches, as shown in Figure 5-6, and pull at the angles and with the forces indicated. Find the x- and y-components of the net force on the box. Hint: Draw a free-body diagram so that you do not leave out a force. 7. You first walk 8.0 km north from home, then walk east until your displacement from home is 10.0 km. How far east did you walk? 8. A child’s swing is held up by two ropes tied to a tree branch that hangs 13.0° from the vertical. If the tension in each rope is 2.28 N, what is the combined force (magnitude and direction) of the two ropes on the swing? 9. Could a vector ever be shorter than one of its components? Equal in length to one of its components? Explain.

17.7 N

20.4 N

55.0° 120.0°

■ Figure 5-6 (Not to scale)

10. In a coordinate system in which the x-axis is east, for what range of angles is the x-component positive? For what range is it negative? You will use these techniques to resolve vectors into their components throughout your study of physics. You will get more practice at it, particularly in the rest of this chapter and the next. Resolving vectors into components allows you to analyze complex systems of vectors without using graphical methods.

5.1 Section Review 11. Distance v. Displacement Is the distance that you walk equal to the magnitude of your displacement? Give an example that supports your conclusion. 12. Vector Difference Subtract vector K from vector L, shown in Figure 5-7. 5.0 4.0

K

■

Figure 5-7

L

M

37.0°

6.0

13. Components Find the components of vector M, shown in Figure 5-7. physicspp.com/self_check_quiz

14. Vector Sum Find the sum of the three vectors shown in Figure 5-7. 15. Commutative Operations The order in which vectors are added does not matter. Mathematicians say that vector addition is commutative. Which ordinary arithmetic operations are commutative? Which are not? 16. Critical Thinking A box is moved through one displacement and then through a second displacement. The magnitudes of the two displacements are unequal. Could the displacements have directions such that the resultant displacement is zero? Suppose the box was moved through three displacements of unequal magnitude. Could the resultant displacement be zero? Support your conclusion with a diagram. Section 5.1 Vectors

125

5.2 Friction

Objectives • Define the friction force. • Distinguish between static and kinetic friction.

Vocabulary kinetic friction static friction coefficient of kinetic friction coefficient of static friction

P

ush your hand across your desktop and feel the force called friction opposing the motion. Push your book across the desk. When you stop pushing, the book will continue moving for a little while, then it will slow down and stop. The frictional force acting on the book gave it an acceleration in the direction opposite to the one in which it was moving. So far, you have neglected friction in solving problems, but friction is all around you. You need it to both start and stop a bicycle and a car. If you have ever walked on ice, you understand the importance of friction.

Static and Kinetic Friction There are two types of friction. Both always oppose motion. When you pushed your book across the desk, it experienced a type of friction that acts on moving objects. This force is known as kinetic friction, and it is exerted on one surface by another when the two surfaces rub against each other because one or both of them are moving. To understand the other kind of friction, imagine trying to push a heavy couch across the floor. You give it a push, but it does not move. Because it does not move, Newton’s laws tell you that there must be a second horizontal force acting on the couch, one that opposes your force and is equal in size. This force is static friction, which is the force exerted on one surface by another when there is no motion between the two surfaces. You might push harder and harder, as shown in Figures 5-8a and 5-8b, but if the couch still does not move, the force of friction must be getting larger. This is because the static friction force acts in response to other forces. Finally, when you push hard enough, as shown in Figure 5-8c, the couch will begin to move. Evidently, there is a limit to how large the static friction force can be. Once your force is greater than this maximum static friction, the couch begins moving and kinetic friction begins to act on it instead of static friction.

■ Figure 5-8 There is a limit to the ability of the static friction force to match the applied force.

a

126

A model for friction forces On what does a frictional force depend? The materials that the surfaces are made of play a role. For example, there is more friction between skis and concrete than there is between skis and snow. It may seem reasonable to think that the force of friction also might depend on either the surface area in contact or the speed of the motion, but experiments have shown that this is not true. The normal force between the two objects does matter, however. The harder one object is pushed against the other, the greater the force of friction that results. b

Chapter 5 Forces in Two Dimensions

c

If you pull a block along a surface at a constant velocity, according to Newton’s laws, the frictional force must be equal and opposite to the force with which you pull. You can pull a block of known mass along a table at a constant velocity and use a spring scale, as shown in Figure 5-9, to measure the force that you exert. You can then stack additional blocks on the block to increase the normal force and repeat the measurement. Plotting the data will yield a graph like the one in Figure 5-10. There is a direct proportion between the kinetic friction force and the normal force. The different lines correspond to dragging the block along different surfaces. Note that the line corresponding to the sandpaper surface has a steeper slope than the line for the highly polished table. You would expect it to be much harder to pull the block along sandpaper than along a polished table, so the slope must be related to the magnitude of the resulting frictional force. The slope of this line, designated k, is called the coefficient of kinetic friction between the two surfaces and relates the frictional force to the normal force, as shown below.

■ Figure 5-9 The spring scale pulls the block with a constant force.

Kinetic Friction Force Ff, kinetic kFN The kinetic friction force is equal to the product of the coefficient of the kinetic friction and the normal force.

The maximum static friction force is related to the normal force in a similar way as the kinetic friction force. Remember that the static friction force acts in response to a force trying to cause a stationary object to start moving. If there is no such force acting on an object, the static friction force is zero. If there is a force trying to cause motion, the static friction force will increase up to a maximum value before it is overcome and motion starts. Static Friction Force Ff, static sFN The static friction force is less than or equal to the product of the coefficient of the static friction and the normal force.

In the equation for the maximum static friction force, s is the coefficient of static friction between the two surfaces, and sFN is the maximum static friction force that must be overcome before motion can begin. In Figure 5-8c, the static friction force is balanced the instant before the couch begins to move.

er

le

pa p

ab

Ro

Sa nd

Kinetic frictional force

Kinetic Frictional Forces v. Normal Force

u

hly

Hig

t gh

hed olis

le

tab

p

Normal force

■

Figure 5-10 There is a linear relationship between the frictional force and the normal force. Section 5.2 Friction

127

Balanced Friction Forces You push a 25.0-kg wooden box across a wooden floor at a constant speed of 1.0 m/s. How much force do you exert on the box? 1

Analyze and Sketch the Problem

Fp

• Identify the forces and establish a coordinate system. • Draw a motion diagram indicating constant v and a 0. • Draw the free-body diagram.

2

Known:

Unknown:

m 25.0 kg v 1.0 m/s a 0.0 m/s2 k 0.20 (Table 5-1)

Fp ?

Ff FN Begin y

v a 0 m/s2

Ff Fnet 0 N F g

x

Solve for the Unknown The normal force is in the y-direction, and there is no acceleration. Math Handbook FN Fg Operations with mg Significant Digits pages 835–836 (25.0 kg)(9.80m/s2) Substitute m 25.0 kg, g 9.80 m/s2 245 N The pushing force is in the x-direction; v is constant, thus there is no acceleration. Fp kmg (0.20)(25.0 kg)(9.80 m/s2) Substitute k 0.20, m 25.0 kg, g 9.80 m/s2 49 N

3

Evaluate the Answer • Are the units correct? Performing dimensional analysis on the units verifies that force is measured in kgm/s2 or N. • Does the sign make sense? The positive sign agrees with the sketch. • Is the magnitude realistic? The force is reasonable for moving a 25.0-kg box.

17. A girl exerts a 36-N horizontal force as she pulls a 52-N sled across a cement sidewalk at constant speed. What is the coefficient of kinetic friction between the sidewalk and the metal sled runners? Ignore air resistance. 18. You need to move a 105-kg sofa to a different location in the room. It takes a force of 102 N to start it moving. What is the coefficient of static friction between the sofa and the carpet? 19. Mr. Ames is dragging a box full of books from his office to his car. The box and books together have a combined weight of 134 N. If the coefficient of static friction between the pavement and the box is 0.55, how hard must Mr. Ames push the box in order to start it moving? 20. Suppose that the sled in problem 17 is resting on packed snow. The coefficient of kinetic friction is now only 0.12. If a person weighing 650 N sits on the sled, what force is needed to pull the sled across the snow at constant speed? 21. Suppose that a particular machine in a factory has two steel pieces that must rub against each other at a constant speed. Before either piece of steel has been treated to reduce friction, the force necessary to get them to perform properly is 5.8 N. After the pieces have been treated with oil, what will be the required force?

128

Chapter 5 Forces in Two Dimensions

Fp

Table 5-1 Typical Coefficients of Friction Surface

s

k

Rubber on dry concrete Rubber on wet concrete Wood on wood Steel on steel (dry) Steel on steel (with oil)

0.80 0.60 0.50 0.78 0.15

0.65 0.40 0.20 0.58 0.06

Note that the equations for the kinetic and maximum static friction forces involve only the magnitudes of the forces. The forces themselves, Ff and FN, are at right angles to each other. Table 5-1 shows coefficients of friction between various surfaces. Although all the listed coefficients are less than 1.0, this does not mean that they must always be less than 1.0. For example, coefficients as large as 5.0 are experienced in drag racing.

Unbalanced Friction Forces If the force that you exert on the 25.0-kg box in Example Problem 3 is doubled, what is the resulting acceleration of the box? 1

Analyze and Sketch the Problem • Draw a motion diagram showing v and a. • Draw the free-body diagram with a doubled Fp. Known: m 25.0 kg k 0.20 v 1.0 m/s

2

Ff

Begin

Unknown:

FN

y

v a

a?

Fnet

Fp 2(49 N) 98 N

Fp

Fg x

Solve for the Unknown The normal force is in the y-direction, and there is no acceleration. FN Fg Substitute Fg mg mg In the x-direction there is an acceleration. So the forces must be unequal. Fnet Fp Ff ma Fp Ff Substitute Fnet ma Fp Ff

a m

Math Handbook

Find Ff and substitute it into the expression for a. Ff kFN kmg Substitute FN mg Fp kmg

a

Isolating a Variable page 845

Substitute Ff k mg

m

98 N (0.20)(25.0 kg)(9.80 25.0 kg

m/s2)

Substitute Fp 98 N, m 25.0 kg, k 0.20, g 9.80 m/s2

2.0 m/s2 3

Evaluate the Answer • Are the units correct? a is measured in m/s2. • Does the sign make sense? In this coordinate system, the sign should be positive. • Is the magnitude realistic? If the force were cut in half, a would be zero.

Section 5.2 Friction

129

22. A 1.4-kg block slides across a rough surface such that it slows down with an acceleration of 1.25 m/s2. What is the coefficient of kinetic friction between the block and the surface? 23. You help your mom move a 41-kg bookcase to a different place in the living room. If you push with a force of 65 N and the bookcase accelerates at 0.12 m/s2, what is the coefficient of kinetic friction between the bookcase and the carpet? Causes of Friction All surfaces, even those that appear to be smooth, are rough at a microscopic level. If you look at a photograph of a graphite crystal magnified by a scanning tunneling microscope, the atomic level surface irregularities of the crystal are revealed. When two surfaces touch, the high points on each are in contact and temporarily bond. This is the origin of both static and kinetic friction. The details of this process are still unknown and are the subject of research in both physics and engineering.

24. A shuffleboard disk is accelerated to a speed of 5.8 m/s and released. If the coefficient of kinetic friction between the disk and the concrete court is 0.31, how far does the disk go before it comes to a stop? The courts are 15.8 m long. 25. Consider the force pushing the box in Example Problem 4. How long would it take for the velocity of the box to double to 2.0 m/s? 26. Ke Min is driving along on a rainy night at 23 m/s when he sees a tree branch lying across the road and slams on the brakes when the branch is 60.0 m in front of him. If the coefficient of kinetic friction between the car’s locked tires and the road is 0.41, will the car stop before hitting the branch? The car has a mass of 2400 kg.

Here are a few important things to remember when dealing with frictional situations. First, friction always acts in a direction opposite to the motion (or in the case of static friction, intended motion). Second, the magnitude of the force of friction depends on the magnitude of the normal force between the two rubbing surfaces; it does not necessarily depend on the weight of either object. Finally, multiplying the coefficient of static friction and the normal force gives you the maximum static friction force. Keep these things in mind as you review this section.

5.2 Section Review 27. Friction In this section, you learned about static and kinetic friction. How are these two types of friction similar? What are the differences between static and kinetic friction? 28. Friction At a wedding reception, you notice a small boy who looks like his mass is about 25 kg running part way across the dance floor, then sliding on his knees until he stops. If the kinetic coefficient of friction between the boy’s pants and the floor is 0.15, what is the frictional force acting on him as he slides? 29. Velocity Derek is playing cards with his friends, and it is his turn to deal. A card has a mass of 2.3 g, and it slides 0.35 m along the table before it stops. If the coefficient of kinetic friction between the card and the table is 0.24, what was the initial speed of the card as it left Derek’s hand? 130

Chapter 5 Forces in Two Dimensions

30. Force The coefficient of static friction between a 40.0-kg picnic table and the ground below it is 0.43. What is the greatest horizontal force that could be exerted on the table while it remains stationary? 31. Acceleration Ryan is moving to a new apartment and puts a dresser in the back of his pickup truck. When the truck accelerates forward, what force accelerates the dresser? Under what circumstances could the dresser slide? In which direction? 32. Critical Thinking You push a 13-kg table in the cafeteria with a horizontal force of 20 N, but it does not move. You then push it with a horizontal force of 25 N, and it accelerates at 0.26 m/s2. What, if anything, can you conclude about the coefficients of static and kinetic friction? physicspp.com/self_check_quiz

5.3 Force and Motion in Two Dimensions

Y

ou have already worked with several situations dealing with forces in two dimensions. For example, when friction acts between two surfaces, you must take into account both the frictional force that is parallel to the surface and the normal force that is perpendicular to it. So far, you have considered only the motion along a level surface. Now you will use your skill in adding vectors to analyze situations in which the forces acting on an object are at angles other than 90°.

• Determine the force that produces equilibrium when three forces act on an object. • Analyze the motion of an object on an inclined plane with and without friction.

Equilibrium Revisited

Objectives

Vocabulary equilibrant

Recall from Chapter 4 that when the net force on an object is zero, the object is in equilibrium. According to Newton’s laws, the object will not accelerate because there is no net force acting on it; an object in equilibrium is motionless or moves with constant velocity. You have already analyzed several equilibrium situations in which two forces acted on an object. It is important to realize that equilibrium can occur no matter how many forces act on an object. As long as the resultant is zero, the net force is zero and the object is in equilibrium. Figure 5-11a shows three forces exerted on a point object. What is the net force acting on the object? Remember that vectors may be moved if you do not change their direction (angle) or length. Figure 5-11b shows the addition of the three forces, A, B, and C. Note that the three vectors form a closed triangle. There is no net force; thus, the sum is zero and the object is in equilibrium. Suppose that two forces are exerted on an object and the sum is not zero. How could you find a third force that, when added to the other two, would add up to zero, and therefore cause the object to be in equilibrium? To find this force, first find the sum of the two forces already being exerted on the object. This single force that produces the same effect as the two individual forces added together is called the resultant force. The force that you need to find is one with the same magnitude as the resultant force, but in the opposite direction. A force that puts an object in equilibrium is called the equilibrant. Figure 5-12 illustrates the procedure for finding this force for two vectors. Note that this general procedure works for any number of vectors.

a A C B

b B

A

C ■

Figure 5-11 An object is in equilibrium when all the forces on it add up to zero.

■ Figure 5-12 The equilibrant is the same magnitude as the resultant, but opposite in direction.

A

B

B

R A

B

A R Equilibrant

Section 5.3 Force and Motion in Two Dimensions

131

Find the equilibrant for the following forces. F1 61.0 N at 17.0° north of east F2 38.0 N at 64.0° north of east F3 54.0 N at 8.0° west of north

y

F4 93.0 N at 53.0° west of north

3

4

2

F5 65.0 N at 21.0° south of west

1

F6 102.0 N at 15.0° west of south

x

10 7

5

F7 26.0 N south

9

F8 77.0 N at 22.0° east of south

8

F9 51.0 N at 33.0° east of south

6

F10 82.0 N at 5.0° south of east

Motion Along an Inclined Plane You have applied Newton’s laws to a variety of equilibrium situations, but only to motions that were either horizontal or vertical. How would you apply them in a situation like the one in Figure 5-13a, in which a skier glides down a slope? Start by identifying the forces acting on the object, the skier, as shown in Figure 5-13b and sketching a free-body diagram. The gravitational force on the skier is in the downward direction toward the center of Earth. There is a normal force perpendicular to the hill, and the frictional forces opposing the skier’s motion are parallel to the hill. The resulting free-body diagram is shown in Figure 5-13c. You can see that, other than the force of friction, only one force acts horizontally or vertically, and you know from experience that the acceleration of the skier will be along the slope. How do you find the net force that causes the skier to accelerate? ■ Figure 5-13 A skier slides down a slope (a). Identify the forces that are acting upon the skier (b) and draw a free-body diagram describing those forces (c). It is important to draw the direction of the normal and the friction forces correctly in order to properly analyze these types of situations.

a

y

System

b

Contact with outside world

Begin

x

Chapter 5 Forces in Two Dimensions

y

FN Fnet Ff a

132

c v

End

Fg

x

Components of Weight for an Object on an Incline A crate weighing 562 N is resting on a plane inclined 30.0° above the horizontal. Find the components of the weight forces that are parallel and perpendicular to the plane. 1

Analyze and Sketch the Problem

y

• Include a coordinate system with the positive x-axis pointing uphill. • Draw the free-body diagram showing Fg, the components Fgx and Fgy , and the angle .

2

Known:

Unknown:

Fg 562 N 30.0°

Fgx ? Fgy ?

y

Fgy

Fg

x

Solve for the Unknown Fgx and Fgy are negative because they point in directions opposite to the positive axes. Fgx = Fg(sin ) = (562 N)(sin 30.0°) Substitute Fg 562, 30.0° = 281 N Fgy Fg(cos ) (562 N)(cos 30.0°) Substitute Fg 562, 30.0° 487 N

3

x

Fgx

Math Handbook Trigonometric Ratios page 855

Evaluate the Answer • Are the units correct? Force is measured in newtons. • Do the signs make sense? The components point in directions opposite to the positive axes. • Are the magnitudes realistic? The values are less than Fg.

33. An ant climbs at a steady speed up the side of its anthill, which is inclined 30.0° from the vertical. Sketch a free-body diagram for the ant. 34. Scott and Becca are moving a folding table out of the sunlight. A cup of lemonade, with a mass of 0.44 kg, is on the table. Scott lifts his end of the table before Becca does, and as a result, the table makes an angle of 15.0° with the horizontal. Find the components of the cup’s weight that are parallel and perpendicular to the plane of the table. 35. Kohana, who has a mass of 50.0 kg, is at the dentist’s office having her teeth cleaned, as shown in Figure 5-14. If the component of her weight perpendicular to the plane of the seat of the chair is 449 N, at what angle is the chair tilted? 36. Fernando, who has a mass of 43.0 kg, slides down the banister at his grandparents’ house. If the banister makes an angle of 35.0° with the horizontal, what is the normal force between Fernando and the banister? 37. A suitcase is on an inclined plane. At what angle, relative to the vertical, will the component of the suitcase’s weight parallel to the plane be equal to half the perpendicular component of its weight?

■

Figure 5-14

Section 5.3 Force and Motion in Two Dimensions

133

Skiing Downhill A 62-kg person on skis is going down a hill sloped at 37°. The coefficient of kinetic friction between the skis and the snow is 0.15. How fast is the skier going 5.0 s after starting from rest? 1

Analyze and Sketch the Problem • Establish a coordinate system. • Draw a free-body diagram showing the skier’s velocity and direction of acceleration. • Draw a motion diagram showing increasing v, and both a and Fnet in the x direction, like the one shown in Figure 5-13.

2

Known:

Unknown:

m 62 kg 37° k 0.15 vi 0.0 m/s t 5.0 s

a ? vf ?

y

y

v Fnet

a

Solve for FN. FN Fgy Fnet, y FN Fgy mg(cos ) x-direction: Solve for a. Fnet, x Fgx Ff

Fgy is negative. It is in the negative direction as defined by the coordinate system. Substitute Fnet, y 0.0 N and rearrange Substitute Fgy mg cos

Math Handbook Isolating a Variable page 845

Ff is negative because it is in the negative direction as defined by the coordinate system. Substitute Fnet, x ma, Fgx mg sin , Ff kFN Substitute a ax because all the acceleration is in the x-direction; substitute FN mg cos

a g(sin kcos ) (9.80 m/s2)(sin 37° (0.15)cos 37°) 4.7 m/s2

Substitute g 9.80 m/s2, 37°, k 0.15

Because vi, a, and t are all known, use the following. vf vi at 0.0 (4.7 m/s2)(5.0 s) Substitute vi 0.0 m/s, a 4.7 m/s2, t 5.0 s 24 m/s

Evaluate the Answer • Are the units correct? Performing dimensional analysis on the units verifies that vf is in m/s and a is in m/s2. • Do the signs make sense? Because vf and a are both in the x direction, the signs do make sense. • Are the magnitudes realistic? The velocity is fast, over 80 km/h (50 mph), but 37° is a steep incline, and the friction between the skis and the snow is not large.

Chapter 5 Forces in Two Dimensions

Fg x

x

There is no acceleration in the y-direction, so ay 0.0 m/s2.

max mg(sin ) kFN mg(sin ) kmg(cos )

134

Ff

Solve for the Unknown y-direction: Fnet, y may 0.0 N

3

FN

38. Consider the crate on the incline in Example Problem 5. Calculate the magnitude of the acceleration. After 4.00 s, how fast will the crate be moving? 39. If the skier in Example Problem 6 were on a 31° downhill slope, what would be the magnitude of the acceleration? 40. Stacie, who has a mass of 45 kg, starts down a slide that is inclined at an angle of 45° with the horizontal. If the coefficient of kinetic friction between Stacie’s shorts and the slide is 0.25, what is her acceleration? 41. After the skier on the 37° hill in Example Problem 6 had been moving for 5.0 s, the friction of the snow suddenly increased and made the net force on the skier zero. What is the new coefficient of friction? The most important decision in problems involving motion along a slope is what coordinate system to use. Because an object’s acceleration is usually parallel to the slope, one axis, usually the x-axis, should be in that direction. The y-axis is perpendicular to the x-axis and perpendicular to the surface of the slope. With this coordinate system, you now have two forces, the normal and frictional forces, in the directions of the coordinate axes; however, the weight is not. This means that when an object is placed on an inclined plane, the magnitude of the normal force between the object and the plane will usually not be equal to the object’s weight. You will need to apply Newton’s laws once in the x-direction and once in the y-direction. Because the weight does not point in either of these directions, you will need to break this vector into its x- and y-components before you can sum your forces in these two directions. Example Problem 5 and Example Problem 6 both showed this procedure.

What’s Your Angle? Prop a board up so that it forms an inclined plane at a 45° angle. Hang a 500-g object from the spring scale. 1. Measure and record the weight of the object. Set the object on the bottom of the board and slowly pull it up the inclined plane at a constant speed. 2. Observe and record the reading on the spring scale. Analyze and Conclude 3. Calculate the component of weight for the 500-g object that is parallel to the inclined plane. 4. Compare the spring-scale reading along the inclined plane with the component of weight parallel to the inclined plane.

5.3 Section Review 42. Forces One way to get a car unstuck is to tie one end of a strong rope to the car and the other end to a tree, then push the rope at its midpoint at right angles to the rope. Draw a free-body diagram and explain why even a small force on the rope can exert a large force on the car. 43. Mass A large scoreboard is suspended from the ceiling of a sports arena by 10 strong cables. Six of the cables make an angle of 8.0° with the vertical while the other four make an angle of 10.0°. If the tension in each cable is 1300.0 N, what is the scoreboard’s mass? 44. Acceleration A 63-kg water skier is pulled up a 14.0° incline by a rope parallel to the incline with a tension of 512 N. The coefficient of kinetic friction is 0.27. What are the magnitude and direction of the skier’s acceleration? physicspp.com/self_check_quiz

45. Equilibrium You are hanging a painting using two lengths of wire. The wires will break if the force is too great. Should you hang the painting as shown in Figures 5-15a or 5-15b? Explain. a

b

■

Figure 5-15

46. Critical Thinking Can the coefficient of friction ever have a value such that a skier would be able to slide uphill at a constant velocity? Explain why or why not. Assume there are no other forces acting on the skier. Section 5.3 Force and Motion in Two Dimensions

135

The Coefficient of Friction Alternate CBL instructions can be found on the Web site. physicspp.com

Static and kinetic friction are forces that are a result of two surfaces in contact with each other. Static friction is the force that must be overcome to cause an object to begin moving, while kinetic friction occurs between two objects in motion relative to each other. The kinetic friction force, Ff, kinetic, is defined by Ff, kinetic k FN, where k is the coefficient of kinetic friction and FN is the normal force acting on the object. The maximum static frictional force, Ff, max static , is defined by Ff, static sFN where s is the coefficient of static friction and FN is the normal force on the object. The maximum static frictional force that must be overcome before movement is able to begin is sFN. If you apply a constant force to pull an object along a horizontal surface at a constant speed, then the frictional force opposing the motion is equal and opposite to the applied force, Fp. Therefore, Fp Ff. The normal force is equal and opposite to the object’s weight when the object is on a horizontal surface and the applied force is horizontal.

QUESTION How can the coefficient of static and kinetic friction be determined for an object on a horizontal surface?

Objectives

Materials

■ Measure the normal and frictional forces acting

pulley C-clamp masking tape wood surface

■ ■ ■ ■

on an object starting in motion and already in motion. Use numbers to calculate s and k. Compare and contrast values of s and k. Analyze the kinetic friction results. Estimate the angle where sliding will begin for an object on an inclined plane.

Safety Precautions

string (1 m) spring scale, 0-5 N wood block

Procedure 1. Check your spring scale to make sure that it reads zero when it is held vertically. If necessary, follow your teacher’s instructions to zero it. 2. Attach the pulley to the edge of the table with a C-clamp. 3. Attach the string to the spring scale hook and the wood block. 4. Measure the weight of the block of wood, or other small object, and record the value as the normal force, FN, in Data Tables 1, 2, and 3. 5. Unhook the string from the spring scale and run it through the pulley. Then reattach it to the spring scale. 6. Move the wood block as far away from the pulley as the string permits, while having it remain on the wood surface. 7. With the spring scale oriented vertically so that a right angle is formed between the wood block, the pulley, and the spring scale, slowly pull up on the spring scale. Observe the force that is necessary to cause the wood block to begin sliding. Record this value for the static frictional force in Data Table 1.

136 Horizons Companies

Material Table Object material Surface material

Data Table 1 FN (N)

Data Table 3 Static Friction Force, Fs (N)

Trial 1

Trial 2

Trial 3

FN (N)

Data Table 4 Trial 2

Trial 3

s

k

(Angle, , when sliding begins on an incline)

*

Kinetic Friction Force, Ff (N) Trial 1

Ff (N)

Average

Data Table 2 FN (N)

Fs (N)

tan *

Average

8. Repeat steps 6 and 7 for two additional trials. 9. Repeat steps 6 and 7. However, once the block begins sliding, pull just hard enough to keep it moving at a constant speed across the other horizontal surface. Record this force as the kinetic frictional force in Data Table 2. 10. Repeat step 9 for two additional trials. 11. Place the block on the end of the surface. Slowly raise one end of the surface to make an incline. Gently tap the block to cause it to move and overcome static friction. If the block stops, replace it at the top of the incline and repeat the procedure. Continue increasing the angle, , between the horizontal and the inclined surface, and tapping the block until it slides at a constant speed down the incline. Record the angle, , in Data Table 4.

Conclude and Apply 1. Compare and Contrast Examine your values for s and k. Explain whether your results are reasonable or not. 2. Use Models Draw a free-body diagram showing the forces acting on the block if it is placed on an incline of angle . Make certain that you include the force due to friction in your diagram. 3. From your diagram, assuming that the angle, , is where sliding begins, what does tan represent? 4. Compare your value for tan (experimental), s , and k.

Going Further

Analyze

Repeat the experiment with additional surfaces that have different characteristics.

1. Average the data for the static frictional force, Fs, max, from the three trials and record the result in the last column of Data Table 1 and in Data Table 3.

Real-World Physics

2. Average the data for the kinetic frictional force, Ff, from the three trials and record the result in the last column of Data Table 2 and in Data Table 3.

If you were downhill skiing and wished to determine the coefficient of kinetic friction between your skis and the slope, how could you do this? Be specific about how you could find a solution to this problem.

3. Use the data in Data Table 3 to calculate the coefficient of static friction, s, and record the value in Data Table 3. 4. Use the data in Data Table 3 to calculate the coefficient of kinetic friction, k, and record the value in Data Table 3.

To find out more about friction, visit the Web site: physicspp.com

5. Calculate tan for your value in Data Table 4. 137

Roller Coasters moving. These organs help maintain balance by providing information to the brain. The brain then sends nerve impulses to the skeletal muscles to contract or relax to maintain balance. The constant change in position during a roller-coaster ride causes the organs of the inner ear to send conflicting messages to the brain. As a result, the skeletal muscles contract and relax throughout the ride. You know that you are moving at high speeds because your eyes see the surroundings move past at high speed. So, designers make use of the surrounding The Force Factor landscape along with Designers of roller twists, turns, tunnels, coasters take into and loops to give the account the magnirider plenty of visual tude of the forces cues. These visual cues, exerted on the rider. along with the mesThey design the sages from the inner coaster in such a way ear, can result in disthat the forces thrill orientation and in the rider without caussome cases, nausea. ing injury or too To enthusiasts the much discomfort. disorientation is part Designers measure of the thrill. the amount of force The thrill of a roller-coaster ride is produced by the forces In order to attract exerted on the rider by acting on the rider and the rider’s reaction to visual cues. visitors, amusement calculating the force parks are constantly factor. The force factor is equal to the force working on designing new rides that take the exerted by the seat on the rider divided by the rider to new thrill levels. As roller-coaster weight of the rider. Suppose the rider weighs technology improves, your most thrilling about 68 kg. When the roller coaster is at the roller-coaster ride may be over the next hill. bottom of a hill, the rider may experience a force factor of 2. That means that at the bottom of the Going Further hill, the rider will feel as though he or she weighs twice as much, or in this case 136 kg. 1. Compare and Contrast Compare and Conversely, at the top of a hill the force factor contrast your experience as a rider in may be 0.5 and the rider will feel as though he the front of a roller coaster versus the or she weighs half his or her normal weight. back of it. Explain your answer in terms Thus, designers create excitement by designing of the forces acting on you. portions that change the rider’s apparent weight.

Why are roller coasters fun? A rollercoaster ride would be no fun at all if not for the forces acting on the coaster car and the rider. What forces do riders experience as they ride a roller coaster? The force of gravity acts on the rider and the coaster car in the downward direction. The seat of the car exerts a force on the rider in the opposite direction. When the coaster car makes a turn, the rider experiences a force in the opposite direction. Also, there are forces present due to the friction between the rider and the seat, the side of the car, and the safety bar.

The Thrill Factors Roller-coaster designers manipulate the way in which the body perceives the external world to create that “thrilling” sensation. For example, the roller coaster moves up the first hill very slowly, tricking the rider into thinking that the hill is higher than it is. The organs of the inner ear sense the position of the head both when it is still and when it is 138

Technology and Society

courtesy of Six Flags Amusement Park

2. Critical Thinking While older roller coasters rely on chain systems to pull the coaster up the first hill, newer ones depend on hydraulic systems to do the same job. Research each of these two systems. What do you think are the advantages and disadvantages of using each system?

5.1 Vectors Vocabulary

Key Concepts

• components (p. 122) • vector resolution (p. 122)

•

When two vectors are at right angles, you can use the Pythagorean theorem to determine the magnitude of the resultant vector. R2 A2 B2

•

The law of cosines and law of sines can be used to find the magnitude of the resultant of any two vectors. R2 A2 B2 2AB cos R A B sin sin a sin b

•

The components of a vector are projections of the component vectors. A A A y opposite side sin ; therefore, Ay A sin hypotenuse A Ry tan1 Rx adjacent side hypotenuse

cos x ; therefore, Ax A cos

( )

•

Vectors can be summed by separately adding the x- and y-components.

5.2 Friction Vocabulary

Key Concepts

• kinetic friction (p. 126) • static friction (p. 126) • coefficient of kinetic friction (p. 127) • coefficient of static friction (p. 127)

• • •

A frictional force acts when two surfaces touch. The frictional force is proportional to the force pushing the surfaces together. The kinetic friction force is equal to the coefficient of kinetic friction times the normal force. Ff, kinetic k FN

•

The static friction force is less than or equal to the coefficient of static friction times the normal force. Ff, static k FN

5.3 Force and Motion in Two Dimensions Vocabulary

Key Concepts

• equilibrant (p. 131)

• • •

The force that must be exerted on an object to cause it to be in equilibrium is called the equilibrant. The equilibrant is found by finding the net force on an object, then applying a force with the same magnitude but opposite direction. An object on an inclined plane has a component of the force of gravity in a direction parallel to the plane; the component can accelerate the object down the plane.

physicspp.com/vocabulary_puzzlemaker

139

Concept Mapping 47. Complete the concept map below with the terms sine, cosine, or tangent to indicate whether each function is positive or negative in each quadrant. Some circles could remain blank, and others can have more than one term.

59. If a coordinate system is set up such that the positive x-axis points in a direction 30° above the horizontal, what should be the angle between the x-axis and the y-axis? What should be the direction of the positive y-axis? (5.3)

60. Explain how you would set up a coordinate system for motion on a hill. (5.3)

Quadrant

61. If your textbook is in equilibrium, what can you say about the forces acting on it? (5.3)

I

II

III

62. Can an object that is in equilibrium be moving?

IV

Explain. (5.3)

63. What is the sum of three vectors that, when placed tip to tail, form a triangle? If these vectors represent forces on an object, what does this imply about the object? (5.3)

64. You are asked to analyze the motion of a book

Mastering Concepts 48. How would you add two vectors graphically? (5.1) 49. Which of the following actions is permissible when you graphically add one vector to another: moving the vector, rotating the vector, or changing the vector’s length? (5.1)

50. In your own words, write a clear definition of the resultant of two or more vectors. Do not explain how to find it; explain what it represents. (5.1)

51. How is the resultant displacement affected when two displacement vectors are added in a different order? (5.1)

52. Explain the method that you would use to subtract two vectors graphically. (5.1)

53. Explain the difference between A and A. (5.1) 54. The Pythagorean theorem usually is written c2 a2 b2. If this relationship is used in vector addition, what do a, b, and c represent? (5.1)

55. When using a coordinate system, how is the angle or direction of a vector determined with respect to the axes of the coordinate system? (5.1)

56. What is the meaning of a coefficient of friction that

placed on a sloping table. (5.3) a. Describe the best coordinate system for analyzing the motion. b. How are the components of the weight of the book related to the angle of the table?

65. For a book on a sloping table, describe what happens to the component of the weight force parallel to the table and the force of friction on the book as you increase the angle that the table makes with the horizontal. (5.3) a. Which components of force(s) increase when the angle increases? b. Which components of force(s) decrease?

Applying Concepts 66. A vector that is 1 cm long represents a displacement of 5 km. How many kilometers are represented by a 3-cm vector drawn to the same scale?

67. Mowing the Lawn If you are pushing a lawn mower across the grass, as shown in Figure 5-16, can you increase the horizontal component of the force that you exert on the mower without increasing the magnitude of the force? Explain. F

Fy

is greater than 1.0? How would you measure it? (5.2)

57. Cars Using the model of friction described in this

Fx

textbook, would the friction between a tire and the road be increased by a wide rather than a narrow tire? Explain. (5.2)

58. Describe a coordinate system that would be suitable for dealing with a problem in which a ball is thrown up into the air. (5.3)

140

Chapter 5 Forces in Two Dimensions For more problems, go to Additional Problems, Appendix B.

■

Figure 5-16

68. A vector drawn 15 mm long represents a velocity of 30 m/s. How long should you draw a vector to represent a velocity of 20 m/s?

69. What is the largest possible displacement resulting from two displacements with magnitudes 3 m and 4 m? What is the smallest possible resultant? Draw sketches to demonstrate your answers.

70. How does the resultant displacement change as the angle between two vectors increases from 0° to 180°?

71. A and B are two sides of a right triangle, where

tan A/B. a. Which side of the triangle is longer if tan is greater than 1.0? b. Which side is longer if tan is less than 1.0? c. What does it mean if tan is equal to 1.0?

72. Traveling by Car A car has a velocity of 50 km/h in a direction 60° north of east. A coordinate system with the positive x-axis pointing east and a positive y-axis pointing north is chosen. Which component of the velocity vector is larger, x or y?

78. TV Towers The transmitting tower of a TV station is held upright by guy wires that extend from the top of the tower to the ground. The force along the guy wires can be resolved into two perpendicular components. Which one is larger?

Mastering Problems 5.1 Vectors 79. Cars A car moves 65 km due east, then 45 km due west. What is its total displacement?

80. Find the horizontal and vertical components of the following vectors, as shown in Figure 5-17. a. E b. F c. A

B(3.0)

73. Under what conditions can the Pythagorean

F(5.0)

theorem, rather than the law of cosines, be used to find the magnitude of a resultant vector?

A(3.0)

C(6.0)

75. Pulling a Cart According to legend, a horse learned Newton’s laws. When the horse was told to pull a cart, it refused, saying that if it pulled the cart forward, according to Newton’s third law, there would be an equal force backwards; thus, there would be balanced forces, and, according to Newton’s second law, the cart would not accelerate. How would you reason with this horse?

76. Tennis When stretching a tennis net between two posts, it is relatively easy to pull one end of the net hard enough to remove most of the slack, but you need a winch to take the last bit of slack out of the net to make the top almost completely horizontal. Why is this true?

77. The weight of a book on an inclined plane can be resolved into two vector components, one along the plane, and the other perpendicular to it. a. At what angle are the components equal? b. At what angle is the parallel component equal to zero? c. At what angle is the parallel component equal to the weight? physicspp.com/chapter_test

D(4.0) ■

74. A problem involves a car moving up a hill, so a coordinate system is chosen with the positive x-axis parallel to the surface of the hill. The problem also involves a stone that is dropped onto the car. Sketch the problem and show the components of the velocity vector of the stone.

E(5.0)

Figure 5-17

81. Graphically find the sum of the following pairs of vectors, whose lengths and directions are shown in Figure 5-17. a. b. c. d.

D and A C and D C and A E and F

82. Graphically add the following sets of vectors, as shown in Figure 5-17. a. A, C, and D b. A, B, and E c. B, D, and F

83. You walk 30 m south and 30 m east. Find the magnitude and direction of the resultant displacement both graphically and algebraically.

84. Hiking A hiker’s trip consists of three segments. Path A is 8.0 km long heading 60.0° north of east. Path B is 7.0 km long in a direction due east. Path C is 4.0 km long heading 315° counterclockwise from east. a. Graphically add the hiker’s displacements in the order A, B, C. b. Graphically add the hiker’s displacements in the order C, B, A. c. What can you conclude about the resulting displacements? Chapter 5 Assessment

141

85. What is the net force acting on the ring in

91. A 225-kg crate is pushed horizontally with a force

Figure 5-18?

of 710 N. If the coefficient of friction is 0.20, calculate the acceleration of the crate. y

92. A force of 40.0 N accelerates a 5.0-kg block at 6.0 m/s2 along a horizontal surface. a. How large is the frictional force? b. What is the coefficient of friction?

400.0 N

500.0 N

40.0°

93. Moving Appliances Your family just had a new

50.0° x

■

Figure 5-18

86. What is the net force acting on the ring in Figure 5-19? y

refrigerator delivered. The delivery man has left and you realize that the refrigerator is not quite in the right position, so you plan to move it several centimeters. If the refrigerator has a mass of 180 kg, the coefficient of kinetic friction between the bottom of the refrigerator and the floor is 0.13, and the static coefficient of friction between these same surfaces is 0.21, how hard do you have to push horizontally to get the refrigerator to start moving?

94. Stopping at a Red Light You are driving a 128 N 128 N

30.0° 64 N

■

x

2500.0-kg car at a constant speed of 14.0 m/s along a wet, but straight, level road. As you approach an intersection, the traffic light turns red. You slam on the brakes. The car’s wheels lock, the tires begin skidding, and the car slides to a halt in a distance of 25.0 m. What is the coefficient of kinetic friction between your tires and the wet road?

Figure 5-19

87. A Ship at Sea A ship at sea is due into a port 500.0 km due south in two days. However, a severe storm comes in and blows it 100.0 km due east from its original position. How far is the ship from its destination? In what direction must it travel to reach its destination?

88. Space Exploration A descent vehicle landing on Mars has a vertical velocity toward the surface of Mars of 5.5 m/s. At the same time, it has a horizontal velocity of 3.5 m/s. a. At what speed does the vehicle move along its descent path? b. At what angle with the vertical is this path?

89. Navigation Alfredo leaves camp and, using a compass, walks 4 km E, then 6 km S, 3 km E, 5 km N, 10 km W, 8 km N, and, finally, 3 km S. At the end of three days, he is lost. By drawing a diagram, compute how far Alfredo is from camp and which direction he should take to get back to camp.

5.2 Friction

5.3 Force and Motion in Two Dimensions 95. An object in equilibrium has three forces exerted on it. A 33.0-N force acts at 90.0° from the x-axis and a 44.0-N force acts at 60.0° from the x-axis. What are the magnitude and direction of the third force?

96. Five forces act on an object: (1) 60.0 N at 90.0°, (2) 40.0 N at 0.0°, (3) 80.0 N at 270.0°, (4) 40.0 N at 180.0°, and (5) 50.0 N at 60.0°. What are the magnitude and direction of a sixth force that would produce equilibrium?

97. Advertising Joe wishes to hang a sign weighing 7.50102 N so that cable A, attached to the store, makes a 30.0° angle, as shown in Figure 5-20. Cable B is horizontal and attached to an adjoining building. What is the tension in cable B?

30.0°

A

B

90. If you use a horizontal force of 30.0 N to slide a 12.0-kg wooden crate across a floor at a constant velocity, what is the coefficient of kinetic friction between the crate and the floor?

142

Chapter 5 Forces in Two Dimensions For more problems, go to Additional Problems, Appendix B.

■

Figure 5-20

98. A street lamp weighs 150 N. It is supported by two wires that form an angle of 120.0° with each other. The tensions in the wires are equal. a. What is the tension in each wire supporting the street lamp? b. If the angle between the wires supporting the street lamp is reduced to 90.0°, what is the tension in each wire?

102. In Figure 5-22, a block of mass M is pushed with a force, F, such that the smaller block of mass m does not slide down the front of it. There is no friction between the larger block and the surface below it, but the coefficient of static friction between the two blocks is s. Find an expression for F in terms of M, m, s, and g.

99. A 215-N box is placed on an inclined plane that makes a 35.0° angle with the horizontal. Find the component of the weight force parallel to the plane’s surface.

m F

M

100. Emergency Room You are shadowing a nurse in the emergency room of a local hospital. An orderly wheels in a patient who has been in a very serious accident and has had severe bleeding. The nurse quickly explains to you that in a case like this, the patient’s bed will be tilted with the head downward to make sure the brain gets enough blood. She tells you that, for most patients, the largest angle that the bed can be tilted without the patient beginning to slide off is 32.0° from the horizontal. a. On what factor or factors does this angle of tilting depend? b. Find the coefficient of static friction between a typical patient and the bed’s sheets.

■

Figure 5-22

Mixed Review 103. The scale in Figure 5-23 is being pulled on by three ropes. What net force does the scale read?

101. Two blocks are connected by a string over a frictionless, massless pulley such that one is resting on an inclined plane and the other is hanging over the top edge of the plane, as shown in Figure 5-21. The hanging block has a mass of 16.0 kg, and the one on the plane has a mass of 8.0 kg. The coefficient of kinetic friction between the block and the inclined plane is 0.23. The blocks are released from rest. a. What is the acceleration of the blocks? b. What is the tension in the string connecting the blocks?

27.0°

75.0 N

27.0°

75.0 N 150.0 N ■ Figure 5-23

104. Sledding A sled with a mass of 50.0 kg is pulled

37.0°

■

Figure 5-21 physicspp.com/chapter_test

along flat, snow-covered ground. The static friction coefficient is 0.30, and the kinetic friction coefficient is 0.10. a. What does the sled weigh? b. What force will be needed to start the sled moving? c. What force is needed to keep the sled moving at a constant velocity? d. Once moving, what total force must be applied to the sled to accelerate it at 3.0 m/s2? Chapter 5 Assessment

143

105. Mythology Sisyphus was a character in Greek mythology who was doomed in Hades to push a boulder to the top of a steep mountain. When he reached the top, the boulder would slide back down the mountain and he would have to start all over again. Assume that Sisyphus slides the boulder up the mountain without being able to roll it, even though in most versions of the myth, he rolled it. a. If the coefficient of kinetic friction between the boulder and the mountainside is 0.40, the mass of the boulder is 20.0 kg, and the slope of the mountain is a constant 30.0°, what is the force that Sisyphus must exert on the boulder to move it up the mountain at a constant velocity? b. If Sisyphus pushes the boulder at a velocity of 0.25 m/s and it takes him 8.0 h to reach the top of the mountain, what is the mythical mountain’s vertical height?

106. Landscaping A tree is being transported on a flatbed trailer by a landscaper, as shown in Figure 5-24. If the base of the tree slides on the trailer, the tree will fall over and be damaged. If the coefficient of static friction between the tree and the trailer is 0.50, what is the minimum stopping distance of the truck, traveling at 55 km/h, if it is to accelerate uniformly and not have the tree slide forward and fall on the trailer?

108. Analyze and Conclude Margaret Mary, Doug, and Kako are at a local amusement park and see an attraction called the Giant Slide, which is simply a very long and high inclined plane. Visitors at the amusement park climb a long flight of steps to the top of the 27° inclined plane and are given canvas sacks. They sit on the sacks and slide down the 70m-long plane. At the time when the three friends walk past the slide, a 135-kg man and a 20-kg boy are each at the top preparing to slide down. “I wonder how much less time it will take the man to slide down than it will take the boy,” says Margaret Mary. “I think the boy will take less time,” says Doug. “You’re both wrong,” says Kako. “They will reach the bottom at the same time.” a. Perform the appropriate analysis to determine who is correct. b. If the man and the boy do not take the same amount of time to reach the bottom of the slide, calculate how many seconds of difference there will be between the two times.

Writing in Physics 109. Investigate some of the techniques used in industry to reduce the friction between various parts of machines. Describe two or three of these techniques and explain the physics of how they work.

110. Olympics In recent years, many Olympic athletes, such as sprinters, swimmers, skiers, and speed skaters, have used modified equipment to reduce the effects of friction and air or water drag. Research a piece of equipment used by one of these types of athletes and the way it has changed over the years. Explain how physics has impacted these changes.

■

Figure 5-24

Cumulative Review 111. Add or subtract as indicated and state the answer with the correct number of significant digits.

Thinking Critically 107. Use Models Using the Example Problems in this chapter as models, write an example problem to solve the following problem. Include the following sections: Analyze and Sketch the Problem, Solve for the Unknown (with a complete strategy), and Evaluate the Answer. A driver of a 975-kg car traveling 25 m/s puts on the brakes. What is the shortest distance it will take for the car to stop? Assume that the road is concrete, the force of friction of the road on the tires is constant, and the tires do not slip.

144

(Chapter 1)

a. b. c. d.

85.26 g 4.7 g 1.07 km 0.608 km 186.4 kg 57.83 kg 60.08 s 12.2 s

112. You ride your bike for 1.5 h at an average velocity of 10 km/h, then for 30 min at 15 km/h. What is your average velocity? (Chapter 3)

113. A 45-N force is exerted in the upward direction on a 2.0-kg briefcase. What is the acceleration of the briefcase? (Chapter 4)

Chapter 5 Forces in Two Dimensions For more problems, go to Additional Problems, Appendix B.

Multiple Choice 1. Two tractors pull against a 1.00103-kg log. If the angle of the tractors’ chains in relation to each other is 18.0°, and each tractor pulls with a force of 8102 N, what forces will they be able to exert? 250 N

1.58103 N

1.52103 N

1.60103 N

6. A string exerts a force of 18 N on a box at an angle of 34° from the horizontal. What is the horizontal component of the force on the box? 10 N 15 N

21.7 N 32 N F 34°

1.00103 kg

9.0° 9.0°

2. An airplane pilot tries to fly directly east with a velocity of 800.0 km/h. If a wind comes from the southwest at 80.0 km/h, what is the relative velocity of the airplane to the surface of Earth? 804 km/h, 5.7° N of E

7. Sukey is riding her bicycle on a path when she comes around a corner and sees that a fallen tree is blocking the way 42 m ahead. If the coefficient of friction between her bicycle’s tires and the gravel path is 0.36, and she is traveling at 50.0 km/h, how much stopping distance will she require? Sukey and her bicycle, together, have a mass of 95 kg.

858 km/h, 3.8° N of E

3.00 m

8.12 m

859 km/h, 4.0° N of E

4.00 m

27.3 m

880 km/h 45° N of E 3. For a winter fair, some students decide to build 30.0-kg wooden pull-carts on sled skids. If two 90.0-kg passengers get in, how much force will the puller have to exert to move a pull-cart? The coefficient of maximum static friction between the cart and the snow is 0.15. 1.8102 3.1102

N

2.1103

N

N

1.4104

N

4. It takes a minimum force of 280 N to move a 50.0-kg crate. What is the coefficient of maximum static friction between the crate and the floor? 0.18

1.8

0.57

5.6

Extended Answer 8. A man starts from a position 310 m north of his car and walks for 2.7 min in a westward direction at a constant velocity of 10 km/h. How far is he from his car when he stops? 9. Jeeves is tired of his 41.2-kg son sliding down the banister, so he decides to apply an extremely sticky paste that increases the coefficient of static friction to 0.72 to the top of the banister. What will be the magnitude of the static friction force on the boy if the banister is at an angle of 52.4° from the horizontal?

Calculators Are Only Machines

5. What is the y-component of a 95.3-N force that is exerted at 57.1° to the horizontal? 51.8 N

114 N

80.0 N

175 N physicspp.com/standardized_test

If your test allows you to use a calculator, use it wisely. Figure out which numbers are relevant, and determine the best way to solve the problem before you start punching keys.

Chapter 5 Standardized Test Practice

145

What You’ll Learn • You will use Newton’s laws and your knowledge of vectors to analyze motion in two dimensions. • You will solve problems dealing with projectile and circular motion. • You will solve relativevelocity problems.

Why It’s Important Almost all types of transportation and amusement-park attractions contain at least one element of projectile or circular motion or are affected by relative velocities. Swinging Around Before this ride starts to move, the seats hang straight down from their supports. When the ride speeds up, the seats swing out at an angle.

Think About This When the swings are moving around the circle at a constant speed, are they accelerating?

physicspp.com 146 Gibson Stock Photography

How can the motion of a projectile be described? Question Can you describe a projectile’s motion in both the horizontal and the vertical directions? Procedure 1. With a marked grid in the background, videotape a ball that is launched with an initial velocity only in the horizontal direction. 2. Make and Use Graphs On a sheet of graph paper, draw the location of the ball every 0.1 s (3 frames). 3. Draw two motion diagrams: one for the ball’s horizontal motion and one for its vertical motion.

Critical Thinking Describe the motion of a horizontally launched projectile.

Analysis How does the vertical motion change as time passes? Does it increase, decrease, or stay the same? How does the horizontal motion change as time passes? Does it increase, decrease, or stay the same?

6.1 Projectile Motion

I

f you observed the movement of a golf ball being hit from a tee, a frog hopping, or a free throw being shot with a basketball, you would notice that all of these objects move through the air along similar paths, as do baseballs, arrows, and bullets. Each path is a curve that moves upward for a distance, and then, after a time, turns and moves downward for some distance. You may be familiar with this curve, called a parabola, from math class. An object shot through the air is called a projectile. A projectile can be a football, a bullet, or a drop of water. After a projectile is launched, what forces are exerted on the projectile? You can draw a free-body diagram of a launched projectile and identify all the forces that are acting on it. No matter what the object is, after a projectile has been given an initial thrust, if you ignore air resistance, it moves through the air only under the force of gravity. The force of gravity is what causes the object to curve downward in a parabolic flight path. Its path through space is called its trajectory. If you know the force of the initial thrust on a projectile, you can calculate its trajectory.

Objectives • Recognize that the vertical and horizontal motions of a projectile are independent. • Relate the height, time in the air, and initial vertical velocity of a projectile using its vertical motion, and then determine the range using the horizontal motion. • Explain how the trajectory of a projectile depends upon the frame of reference from which it is observed.

Vocabulary projectile trajectory

Section 6.1 Projectile Motion

147

Horizons Companies

Independence of Motion in Two Dimensions

Over the Edge Obtain two balls, one twice the mass of the other. 1. Predict which ball will hit the floor first when you roll them over the surface of a table and let them roll off the edge. 2. Predict which ball will hit the floor furthest from the table. 3. Explain your predictions. 4. Test your predictions. Analyze and Conclude 5. Does the mass of the ball affect its motion? Is mass a factor in any of the equations for projectile motion?

Think about two softball players warming up for a game, tossing a ball back and forth. What does the path of the ball through the air look like? It looks like a parabola, as you just learned. Imagine that you are standing directly behind one of the players and you are watching the softball as it is being tossed. What would the motion of the ball look like? You would see it go up and back down, just like any object that is tossed straight up in the air. If you were watching the softball from a hot-air balloon high above the field, what motion would you see then? You would see the ball move from one player to the other at a constant speed, just like any object that is given an initial horizontal velocity, such as a hockey puck sliding across ice. The motion of projectiles is a combination of these two motions. Why do projectiles behave in this way? After a softball leaves a player’s hand, what forces are exerted on the ball? If you ignore air resistance, there are no contact forces on the ball. There is only the field force of gravity in the downward direction. How does this affect the ball’s motion? Gravity causes the ball to have a downward acceleration. Figure 6-1 shows the trajectories of two softballs. One was dropped and the other was given an initial horizontal velocity of 2.0 m/s. What is similar about the two paths? Look at their vertical positions. During each flash from the strobe light, the heights of the two softballs are the same. Because the change in vertical position is the same for both, their average vertical velocities during each interval are also the same. The increasingly large distance traveled vertically by the softballs, from one time interval to the next, shows that they are accelerated downward due to the force of gravity. Notice that the horizontal motion of the launched ball does not affect its vertical motion. A projectile launched horizontally has no initial vertical velocity. Therefore, its vertical motion is like that of an object dropped from rest. The downward velocity increases regularly because of the acceleration due to gravity.

■ Figure 6-1 The ball on the right was given an initial horizontal velocity. The ball on the left was dropped at the same time from rest. Note that the vertical positions of the two objects are the same during each flash.

148

Chapter 6 Motion in Two Dimensions

a

b

Begin

Begin

vx ax 0

ay vy

y

x

Separate motion diagrams for the horizontal and vertical motions are shown in Figure 6-2a. The vertical-motion diagram represents the motion of the dropped ball. The horizontal-motion diagram shows the constant velocity in the x-direction of the launched ball. This constant velocity in the horizontal direction is exactly what should be expected because there is no horizontal force acting on the ball. In Figure 6-2b, the horizontal and vertical components are added to form the total velocity vector for the projectile. You can see how the combination of constant horizontal velocity and uniform vertical acceleration produces a trajectory that has a parabolic shape.

■ Figure 6-2 A object’s motion can be broken into its x- and y-components (a). When the horizontal and vertical components of the ball’s velocity are combined (b), the resultant vectors are tangent to a parabola.

Motion in Two Dimensions Projectile motion in two dimensions can be determined by breaking the problem into two connected one-dimensional problems. 1. Divide the projectile motion into a vertical motion problem and a horizontal motion problem. 2. The vertical motion of a projectile is exactly that of an object dropped or thrown straight up or straight down. A gravitational force acts on the object and accelerates it by an amount, g. Review Section 3.3 on free fall to refresh your problem-solving skills for vertical motion. 3. Analyzing the horizontal motion of a projectile is the same as solving a constant velocity problem. No horizontal force acts on a projectile when drag due to air resistance is neglected. Consequently, there are no forces acting in the horizontal direction and therefore, no horizontal acceleration; ax 0.0 m/s. To solve, use the same methods that you learned in Section 2.4. 4. Vertical motion and horizontal motion are connected through the variable of time.

The time from the launch of the projectile to the time it hits the target is the same for both vertical motion and horizontal motion. Therefore, solving for time in one of the dimensions, vertical or horizontal, automatically gives you time for the other dimension.

Section 6.1 Projectile Motion

149

1. A stone is thrown horizontally at a speed of 5.0 m/s from the top of a cliff that is 78.4 m high. a. How long does it take the stone to reach the bottom of the cliff? b. How far from the base of the cliff does the stone hit the ground? c. What are the horizontal and vertical components of the stone’s velocity just before it hits the ground? 2. Lucy and her friend are working at an assembly plant making wooden toy giraffes. At the end of the line, the giraffes go horizontally off the edge of the conveyor belt and fall into a box below. If the box is 0.6 m below the level of the conveyor belt and 0.4 m away from it, what must be the horizontal velocity of giraffes as they leave the conveyor belt? 3. You are visiting a friend from elementary school who now lives in a small town. One local amusement is the ice-cream parlor, where Stan, the short-order cook, slides his completed ice-cream sundaes down the counter at a constant speed of 2.0 m/s to the servers. (The counter is kept very well polished for this purpose.) If the servers catch the sundaes 7.0 cm from the edge of the counter, how far do they fall from the edge of the counter to the point at which the servers catch them?

Projectiles Launched at an Angle When a projectile is launched at an angle, the initial velocity has a vertical component, as well as a horizontal component. If the object is launched upward, like a ball tossed straight up in the air, it rises with slowing speed, reaches the top of its path, and descends with increasing speed. Figure 6-3a shows the separate vertical- and horizontal-motion diagrams for the trajectory. In the coordinate system, the positive x-axis is horizontal and the positive y-axis is vertical. Note the symmetry. At each point in the vertical direction, the velocity of the object as it is moving upward has the same magnitude as when it is moving downward. The only difference is that the directions of the two velocities are opposite. Figure 6-3b defines two quantities associated with the trajectory. One is the maximum height, which is the height of the projectile when the vertical velocity is zero and the projectile has only its horizontal-velocity component. The other quantity depicted is the range, R, which is the horizontal distance that the projectile travels. Not shown is the flight time, which is how much time the projectile is in the air. For football punts, flight time often is called hang time.

■

Figure 6-3 The vector sum of vx and vy at each position points in the direction of the flight.

a

vx ax 0

b ay

ay Maximum height

vy y

x

° Begin Range

150

Chapter 6 Motion in Two Dimensions

The Flight of a Ball A ball is launched at 4.5 m/s at 66° above the horizontal. What are the maximum height and flight time of the ball? 1

y

Analyze and Sketch the Problem • Establish a coordinate system with the initial position of the ball at the origin. • Show the positions of the ball at the beginning, at the maximum height, and at the end of the flight. • Draw a motion diagram showing v, a, and Fnet. Known: yi 0.0 m i 66°

vi 4.5 m/s 2

ymax vi °

Unknown:

R

ymax ? t?

ay g

v a vy0

Solve for the Unknown Find the y-component of vi . vyi vi(sin i) (4.5 m/s)(sin 66°) 4.1 m/s Find an expression for time. vy vyi ay t vyi gt t

x

Begin

vyi vy g

Fg Fnet

v0 vx0

Substitute vi 4.5 m/s, i 66°

Substitute ay g Solve for t.

Solve for the maximum height. 1 ymax yi + vyit + at 2

2 vyi vy vyi vy 2 vyi vy 1 yi + vyi + (g) Substitute t , a g g g g 2 4.1 m/s 0.0 m/s 1 4.1 m/s 0.0 m/s 2 0.0 m + (4.1 m/s) + (9.80 m/s2) 9.80 m/s2 2 9.80 m/s2

(

)

(

(

0.86 m

)

)

(

Substitute yi 0.0 m, vyi 4.1 m/s, vy 0.0 m/s at ymax , g 9.80 m/s2

)

Solve for the time to return to the launching height. 1 yf yi vyit at 2 2

1 2

0.0 m 0.0 m + vyit gt 2 v 4( g)(0.0 m) vyi

t

1 2

2 yi

(

1 2

)

2 g

vy i vyi

Substitute yf 0.0 m, yi 0.0 m, a g

Use the quadratic formula to solve for t.

Math Handbook

g

2vyi

g (2)(4.1 m/s) (9.8 0 m/s2)

0 is the time the ball left the launch, so use this solution.

Quadratic Formula page 846

Substitute vyi 4.1 m/s, g 9.80 m/s2

0.84 s 3

Evaluate the Answer • Are the units correct? Dimensional analysis verifies that the units are correct. • Do the signs make sense? All should be positive. • Are the magnitudes realistic? 0.84 s is fast, but an initial velocity of 4.5 m/s makes this time reasonable.

Section 6.1 Projectile Motion

151

4. A player kicks a football from ground level with an initial velocity of 27.0 m/s, 30.0° above the horizontal, as shown in Figure 6-4. Find each of the following. Assume that air resistance is negligible. a. the ball’s hang time

25

b. the ball’s maximum height

Trajectory

c. the ball’s range 5. The player in problem 4 then kicks the ball with the same speed, but at 60.0° from the horizontal. What is the ball’s hang time, range, and maximum height?

y (m)

60.0°

30.0° 0

60

x (m)

6. A rock is thrown from a 50.0-m-high cliff with an initial velocity of 7.0 m/s at an angle of 53.0° above the horizontal. Find the velocity vector for when it hits the ground below.

■

Figure 6-4

Trajectories Depend upon the Viewer Suppose you toss a ball up and catch it while riding in a bus. To you, the ball would seem to go straight up and straight down. But what would an observer on the sidewalk see? The observer would see the ball leave your hand, rise up, and return to your hand, but because the bus would be moving, your hand also would be moving. The bus, your hand, and the ball would all have the same horizontal velocity. Thus, the trajectory of the ball would be similar to that of the ball in Example Problem 1. Air resistance So far, air resistance has been ignored in the analysis of projectile motion. While the effects of air resistance are very small for some projectiles, for others, the effects are large and complex. For example, dimples on a golf ball reduce air resistance and maximize its range. In baseball, the spin of the ball creates forces that can deflect the ball. For now, just remember that the force due to air resistance does exist and it can be important.

6.1 Section Review 7. Projectile Motion Two baseballs are pitched horizontally from the same height, but at different speeds. The faster ball crosses home plate within the strike zone, but the slower ball is below the batter’s knees. Why does the faster ball not fall as far as the slower one? 8. Free-Body Diagram An ice cube slides without friction across a table at a constant velocity. It slides off the table and lands on the floor. Draw free-body and motion diagrams of the ice cube at two points on the table and at two points in the air. 9. Projectile Motion A softball is tossed into the air at an angle of 50.0° with the vertical at an initial velocity of 11.0 m/s. What is its maximum height? 152

Chapter 6 Motion in Two Dimensions

10. Projectile Motion A tennis ball is thrown out a window 28 m above the ground at an initial velocity of 15.0 m/s and 20.0° below the horizontal. How far does the ball move horizontally before it hits the ground? 11. Critical Thinking Suppose that an object is thrown with the same initial velocity and direction on Earth and on the Moon, where g is one-sixth that on Earth. How will the following quantities change? a. vx b. the object’s time of flight c. ymax d. R physicspp.com/self_check_quiz

6.2 Circular Motion

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onsider an object moving in a circle at a constant speed, such as a stone being whirled on the end of a string or a fixed horse on a merry-go-round. Are these objects accelerating? At first, you might think that they are not because their speeds do not change. However, remember that acceleration is the change in velocity, not just the change in speed. Because their direction is changing, the objects must be accelerating.

• Explain why an object moving in a circle at a constant speed is accelerated. • Describe how centripetal acceleration depends upon the object’s speed and the radius of the circle.

Describing Circular Motion Uniform circular motion is the movement of an object or particle trajectory at a constant speed around a circle with a fixed radius. The positions of an object in uniform circular motion, relative to the center of the circle, are given by the position vectors r1 and r2, shown in Figure 6-5a. As the object moves around the circle, the length of each position vector does not change, but its direction does. To find the object’s velocity, you need to find its displacement vector over a time interval. The change in position, or the object’s displacement, is represented by r. Figure 6-5b shows two position vectors: r1 at the beginning of a time interval, and r2 at the end of the time interval. Remember that a position vector is a displacement vector with its tail at the origin. In the vector diagram, r1 and r2 are subtracted to give the resultant r, the displacement during the time interval. You know that a moving object’s average velocity is d/t, so for an object in circular motion, v r/t. The velocity vector has the same direction as the displacement, but a different length. You can see in Figure 6-6a that the velocity is at right angles to the position vector, which is tangent to its circular path. As the velocity vector moves around the circle, its direction changes but its length remains the same. What is the direction of the object’s acceleration? Figure 6-6a shows the velocity vectors v1 and v2 at the beginning and end of a time interval. The difference in the two vectors, v, is found by subtracting the vectors, as shown in Figure 6-6b. The average acceleration, a v/t, is in the same direction as v; that is, toward the center of the circle. Repeat this process for several other time intervals when the object is in different locations on the circle. As the object moves around the circle, the direction of the acceleration vector changes, but its length remains the same. Notice that the acceleration vector of an object in uniform circular motion always points in toward the center of the circle. For this reason, the acceleration of such an object is called center-seeking or centripetal acceleration. v1

a

a r

v2

Objectives

• Identify the force that causes centripetal acceleration.

Vocabulary uniform circular motion centripetal acceleration centripetal force

v1

a r1

r2

v2

r

b r1 r2

■ Figure 6-5 The displacement, r, of an object in circular motion, divided by the time interval in which the displacement occurs, is the object’s average velocity during that time interval.

b v1 v2

v

■ Figure 6-6 The direction of the change in velocity is toward the center of the circle, and so the acceleration vector also points to the center of the circle.

Section 6.2 Circular Motion

153

Centripetal Acceleration Space Elevators Scientists are considering the use of space elevators as a low-cost transportation system to space. A cable would be anchored to a station at Earth’s equator, and the cable would extend almost 35,800 km from Earth’s surface. The cable would be attached to a counterweight and would stay extended due to centripetal force. Special magnetically powered vehicles would then travel along the cable.

What is the magnitude of an object’s centripetal acceleration? Compare the triangle made from the position vectors in Figure 6-5b with the triangle made by the velocity vectors in Figure 6-6b. The angle between r1 and r2 is the same as that between v1 and v2. Therefore, the two triangles formed by subtracting the two sets of vectors are similar triangles, and the ratios of the lengths of two corresponding sides are equal. Thus, r/r v/v. The equation is not changed if both sides are divided by t. r v rt vt

However, v r/t and a v/t. 1 r 1 v r t v t

Substituting v r/t in the left-hand side and a v/t in the right-hand side gives the following equation. v r

a v

Solve this equation for acceleration and give it the special symbol ac , for centripetal acceleration. Centripetal Acceleration

v2

ac r

Centripetal acceleration always points to the center of the circle. Its magnitude is equal to the square of the speed, divided by the radius of motion.

■ Figure 6-7 When the thrower lets go, the hammer initially moves in a straight line that is tangent to the point of release. Then it follows a trajectory like that of any object released into the air with an initial horizontal velocity.

How can you measure the speed of an object moving in a circle? One way is to measure its period, T, the time needed for the object to make one complete revolution. During this time, the object travels a distance equal to the circumference of the circle, 2 r. The object’s speed, then, is represented by v 2 r/T. If this expression is substituted for v in the equation for centripetal acceleration, the following equation is obtained. 2 r 2 T

4 2r T

ac 2 r

Because the acceleration of an object moving in a circle is always in the direction of the net force acting on it, there must be a net force toward the center of the circle. This force can be provided by any number of agents. For Earth circling the Sun, the force is the Sun’s gravitational force on Earth, as you’ll learn in Chapter 7. When a hammer thrower swings the hammer, as in Figure 6-7, the force is the tension in the chain attached to the massive ball. When an object moves in a circle, the net force toward the center of the circle is called the centripetal force. To accurately analyze centripetal acceleration situations, you must identify the agent of the force that causes the acceleration. Then you can apply Newton’s second law for the component in the direction of the acceleration in the following way. Newton’s Second Law for Circular Motion

Fnet mac

The net centripetal force on an object moving in a circle is equal to the object’s mass, times the centripetal acceleration.

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Chapter 6 Motion in Two Dimensions

Denis Charlet/AFP/Getty Images

When solving problems, you have found it useful to choose a coordinate system with one axis in the direction of the acceleration. For circular motion, the direction of the acceleration is always toward the center of the circle. Rather than labeling this axis x or y, call it c, for centripetal acceleration. The other axis is in the direction of the velocity, tangent to the circle. It is labeled tang for tangential. You will apply Newton’s second law in these directions, just as you did in the two-dimensional problems in Chapter 5. Remember that centripetal force is just another name for the net force in the centripetal direction. It is the sum of all the real forces, those for which you can identify agents that act along the centripetal axis. In the case of the hammer thrower in Figure 6-7, in what direction does the hammer fly when the chain is released? Once the contact force of the chain is gone, there is no force accelerating the hammer toward the center of the circle, so the hammer flies off in the direction of its velocity, which is tangent to the circle. Remember, if you cannot identify the agent of the force, then it does not exist.

Uniform Circular Motion A 13-g rubber stopper is attached to a 0.93-m string. The stopper is swung in a horizontal circle, making one revolution in 1.18 s. Find the tension force exerted by the string on the stopper. 1

Analyze and Sketch the Problem

r

m

• Draw a free-body diagram for the swinging stopper. • Include the radius and the direction of motion. • Establish a coordinate system labeled tang and c. The directions of a and FT are parallel to c.

2

Known:

Unknown:

m 13 g r 0.93 m T 1.18 s

FT ?

v2 a

v1

Solve for the Unknown

FT

tang

c

Find the centripetal acceleration. 4 2r T 4 2(0.93 m) (1.18 s)2

ac 2

Substitute r 0.93 m, T 1.18 s

26 m/s2 Use Newton’s second law to find the tension in the string. FT mac (0.013 kg)(26 m/s2) Substitute m 0.013 kg, ac 26 m/s2 0.34 N 3

Math Handbook Operations with Significant Digits pages 835–836

Evaluate the Answer • Are the units correct? Dimensional analysis verifies that a is in m/s2 and F is in N. • Do the signs make sense? The signs should all be positive. • Are the magnitudes realistic? The force is almost three times the weight of the stopper, and the acceleration is almost three times that of gravity, which is reasonable for such a light object.

Section 6.2 Circular Motion

155

12. A runner moving at a speed of 8.8 m/s rounds a bend with a radius of 25 m. What is the centripetal acceleration of the runner, and what agent exerts force on the runner? 13. A car racing on a flat track travels at 22 m/s around a curve with a 56-m radius. Find the car’s centripetal acceleration. What minimum coefficient of static friction between the tires and road is necessary for the car to round the curve without slipping? 14. An airplane traveling at 201 m/s makes a turn. What is the smallest radius of the circular path (in km) that the pilot can make and keep the centripetal acceleration under 5.0 m/s2? 15. A 45-kg merry-go-round worker stands on the ride’s platform 6.3 m from the center. If her speed as she goes around the circle is 4.1 m/s, what is the force of friction necessary to keep her from falling off the platform?

Path of car

Path of passenger without car

■ Figure 6-8 The passenger would move forward in a straight line if the car did not exert an inward force.

A Nonexistent Force If a car makes a sharp left turn, a passenger on the right side might be thrown against the right door. Is there an outward force on the passenger? Consider a similar situation. If a car in which you are riding stops suddenly, you will be thrown forward into your safety belt. Is there a forward force on you? No, because according to Newton’s first law, you will continue moving with the same velocity unless there is a net force acting on you. The safety belt applies the force that accelerates you to a stop. Figure 6-8 shows a car turning to the left as viewed from above. A passenger in the car would continue to move straight ahead if it were not for the force of the door acting in the direction of the acceleration; that is, toward the center of the circle. Thus, there is no outward force on the passenger. The socalled centrifugal, or outward force, is a fictitious, nonexistent force. Newton’s laws are able to explain motion in both straight lines and circles.

6.2 Section Review 16. Uniform Circular Motion What is the direction of the force that acts on the clothes in the spin cycle of a washing machine? What exerts the force? 17. Free-Body Diagram You are sitting in the backseat of a car going around a curve to the right. Sketch motion and free-body diagrams to answer the following questions. a. What is the direction of your acceleration? b. What is the direction of the net force that is acting on you? c. What exerts this force? 18. Centripetal Force If a 40.0-g stone is whirled horizontally on the end of a 0.60-m string at a speed of 2.2 m/s, what is the tension in the string? 156

Chapter 6 Motion in Two Dimensions

19. Centripetal Acceleration A newspaper article states that when turning a corner, a driver must be careful to balance the centripetal and centrifugal forces to keep from skidding. Write a letter to the editor that critiques this article. 20. Centripetal Force A bowling ball has a mass of 7.3 kg. If you move it around a circle with a radius of 0.75 m at a speed of 2.5 m/s, what force would you have to exert on it? 21. Critical Thinking Because of Earth’s daily rotation, you always move with uniform circular motion. What is the agent that supplies the force that accelerates you? How does this motion affect your apparent weight? physicspp.com/self_check_quiz

6.3 Relative Velocity

S

uppose that you are in a school bus that is traveling at a velocity of 8 m/s in a positive direction. You walk with a velocity of 3 m/s toward the front of the bus. If a friend of yours is standing on the side of the road watching the bus with you on it go by, how fast would your friend say that you are moving? If the bus is traveling at 8 m/s, this means that the velocity of the bus is 8 m/s, as measured by your friend in a coordinate system fixed to the road. When you are standing still, your velocity relative to the road is also 8 m/s, but your velocity relative to the bus is zero. Walking at 3 m/s toward the front of the bus means that your velocity is measured relative to the bus. The problem can be rephrased as follows: Given the velocity of the bus relative to the road and your velocity relative to the bus, what is your velocity relative to the road? A vector representation of this problem is shown in Figure 6-9a. After studying it, you will find that your velocity relative to the street is 11 m/s, the sum of 8 m/s and 3 m/s. Suppose that you now walk at the same speed toward the rear of the bus. What would be your velocity relative to the road? Figure 6-9b shows that because the two velocities are in opposite directions, the resultant velocity is 5 m/s, the difference between 8 m/s and 3 m/s. You can see that when the velocities are along the same line, simple addition or subtraction can be used to determine the relative velocity. Take a closer look at how these results were obtained and see if you can find a mathematical rule to describe how velocities are combined in these relative-velocity situations. For the above situation, you can designate the velocity of the bus relative to the road as vb/r, your velocity relative to the bus as vy/b, and the velocity of you relative to the road as vy/r. To find the velocity of you relative to the road in both cases, you vectorially added the velocities of you relative to the bus and the bus relative to the road. Mathematically, this is represented as vy/b vb/r vy/r. The more general form of this equation is as follows. Relative Velocity

va/b vb/c va/c

The relative velocity of object a to object c is the vector sum of object a’s velocity relative to object b and object b’s velocity relative to object c.

Objectives • Analyze situations in which the coordinate system is moving. • Solve relative-velocity problems.

a vbus relative to street vyou relative to bus vyou relative to street

b vbus relative to street vyou relative to bus vyou relative to street ■ Figure 6-9 When a coordinate system is moving, two velocities are added if both motions are in the same direction and one is subtracted from the other if the motions are in opposite directions.

Phillipe whirls a stone of mass m on a rope in a perfect horizontal circle above his head such that the stone is at a height, h, above the ground. The circle has a radius of r, and the tension in the rope is T. Suddenly the rope breaks and the stone falls to the ground. The stone travels a horizontal distance, s, from the time the rope breaks until it impacts the ground. Find a mathematical expression for s in terms of T, r, m, and h. Does your expression change if Phillipe is walking 0.50 m/s relative to the ground?

Section 6.3 Relative Velocity

157

vair relative to ground

vplane relative to air

vplane relative to ground

■

Figure 6-10 The plane’s velocity relative to the ground can be obtained by vector addition.

This method for adding relative velocities also applies to motion in two dimensions. For example, airline pilots cannot expect to reach their destinations by simply aiming their planes along a compass direction. They must take into account the plane’s speed relative to the air, which is given by their airspeed indicators, and their direction of flight relative to the air. They also must consider the velocity of the wind at the altitude they are flying relative to the ground. These two vectors must be combined, as shown in Figure 6-10, to obtain the velocity of the airplane relative to the ground. The resultant vector tells the pilot how fast and in what direction the plane must travel relative to the ground to reach its destination. A similar situation occurs for boats traveling on water with a flowing current.

Relative Velocity of a Marble Ana and Sandra are riding on a ferry boat that is traveling east at a speed of 4.0 m/s. Sandra rolls a marble with a velocity of 0.75 m/s north, straight across the deck of the boat to Ana. What is the velocity of the marble relative to the water? 1

Analyze and Sketch the Problem • Establish a coordinate system. • Draw vectors to represent the velocities of the boat relative to the water and the marble relative to the boat.

2

Known:

Unknown:

vb/w 4.0 m/s vm/b 0.75 m/s

vm/w ?

Solve for the Unknown Because the two velocities are at right angles, use the Pythagorean theorem. vm/w2 vb/w2 vm/b2

vm/b

y

vm/w vb/w

x

vm/w vb/w2 vm/b2 2 (4.0 m/s) (0.75 m/s)2

Substitute vb/w 4.0 m/s, vm/b 0.75 m/s

4.1 m/s

Math Handbook Inverses of Sine, Cosine, and Tangent page 856

Find the angle of the marble’s motion.

( ) ( ) v vb/w

/b tan1 m

0.75 m/s tan1 4.0 m/s

Substitute vb/w 4.0 m/s, vm/b 0.75 m/s

11° north of east The marble is traveling 4.1 m/s at 11° north of east. 3

Evaluate the Answer • Are the units correct? Dimensional analysis verifies that the velocity is in m/s. • Do the signs make sense? The signs should all be positive. • Are the magnitudes realistic? The resulting velocity is of the same order of magnitude as the velocities given in the problem.

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Chapter 6 Motion in Two Dimensions

22. You are riding in a bus moving slowly through heavy traffic at 2.0 m/s. You hurry to the front of the bus at 4.0 m/s relative to the bus. What is your speed relative to the street? 23. Rafi is pulling a toy wagon through the neighborhood at a speed of 0.75 m/s. A caterpillar in the wagon is crawling toward the rear of the wagon at a rate of 2.0 cm/s. What is the caterpillar’s velocity relative to the ground? 24. A boat is rowed directly upriver at a speed of 2.5 m/s relative to the water. Viewers on the shore see that the boat is moving at only 0.5 m/s relative to the shore. What is the speed of the river? Is it moving with or against the boat? 25. An airplane flies due north at 150 km/h relative to the air. There is a wind blowing at 75 km/h to the east relative to the ground. What is the plane’s speed relative to the ground?

Another example of combined relative velocities is the navigation of migrating neotropical songbirds. In addition to knowing in which direction to fly, a bird must account for its speed relative to the air and its direction relative to the ground. If a bird tries to fly over the Gulf of Mexico into too strong a headwind, it will run out of energy before it reaches the other shore and will perish. Similarly, the bird must account for crosswinds or it will not reach its destination. You can add relative velocities even if they are at arbitrary angles by using the graphical methods that you learned in Chapter 5. Remember that the key to properly analyzing a two-dimensional relative-velocity situation is drawing the proper triangle to represent the three velocities. Once you have this triangle, you simply apply your knowledge of vector addition from Chapter 5. If the situation contains two velocities that are perpendicular to each other, you can find the third by applying the Pythagorean theorem; however, if the situation has no right angles, you will need to use one or both of the laws of sines and cosines.

Biology Connection

6.3 Section Review 26. Relative Velocity A fishing boat with a maximum speed of 3 m/s relative to the water is in a river that is flowing at 2 m/s. What is the maximum speed the boat can obtain relative to the shore? The minimum speed? Give the direction of the boat, relative to the river’s current, for the maximum speed and the minimum speed relative to the shore. 27. Relative Velocity of a Boat A powerboat heads due northwest at 13 m/s relative to the water across a river that flows due north at 5.0 m/s. What is the velocity (both magnitude and direction) of the motorboat relative to the shore? 28. Relative Velocity An airplane flies due south at 175 km/h relative to the air. There is a wind blowing at 85 km/h to the east relative to the ground. What are the plane’s speed and direction relative to the ground? physicspp.com/self_check_quiz

29. A Plane’s Relative Velocity An airplane flies due north at 235 km/h relative to the air. There is a wind blowing at 65 km/h to the northeast relative to the ground. What are the plane’s speed and direction relative to the ground? 30. Relative Velocity An airplane has a speed of 285 km/h relative to the air. There is a wind blowing at 95 km/h at 30.0° north of east relative to Earth. In which direction should the plane head to land at an airport due north of its present location? What is the plane’s speed relative to the ground? 31. Critical Thinking You are piloting a boat across a fast-moving river. You want to reach a pier directly opposite your starting point. Describe how you would navigate the boat in terms of the components of your velocity relative to the water. Section 6.3 Relative Velocity

159

On Target In this activity, you will analyze several factors that affect the motion of a projectile and use your understanding of these factors to predict the path of a projectile. Finally, you will design a projectile launcher and hit a target a known distance away.

QUESTION What factors affect the path of a projectile?

Objectives

Procedure

■ Formulate models and then summarize the

1. Brainstorm and list as many factors as you are able to think of that may affect the path of a projectile.

factors that affect the motion of a projectile. ■ Use models to predict where a projectile will land.

Safety Precautions

2. Create a design for your projectile launcher and decide what object will be your projectile shot by your launcher. 3. Taking the design of your launcher into account, determine which two factors are most likely to have a significant effect on the flight path of your projectile.

Possible Materials duct tape plastic ware rubber bands paper clips paper masking tape wood blocks nails

160 Horizons Companies

hammer PVC tubing handsaw scissors coat hanger chicken wire wire cutter

4. Check the design of your launcher and discuss your two factors with your teacher and make any necessary changes to your setup before continuing. 5. Create a method for determining what effect these two factors will have on the path of your projectile. 6. Have your teacher approve your method before collecting data.

Data Table 1 Launch Angle (deg)

Distance Projectile Travels (cm)

Data Table 2 Distance Rubber Band Is Pulled Back (cm)

Distance Projectile Travels (cm)

Analyze

Going Further

1. Make and Use Graphs Make graphs of your data to help you predict how to use your launcher to hit a target.

1. How might your data have varied if you did this experiment outside? Would there be any additional factors affecting the motion of your projectile?

2. Analyze What are the relationships between each variable you have tested and the distance the projectile travels?

Conclude and Apply 1. What were the main factors influencing the path of the projectile? 2. Predict the conditions necessary to hit a target provided by your teacher. 3. Explain If you have a perfect plan and still miss the target on your first try, is there a problem with the variability of laws of physics? Explain. 4. Launch your projectile at the target. If you miss, make the necessary adjustments and try again.

2. How might the results of your experiment be different if the target was elevated above the height of the launcher? 3. How might your experiment differ if the launcher was elevated above the height of the target?

Real-World Physics 1. When a kicker attempts a field goal, do you think it is possible for him to miss because he kicked it too high? Explain. 2. If you wanted to hit a baseball as far as possible, what would be the best angle to hit the ball?

To find out more about projectile motion, visit the Web site: physicspp.com

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Spinning Space Stations There is a lot going on aboard the International Space Station (ISS). Scientists from different countries are conducting experiments and making observations. They have seen water drops form as floating spheres and have grown peas in space to test whether crops can be grown in weightlessness. One goal of the ISS is to examine the effects on the human body when living in space for prolonged periods of time. If negative health effects can be identified, perhaps they can be prevented. This could give humans the option of living in space for long periods of time. Harmful effects of weightlessness have been observed. On Earth, muscles have gravity to push and pull against. Muscles weaken from disuse if this resistance is removed. Bones can weaken for the same reason. Also, blood volume can decrease. On Earth, gravity pulls blood downward so it collects in the lower legs. In weightlessness, the blood can more easily collect in an astronaut’s head. The brain senses the extra blood and sends a signal to make less of it. Long-term life in space is hindered by the practical challenges of weightlessness as well. Imagine how daily life would change. Everything must be strapped or bolted down. You would have to be strapped down to a bed to sleep in one. Your life would be difficult in a space station unless the space station could be modified to simulate gravity. How could this be done?

The Rotating Space Station Have you ever been on a human centrifuge—a type of amusement park ride that uses centripetal force? Everyone stands against the walls of a big cylinder. Then the cylinder begins to rotate faster and faster until the riders are pressed against the walls. Because of the centripetal acceleration, the riders are held there so that even when the floor drops down they are held securely against the walls of the whirling container. A space station could be designed that uses the effects of centripetal motion as a replacement for gravity. Imagine a space station in the form of a large ring. The space station and all the objects and occupants inside would float weightlessly inside. If the ring were made to spin, unattached objects would be held against the ring’s outer edge because of the centripetal motion. If the space station spun 162

Future Technology

This is an artist’s rendition of a rotating space station.

at the right rate and if it had the right diameter, the centripetal motion would cause the occupants to experience a force of the same magnitude as gravity. The down direction in the space station would be what an observer outside the station would see as radially outward, away from the ring’s center. Centripetal acceleration is directly proportional to the distance from the center of a rotating object. A rotating space station could be built in the form of concentric rings, each ring experiencing a different gravity. The innermost rings would experience the smallest gravity, while outermost rings would experience the largest force. You could go from floating peacefully in a low-gravity ring to standing securely in the simulated Earth-gravity ring.

Going Further 1. Research What factors must engineers take into account in order to make a rotating space station that can simulate Earth’s gravity? 2. Apply You are an astronaut aboard a rotating space station. You feel pulled by gravity against the floor. Explain what is really going on in terms of Newton’s laws and centripetal force. 3. Critical Thinking What benefits does a rotating space station offer its occupants? What are the negative features?

6.1 Projectile Motion Vocabulary

Key Concepts

• projectile (p. 147) • trajectory (p. 147)

• • • • • •

The vertical and horizontal motions of a projectile are independent. The vertical motion component of a projectile experiences a constant acceleration. When there is no air resistance, the horizontal motion component does not experience an acceleration and has constant velocity. Projectile problems are solved by first using the vertical motion to relate height, time in the air, and initial vertical velocity. Then the distance traveled horizontally is found. The range of a projectile depends upon the acceleration due to gravity and upon both components of the initial velocity. The curved flight path that is followed by a projectile is called a parabola.

6.2 Circular Motion Vocabulary

Key Concepts

• uniform circular motion

•

(p. 153)

• centripetal acceleration

•

(p. 153)

An object moving in a circle at a constant speed accelerates toward the center of the circle, and therefore, it has centripetal acceleration. Centripetal acceleration depends directly on the square of the object’s speed and inversely on the radius of the circle.

• centripetal force (p. 154)

v2 r

ac

•

The centripetal acceleration for an object traveling in a circle can also be expressed as a function of its period, T. 4 2r T

ac 2

•

A net force must be exerted toward the circle’s center to cause centripetal acceleration. Fnet mac

•

The velocity vector of an object with a centripetal acceleration is always tangent to the circular path.

6.3 Relative Velocity Key Concepts

•

Vector addition can be used to solve problems involving relative velocities. va/b vb/c va/c

•

The key to properly analyzing a two-dimensional relative-velocity problem is drawing the proper triangle to represent all three velocity vectors.

physicspp.com/vocabulary_puzzlemaker

163

Concept Mapping 32. Use the following terms to complete the concept map below: constant speed, horizontal part of projectile motion, constant acceleration, relative-velocity motion, uniform circular motion.

37. To obtain uniform circular motion, how must the net force depend on the speed of the moving object? (6.2)

38. If you whirl a yo-yo about your head in a horizontal circle, in what direction must a force act on the yo-yo? What exerts the force? (6.2)

39. Why is it that a car traveling in the opposite

Categories of Motion

direction as the car in which you are riding on the freeway often looks like it is moving faster than the speed limit? (6.3) constant velocity

Applying Concepts 40. Projectile Motion Analyze how horizontal motion can be uniform while vertical motion is accelerated. How will projectile motion be affected when drag due to air resistance is taken into consideration?

vertical part of projectile motion

41. Baseball A batter hits a pop-up straight up over

Mastering Concepts 33. Consider the trajectory of the cannonball shown in Figure 6-11. (6.1) a. Where is the magnitude of the vertical-velocity component largest? b. Where is the magnitude of the horizontalvelocity component largest? c. Where is the vertical-velocity smallest? d. Where is the magnitude of the acceleration smallest?

A

B

C D E

■

Figure 6-11

34. A student is playing with a radio-controlled race car on the balcony of a sixth-floor apartment. An accidental turn sends the car through the railing and over the edge of the balcony. Does the time it takes the car to fall depend upon the speed it had when it left the balcony? (6.1)

35. An airplane pilot flying at constant velocity and altitude drops a heavy crate. Ignoring air resistance, where will the plane be relative to the crate when the crate hits the ground? Draw the path of the crate as seen by an observer on the ground. (6.1)

36. Can you go around a curve with the following accelerations? Explain. a. zero acceleration b. constant acceleration (6.2)

164

home plate at an initial velocity of 20 m/s. The ball is caught by the catcher at the same height that it was hit. At what velocity does the ball land in the catcher’s mitt? Neglect air resistance.

42. Fastball In baseball, a fastball takes about 12 s to

reach the plate. Assuming that such a pitch is thrown horizontally, compare the distance the ball falls in the first 14 s with the distance it falls in the second 14 s.

43. You throw a rock horizontally. In a second horizontal throw, you throw the rock harder and give it even more speed. a. How will the time it takes the rock to hit the ground be affected? Ignore air resistance. b. How will the increased speed affect the distance from where the rock left your hand to where the rock hits the ground?

44. Field Biology A zoologist standing on a cliff aims a tranquilizer gun at a monkey hanging from a distant tree branch. The barrel of the gun is horizontal. Just as the zoologist pulls the trigger, the monkey lets go and begins to fall. Will the dart hit the monkey? Ignore air resistance.

45. Football A quarterback throws a football at 24 m/s at a 45° angle. If it takes the ball 3.0 s to reach the top of its path and the ball is caught at the same height at which it is thrown, how long is it in the air? Ignore air resistance.

46. Track and Field You are working on improving your performance in the long jump and believe that the information in this chapter can help. Does the height that you reach make any difference to your jump? What influences the length of your jump?

Chapter 6 Motion in Two Dimensions For more problems, go to Additional Problems, Appendix B.

47. Imagine that you are sitting in a car tossing a ball straight up into the air. a. If the car is moving at a constant velocity, will the ball land in front of, behind, or in your hand? b. If the car rounds a curve at a constant speed, where will the ball land?

48. You swing one yo-yo around your head in a

53. A dart player throws a dart horizontally at 12.4 m/s. The dart hits the board 0.32 m below the height from which it was thrown. How far away is the player from the board?

54. The two baseballs in Figure 6-13 were hit with the same speed, 25 m/s. Draw separate graphs of y versus t and x versus t for each ball.

horizontal circle. Then you swing another yo-yo with twice the mass of the first one, but you don’t change the length of the string or the period. How do the tensions in the strings differ?

A

49. Car Racing The curves on a race track are banked to make it easier for cars to go around the curves at high speeds. Draw a free-body diagram of a car on a banked curve. From the motion diagram, find the direction of the acceleration. a. What exerts the force in the direction of the acceleration? b. Can you have such a force without friction?

50. Driving on the Highway Explain why it is that when you pass a car going in the same direction as you on the freeway, it takes a longer time than when you pass a car going in the opposite direction.

Mastering Problems

60°

yA

B yB

30° xA xB ■

Figure 6-13

55. Swimming You took a running leap off a highdiving platform. You were running at 2.8 m/s and hit the water 2.6 s later. How high was the platform, and how far from the edge of the platform did you hit the water? Ignore air resistance.

56. Archery An arrow is shot at 30.0° above the

6.1 Projectile Motion 51. You accidentally throw your car keys horizontally at 8.0 m/s from a cliff 64-m high. How far from the base of the cliff should you look for the keys?

52. The toy car in Figure 6-12 runs off the edge of a table that is 1.225-m high. The car lands 0.400 m from the base of the table. a. How long did it take the car to fall? b. How fast was the car going on the table?

horizontal. Its velocity is 49 m/s, and it hits the target. a. What is the maximum height the arrow will attain? b. The target is at the height from which the arrow was shot. How far away is it?

57. Hitting a Home Run A pitched ball is hit by a batter at a 45° angle and just clears the outfield fence, 98 m away. If the fence is at the same height as the pitch, find the velocity of the ball when it left the bat. Ignore air resistance.

58. At-Sea Rescue An airplane traveling 1001 m above

v

the ocean at 125 km/h is going to drop a box of supplies to shipwrecked victims below. a. How many seconds before the plane is directly overhead should the box be dropped? b. What is the horizontal distance between the plane and the victims when the box is dropped? 1.225 m

59. Diving Divers in Acapulco dive from a cliff that is 61 m high. If the rocks below the cliff extend outward for 23 m, what is the minimum horizontal velocity a diver must have to clear the rocks?

60. Jump Shot A basketball player is trying to make 0.400 m ■

Figure 6-12 physicspp.com/chapter_test

a half-court jump shot and releases the ball at the height of the basket. Assuming that the ball is launched at 51.0°, 14.0 m from the basket, what speed must the player give the ball? Chapter 6 Assessment

165

6.2 Circular Motion 61. Car Racing A 615-kg racing car completes one lap in 14.3 s around a circular track with a radius of 50.0 m. The car moves at a constant speed. a. What is the acceleration of the car? b. What force must the track exert on the tires to produce this acceleration?

62. Hammer Throw An athlete whirls a 7.00-kg hammer

67. A 75-kg pilot flies a plane in a loop as shown in Figure 6-15. At the top of the loop, when the plane is completely upside-down for an instant, the pilot hangs freely in the seat and does not push against the seat belt. The airspeed indicator reads 120 m/s. What is the radius of the plane’s loop? vtang 120 m/s

1.8 m from the axis of rotation in a horizontal circle, as shown in Figure 6-14. If the hammer makes one revolution in 1.0 s, what is the centripetal acceleration of the hammer? What is the tension in the chain? vtang

■

Figure 6-15

6.3 Relative Velocity 68. Navigating an Airplane An airplane flies at ■

Figure 6-14

63. A coin is placed on a vinyl stereo record that is

making 3313 revolutions per minute on a turntable. a. In what direction is the acceleration of the coin? b. Find the magnitude of the acceleration when the coin is placed 5.0, 10.0, and 15.0 cm from the center of the record. c. What force accelerates the coin? d. At which of the three radii in part b would the coin be most likely to fly off the turntable? Why?

64. A rotating rod that is 15.3 cm long is spun with its axis through one end of the rod so that the other end of the rod has a speed of 2010 m/s (4500 mph). a. What is the centripetal acceleration of the end of the rod? b. If you were to attach a 1.0-g object to the end of the rod, what force would be needed to hold it on the rod?

65. Friction provides the force needed for a car to travel

200.0 km/h relative to the air. What is the velocity of the plane relative to the ground if it flies during the following wind conditions? a. a 50.0-km/h tailwind b. a 50.0-km/h headwind

69. Odina and LaToya are sitting by a river and decide to have a race. Odina will run down the shore to a dock, 1.5 km away, then turn around and run back. LaToya will also race to the dock and back, but she will row a boat in the river, which has a current of 2.0 m/s. If Odina’s running speed is equal to LaToya’s rowing speed in still water, which is 4.0 m/s, who will win the race? Assume that they both turn instantaneously.

70. Crossing a River You row a boat, such as the one in Figure 6-16, perpendicular to the shore of a river that flows at 3.0 m/s. The velocity of your boat is 4.0 m/s relative to the water. a. What is the velocity of your boat relative to the shore? b. What is the component of your velocity parallel to the shore? Perpendicular to it?

66. A carnival clown rides a motorcycle down a ramp and around a vertical loop. If the loop has a radius of 18 m, what is the slowest speed the rider can have at the top of the loop to avoid falling? Hint: At this slowest speed, the track exerts no force on the motorcycle at the top of the loop.

166

vb

around a flat, circular race track. What is the maximum speed at which a car can safely travel if the radius of the track is 80.0 m and the coefficient of friction is 0.40? vw

■

Chapter 6 Motion in Two Dimensions For more problems, go to Additional Problems, Appendix B.

Figure 6-16

71. Studying the Weather A weather station releases a balloon to measure cloud conditions that rises at a constant 15 m/s relative to the air, but there is also a wind blowing at 6.5 m/s toward the west. What are the magnitude and direction of the velocity of the balloon?

72. Boating You are boating on a river that flows toward the east. Because of your knowledge of physics, you head your boat 53° west of north and have a velocity of 6.0 m/s due north relative to the shore. a. What is the velocity of the current? b. What is the speed of your boat relative to the water?

73. Air Travel You are piloting a small plane, and you want to reach an airport 450 km due south in 3.0 h. A wind is blowing from the west at 50.0 km/h. What heading and airspeed should you choose to reach your destination in time?

78. A 1.13-kg ball is swung vertically from a 0.50-m cord in uniform circular motion at a speed of 2.4 m/s. What is the tension in the cord at the bottom of the ball’s motion?

79. Banked Roads Curves on roads often are banked to help prevent cars from slipping off the road. If the posted speed limit for a particular curve of radius 36.0 m is 15.7 m/s (35 mph), at what angle should the road be banked so that cars will stay on a circular path even if there were no friction between the road and the tires? If the speed limit was increased to 20.1 m/s (45 mph), at what angle should the road be banked?

80. The 1.45-kg ball in Figure 6-17 is suspended from a 0.80-m string and swung in a horizontal circle at a constant speed such that the string makes an angle of 14.0° with the vertical. a. What is the tension in the string? b. What is the speed of the ball?

Mixed Review 74. Early skeptics of the idea of a rotating Earth said that the fast spin of Earth would throw people at the equator into space. The radius of Earth is about 6.38103 km. Show why this idea is wrong by calculating the following. a. the speed of a 97-kg person at the equator b. the force needed to accelerate the person in the circle c. the weight of the person d. the normal force of Earth on the person, that is, the person’s apparent weight

75. Firing a Missile An airplane, moving at 375 m/s relative to the ground, fires a missile forward at a speed of 782 m/s relative to the plane. What is the speed of the missile relative to the ground?

76. Rocketry A rocket in outer space that is moving at a speed of 1.25 km/s relative to an observer fires its motor. Hot gases are expelled out the back at 2.75 km/s relative to the rocket. What is the speed of the gases relative to the observer?

77. Two dogs, initially separated by 500.0 m, are running towards each other, each moving with a constant speed of 2.5 m/s. A dragonfly, moving with a constant speed of 3.0 m/s, flies from the nose of one dog to the other, then turns around instantaneously and flies back to the other dog. It continues to fly back and forth until the dogs run into each other. What distance does the dragonfly fly during this time? physicspp.com/chapter_test

14.0°

■

Figure 6-17

81. A baseball is hit directly in line with an outfielder at an angle of 35.0° above the horizontal with an initial velocity of 22.0 m/s. The outfielder starts running as soon as the ball is hit at a constant velocity of 2.5 m/s and barely catches the ball. Assuming that the ball is caught at the same height at which it was hit, what was the initial separation between the hitter and outfielder? Hint: There are two possible answers.

82. A Jewel Heist You are serving as a technical consultant for a locally produced cartoon. In one episode, two criminals, Shifty and Lefty, have stolen some jewels. Lefty has the jewels when the police start to chase him, and he runs to the top of a 60.0-m tall building in his attempt to escape. Meanwhile, Shifty runs to the convenient hot-air balloon 20.0 m from the base of the building and untethers it, so it begins to rise at a constant speed. Lefty tosses the bag of jewels horizontally with a speed of 7.3 m/s just as the balloon begins its ascent. What must the velocity of the balloon be for Shifty to easily catch the bag? Chapter 6 Assessment

167

Thinking Critically 83. Apply Concepts Consider a roller-coaster loop like the one in Figure 6-18. Are the cars traveling through the loop in uniform circular motion? Explain.

■

Figure 6-18

84. Use Numbers A 3-point jump shot is released 2.2 m above the ground and 6.02 m from the basket. The basket is 3.05 m above the floor. For launch angles of 30° and 60°, find the speed the ball needs to be thrown to make the basket.

85. Analyze For which angle in problem 84 is it more important that the player get the speed right? To explore this question, vary the speed at each angle by 5 percent and find the change in the range of the attempted shot.

86. Apply Computers and Calculators A baseball player hits a belt-high (1.0 m) fastball down the left-field line. The ball is hit with an initial velocity of 42.0 m/s at 26°. The left-field wall is 96.0 m from home plate at the foul pole and is 14-m high. Write the equation for the height of the ball, y, as a function of its distance from home plate, x. Use a computer or graphing calculator to plot the path of the ball. Trace along the path to find how high above the ground the ball is when it is at the wall. a. Is the hit a home run? b. What is the minimum speed at which the ball could be hit and clear the wall? c. If the initial velocity of the ball is 42.0 m/s, for what range of angles will the ball go over the wall?

87. Analyze Albert Einstein showed that the rule you learned for the addition of velocities does not work for objects moving near the speed of light. For example, if a rocket moving at velocity vA releases a missile that has velocity vB relative to the rocket, then the velocity of the missile relative to an observer that is at rest is given by v (vA vB )/(1 vAvB /c2), where c is the speed of light, 3.00108 m/s. This formula gives the correct values for objects moving at slow speeds as well. Suppose a rocket moving at 11 km/s shoots a laser beam out in front of it. What speed would an unmoving observer find for the laser light? Suppose that a rocket moves at a speed c/2, half the speed of light, and shoots a missile forward at a speed of c/2 relative to the rocket. How fast would the missile be moving relative to a fixed observer?

168

88. Analyze and Conclude A ball on a light string moves in a vertical circle. Analyze and describe the motion of this system. Be sure to consider the effects of gravity and tension. Is this system in uniform circular motion? Explain your answer.

Writing in Physics 89. Roller Coasters If you take a look at vertical loops on roller coasters, you will notice that most of them are not circular in shape. Research why this is so and explain the physics behind this decision by the coaster engineers.

90. Many amusement-park rides utilize centripetal acceleration to create thrills for the park’s customers. Choose two rides other than roller coasters that involve circular motion and explain how the physics of circular motion creates the sensations for the riders.

Cumulative Review 91. Multiply or divide, as indicated, using significant digits correctly. (Chapter 1) a. (5108 m)(4.2107 m) b. (1.67102 km)(8.5106 km) c. (2.6104 kg)/(9.4103 m3) d. (6.3101 m)/(3.8102 s)

92. Plot the data in Table 6-1 on a position-time graph. Find the average velocity in the time interval between 0.0 s and 5.0 s. (Chapter 3)

Table 6-1 Position v. Time Clock Reading t (s)

Position d (m)

0.0 1.0 2.0 3.0 4.0 5.0

30 30 35 45 60 70

93. Carlos and his older brother Ricardo are at the grocery store. Carlos, with mass 17.0 kg, likes to hang on the front of the cart while Ricardo pushes it, even though both boys know this is not safe. Ricardo pushes the cart, with mass 12.4 kg, with his brother hanging on it such that they accelerate at a rate of 0.20 m/s2. (Chapter 4) a. With what force is Ricardo pushing? b. What is the force the cart exerts on Carlos?

Chapter 6 Motion in Two Dimensions For more problems, go to Additional Problems, Appendix B.

Bill Aron/PhotoEdit

Multiple Choice 1. A 1.60-m-tall girl throws a football at an angle of 41.0° from the horizontal and at an initial velocity of 9.40 m/s. How far away from the girl will it land? 4.55 m

8.90 m

5.90 m

10.5 m

7. An orange is dropped at the same time a bullet is shot from a gun. Which of the following is true? The acceleration due to gravity is greater for the orange because the orange is heavier. Gravity acts less on the bullet than on the orange because the bullet is moving so fast. The velocities will be the same.

2. A dragonfly is sitting on a merry-go-round 2.8 m from the center. If the tangential velocity of the ride is 0.89 m/s, what is the centripetal acceleration of the dragonfly? 0.11 m/s2

0.32 m/s2

0.28 m/s2

2.2 m/s2

3. The centripetal force on a 0.82-kg object on the end of a 2.0-m massless string being swung in a horizontal circle is 4.0 N. What is the tangential velocity of the object? 2.8 m/s2

4.9 m/s2

3.1 m/s2

9.8 m/s2

The two objects will hit the ground at the same time.

Extended Answer 8. A colorfully feathered lead cannonball is shot horizontally out of a circus cannon 25 m/s from the high-wire platform on one side of a circus ring. If the high-wire platform is 52 m above the 80-m diameter ring, will the performers need to adjust their cannon (will the ball land inside the ring, or past it)? v

4. A 1000-kg car enters an 80.0-m-radius curve at 20.0 m/s. What centripetal force must be supplied by friction so the car does not skid? 5.0 N

5.0103 N

2.5102 N

1.0103 N

52 m

80 m (Not to scale)

5. A jogger on a riverside path sees a rowing team coming toward him. If the jogger is moving at 10 km/h, and the boat is moving at 20 km/h, how quickly does the jogger approach the boat? 3 m/s

40 m/s

8 m/s

100 m/s

6. What is the maximum height obtained by a 125-g apple that is slung from a slingshot at an angle of 78° from the horizontal with an initial velocity of 18 m/s? 0.70 m

32 m

16 m

33 m physicspp.com/standardized_test

9. A mythical warrior swings a 5.6-kg mace on the end of a magically massless 86-cm chain in a horizontal circle above his head. The mace makes one full revolution in 1.8 s. Find the tension in the magical chain.

Practice Under Testlike Conditions Answer all of the questions in the time provided without referring to your book. Did you complete the test? Could you have made better use of your time? What topics do you need to review?

Chapter 6 Standardized Test Practice

169

What You’ll Learn • You will learn the nature of gravitational force. • You will relate Kepler’s laws of planetary motion to Newton’s laws of motion. • You will describe the orbits of planets and satellites using the law of universal gravitation.

Why It’s Important Kepler’s laws and the law of universal gravitation will help you understand the motion of planets and satellites. Comets Comet Hale-Bopp was discovered by Alan Hale and Thomas Bopp in 1995. The comet entered the inner solar system in 1997 and was visible from Joshua Tree National Park in California, providing spectacular views of its white dust tail and blue ion tail.

Think About This Comets orbit the Sun just as planets and stars do. How can you describe the orbit of a comet such as Hale-Bopp?

physicspp.com 170

Wally Pacholka/Astropics.com

Can you model Mercury’s motion? Question Do planets in our solar system have circular orbits or do they travel in some other path? Procedure 1. Use the data table to plot the orbit of Mercury using the scale 10 cm 1 AU. Note that one astronomical unit, AU, is Earth’s distance from the Sun. 1 AU is equal to 1.5108 km. 2. Calculate the distance in cm for each distance measured in AU. 3. Mark the center of your paper and draw a horizontal zero line and a vertical zero line going through it. 4. Place your protractor on the horizontal line and center it on the center point. Measure the degrees and place a mark. 5. Place a ruler connecting the center and the angle measurement. Mark the distance in centimeters for the corresponding angle. You will need to place the protractor on the vertical zero line for certain angle measurements.

6. Once you have marked all the data points, draw a line connecting them. Analysis

Mercury’s Orbit

Describe the shape of Mercury’s orbit. Draw a line going through the Sun that represents the longest axis of the orbit, called the major axis. Critical Thinking How does the orbit of Mercury compare to the orbit of comet Hale-Bopp, shown on page 170?

(°)

d (AU)

4 61 122 172 209 239 266 295 330 350

0.35 0.31 0.32 0.38 0.43 0.46 0.47 0.44 0.40 0.37

7.1 Planetary Motion and Gravitation

S

ince ancient times, the Sun, Moon, planets, and stars had been assumed to revolve around Earth. Nicholas Copernicus, a Polish astronomer, noticed that the best available observations of the movements of planets and stars did not fully agree with the Earth-centered model. The results of his many years of work were published in 1543, when Copernicus was on his deathbed. His book showed that the motion of planets is much more easily understood by assuming that Earth and other planets revolve around the Sun. Tycho Brahe was born a few years after the death of Copernicus. As a boy of 14 in Denmark, Brahe observed an eclipse of the Sun on August 21, 1560, and vowed to become an astronomer. Brahe studied astronomy as he traveled throughout Europe for five years. He did not use telescopes. Instead, he used huge instruments that he designed and built in his own shop on the Danish island of Hven. He spent the next 20 years carefully recording the exact positions of the planets and stars. Brahe concluded that the Sun and the Moon orbit Earth and that all other planets orbit the Sun.

Objectives • Relate Kepler’s laws to the law of universal gravitation. • Calculate orbital speeds and periods. • Describe the importance of Cavendish’s experiment.

Vocabulary Kepler’s first law Kepler’s second law Kepler’s third law gravitational force law of universal gravitation

Section 7.1 Planetary Motion and Gravitation

171

a

■ Figure 7-1 Among the huge astronomical instruments that Tycho Brahe constructed to use on Hven (a) were an astrolabe (b) and a sextant (c).

b

c

Kepler’s Laws Johannes Kepler, a 29-year-old German, became one of Brahe’s assistants when he moved to Prague. Brahe trained his assistants to use instruments, such as those shown in Figure 7-1. Upon his death in 1601, Kepler inherited 30 years’ worth of Brahe’s observations. He studied Brahe’s data and was convinced that geometry and mathematics could be used to explain the number, distance, and motion of the planets. Kepler believed that the Sun exerted a force on the planets and placed the Sun at the center of the system. After several years of careful analysis of Brahe’s data on Mars, Kepler discovered the laws that describe the motion of every planet and satellite. Kepler’s first law states that the paths of the planets are ellipses, with the Sun at one focus. An ellipse has two foci, as shown in Figure 7-2. Like planets and stars, comets also orbit the Sun in elliptical orbits. Comets are divided into two groups—long-period comets and short-period comets— based on orbital periods, each of which is the time it takes the comet to complete one revolution. Long-period comets have orbital periods longer than 200 years, and short-period comets have orbital periods shorter than 200 years. Comet Hale-Bopp, with a period of 2400 years, is an example of a long-period comet. Comet Halley, with a period of 76 years, is an example of a short-period comet.

■ Figure 7-2 Planets orbit the Sun in elliptical orbits with the Sun at one focus. (Illustration not to scale)

Foci

Planet

Sun

Ellipse

172

Chapter 7 Gravitation

Newberry Library/Stock Montage

Kepler found that the planets 11 12 move faster when they are closer to the Sun and slower when they Planet are farther away from the Sun. Thus, Kepler’s second law states that an imaginary line from the Sun to a planet sweeps out equal areas in Sun equal time intervals, as illustrated in 1 Figure 7-3. Kepler also found that there is a mathematical relationship between periods of planets and their mean 2 distances away from the Sun. 3 Kepler’s third law states that the square of the ratio of the periods of any two planets revolving about the Sun is equal to the cube of the ratio of their average distances from the Sun. Thus, if the periods of the planets are TA and TB, and their average distances from the Sun are rA and rB, Kepler’s third law can be expressed as follows.

Kepler’s Third Law

10 9 8

7

6 5 4 ■ Figure 7-3 A planet moves fastest when it is close to the Sun and slowest when it is farther from the Sun. Equal areas are swept out in equal amounts of time. (Illustration not to scale)

T r TA 2 B

rA 3 B

The squared quantity of the period of object A divided by the period of object B, is equal to the cubed quantity of object A’s average distance from the Sun, divided by object B’s average distance from the Sun.

Note that the first two laws apply to each planet, moon, and satellite individually. The third law, however, relates the motion of several objects about a single body. For example, it can be used to compare the planets’ distances from the Sun, shown in Table 7-1, to their periods about the Sun. It also can be used to compare distances and periods of the Moon and artificial satellites orbiting Earth.

Table 7-1 Planetary Data Name Sun Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

Average Radius (m)

Mass (kg)

6.96108 2.44106 6.05106 6.38106 3.40106 7.15107 6.03107 2.56107 2.48107

1.991030 3.301023 4.871024 5.981024 6.421023 1.901027 5.691026 8.681025 1.021026

1.20106

1.251022

Mean Distance From Sun (m) — 5.791010 1.081011 1.501011 2.281011 7.781011 1.431012 2.871012 4.501012 5.871012

Section 7.1 Planetary Motion and Gravitation

173

Callisto’s Distance from Jupiter Galileo measured the orbital sizes of Jupiter’s moons using the diameter of Jupiter as a unit of measure. He found that Io, the closest moon to Jupiter, had a period of 1.8 days and was 4.2 units from the center of Jupiter. Callisto, the fourth moon from Jupiter, had a period of 16.7 days. Using the same units that Galileo used, predict Callisto’s distance from Jupiter. 1

Analyze and Sketch the Problem • Sketch the orbits of Io and Callisto. • Label the radii. Known: TC 16.7 days T I 1.8 days r I 4.2 units

2

Unknown: rC = ?

Jupiter

Io

rI

Solve for the Unknown

rC

Callisto

Solve Kepler’s third law for r C . TC 2 TI

r 3 C rI

( ) ( ) r r ( ) r r ( ) 3

3

C

I

3

C

TC 2 TI

Math Handbook Isolating a Variable page 845

TC 2 3 I TI

) units) (4.2 ( 1.8 day s 3

3

16.7 days 2

Substitute r I 4.2 units, T C 16.7 days, T I 1.8 days

6.41 03 unit s3 3

19 units 3

Evaluate the Answer • Are the units correct? r C should be in Galileo’s units, like r I. • Is the magnitude realistic? The period is large, so the radius should be large.

1. If Ganymede, one of Jupiter’s moons, has a period of 32 days, how many units are there in its orbital radius? Use the information given in Example Problem 1. 2. An asteroid revolves around the Sun with a mean orbital radius twice that of Earth’s. Predict the period of the asteroid in Earth years. 3. From Table 7-1, on page 173, you can find that, on average, Mars is 1.52 times as far from the Sun as Earth is. Predict the time required for Mars to orbit the Sun in Earth days. 4. The Moon has a period of 27.3 days and a mean distance of 3.90105 km from the center of Earth. a. Use Kepler’s laws to find the period of a satellite in orbit 6.70103 km from the center of Earth. b. How far above Earth’s surface is this satellite? 5. Using the data in the previous problem for the period and radius of revolution of the Moon, predict what the mean distance from Earth's center would be for an artificial satellite that has a period of exactly 1.00 day.

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Chapter 7 Gravitation

Newton’s Law of Universal Gravitation In 1666, 45 years after Kepler published his work, Newton began his studies of planetary motion. He found that the magnitude of the force, F, on a planet due to the Sun varies inversely with the square of the distance, r, between the centers of the planet and the Sun. That is, F is proportional to 1/r 2. The force, F, acts in the direction of the line connecting the centers of the two objects. It is quoted that the sight of a falling apple made Newton wonder if the force that caused the apple to fall might extend to the Moon, or even beyond. He found that both the apple's and Moon’s accelerations agreed with the 1/r 2 relationship. According to his own third law, the force Earth exerts on the apple is exactly the same as the force the apple exerts on Earth. The force of attraction between two objects must be proportional to the objects’ masses, and is known as the gravitational force. Newton was confident that the same force of attraction would act between any two objects, anywhere in the universe. He proposed his law of universal gravitation, which states that objects attract other objects with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This can be represented by the following equation.

According to Newton’s equation, F is directly proportional to m1 and m2. Thus, if the mass of a planet near the Sun were doubled, the force of attraction would be doubled. Use the Connecting Math to Physics feature below to examine how changing one variable affects another. Figure 7-4 illustrates the inverse square law graphically.

80

60 Force (N)

m m r The gravitational force is equal to the universal gravitational constant, times the mass of object 1, times the mass of object 2, divided by the distance between the centers of the objects, squared.

Law of Universal Gravitation F G 122

Force v. Distance Inverse Square Law

40

20

0.5 Distance (m)

■ Figure 7-4 The change in gravitational force with distance follows the inverse square law.

Direct and Inverse Relationships Newton’s law of universal gravitation

has both direct and inverse relationships. 1 r

F 2

F m1m2 Change

Result

Change

Result

2m1m2

2F

2r

1 F 4

3m1m2

3F

3r

1 F 9

2m1 3m2

6F

1 r 2

4F

1 m1m2 2

1 F 2

1 r 3

9F

Section 7.1 Planetary Motion and Gravitation

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Universal Gravitation and Kepler’s Third Law Sun

r

mp

Planet

■ Figure 7-5 A planet with mass mp and orbital radius r orbits the Sun with mass mS. (Illustration not to scale)

Newton stated his law of universal gravitation in terms that applied to the motion of planets about the Sun. This agreed with Kepler’s third law and confirmed that Newton’s law fit the best observations of the day. Consider a planet orbiting the Sun, as shown in Figure 7-5. Newton’s second law of motion, Fnet ma, can be written as Fnet mpac, where F is the gravitational force, mp is the mass of the planet, and ac is the centripetal acceleration of the planet. For simplicity, assume circular orbits. Recall from your study of uniform circular motion in Chapter 6 that, for a circular orbit, ac 42r/T2. This means that Fnet mpac may now be written as Fnet mp42r/T2. In this equation, T is the time required for the planet to make one complete revolution about the Sun. If you set the right side of this equation equal to the right side of the law of universal gravitation, you arrive at the following result: mp4 2r

mSmp

G2 2 r

T 2 4 T2 r3 GmS

Thus, T

4 2 r3 G mS

The period of a planet orbiting the Sun can be expressed as follows.

Gm 3

r Period of a Planet Orbiting the Sun T 2 S

The period of a planet orbiting the Sun is equal to 2 times the square root of the orbital radius cubed, divided by the product of the universal gravitational constant and the mass of the Sun.

Squaring both sides makes it apparent that this equation is Kepler’s third law of planetary motion: the square of the period is proportional to the cube of the distance that separates the masses. The factor 42/GmS depends on the mass of the Sun and the universal gravitational constant. Newton found that this derivation applied to elliptical orbits as well.

Astronomers have detected three planets that orbit the star Upsilon Andromedae. Planet B has an average orbital radius of 0.059 AU and a period of 4.6170 days. Planet C has an average orbital radius of 0.829 AU and a period of 241.5 days. Planet D has an average orbital radius of 2.53 AU and a period of 1284 days. (Distances are given in astronomical units (AU)—Earth’s average distance from the Sun. The distance from Earth to the Sun is 1.00 AU.) 1. Do these planets obey Kepler’s third law? 2. Find the mass of the star Upsilon Andromedae in units of the Sun’s mass.

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D

B

C

rC

rB

Upsilon Andromedae

rD

Measuring the Universal Gravitational Constant How large is the constant, G? As you know, the force of gravitational attraction between two objects on Earth is relatively small. The slightest attraction, even between two massive bowling balls, is difficult to detect. In fact, it took 100 years from the time of Newton’s work for scientists to develop an apparatus that was sensitive enough to measure the force of gravitational attraction. Cavendish’s experiment In 1798, Englishman Henry Cavendish used equipment similar to the apparatus shown in Figure 7-6 to measure the gravitational force between two objects. The apparatus had a horizontal rod with two small lead spheres attached to each end. The rod was suspended at its midpoint by a thin wire so that it could rotate. Because the rod was suspended by a thin wire, the rod and spheres were very sensitive to horizontal forces. To measure G, Cavendish placed two large lead spheres in a fixed position, close to each of the two small spheres, as shown in Figure 7-7. The force of attraction between the large and the small spheres caused the rod to rotate. When the force required to twist the wire equaled the gravitational force between the spheres, the rod stopped rotating. By measuring the angle through which the rod turned, Cavendish was able to calculate the attractive force between the objects. The angle through which the rod turned is measured using the beam of light that is reflected from the mirror. He measured the masses of the spheres and the distance between their centers. Substituting these values for force, mass, and distance into Newton’s law of universal gravitation, he found an experimental value for G: when m1 and m2 are measured in kilograms, r in meters, and F in newtons, then G 6.671011 Nm2/kg2.

■ Figure 7-6 Modern Cavendish balances are used to measure the gravitational forces between two objects.

Large lead sphere Mirror

Small lead sphere Direction of rotation

■ Figure 7-7 When the large lead spheres are placed near the small lead spheres, the gravitational attraction between the spheres causes the rod to rotate. The rotation is measured with the help of the reflected light ray.

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The importance of G Cavendish’s experiment often is called “weighing Earth,” because his experiment helped determine Earth’s mass. Once the value of G is known, not only the mass of Earth, but also the mass of the Sun can be determined. In addition, the gravitational force between any two objects can be calculated using Newton’s law of universal gravitation. For example, the attractive gravitational force, Fg, between two bowling balls of mass 7.26 kg, with their centers separated by 0.30 m, can be calculated as follows: (6.671011 Nm2/kg2)(7.26 kg)(7.26 kg) (0.30 m)

Fg 3.9108 N 2 You know that on Earth’s surface, the weight of an object of mass m is a measure of Earth’s gravitational attraction: Fg mg. If Earth’s mass is represented by mE and Earth’s radius is represented by rE, the following is true: m m rE

m rE

E Fg GE 2 2 mg, and so g G

This equation can be rearranged to solve for mE. gr 2 G

mE E Using rE 6.38106 m, g 9.80 m/s2, and G 6.671011 Nm2/kg2, the following result is obtained for Earth’s mass: 2

6

2

(9.80 m/s )(6.3810 m) 5.981024 kg mE 11 2 6.6710

Nm/kg

When you compare the mass of Earth to that of a bowling ball, you can see why the gravitational attraction between everyday objects is not easily observed. Cavendish’s experiment determined the value of G, confirmed Newton’s prediction that a gravitational force exists between two objects, and helped calculate the mass of Earth.

7.1 Section Review 6. Neptune’s Orbital Period Neptune orbits the Sun with an orbital radius of 4.4951012 m, which allows gases, such as methane, to condense and form an atmosphere, as shown in Figure 7-8. If the mass of the Sun is 1.991030 kg, calculate the period of Neptune’s orbit.

9. Universal Gravitational Constant Cavendish did his experiment using lead spheres. Suppose he had replaced the lead spheres with copper spheres of equal mass. Would his value of G be the same or different? Explain.

■

Figure 7-8

7. Gravity If Earth began to shrink, but its mass remained the same, what would happen to the value of g on Earth’s surface? 8. Gravitational Force What is the gravitational force between two 15-kg packages that are 35 cm apart? What fraction is this of the weight of one package? 178

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10. Laws or Theories? Kepler’s three statements and Newton’s equation for gravitational attraction are called “laws.” Were they ever theories? Will they ever become theories? 11. Critical Thinking Picking up a rock requires less effort on the Moon than on Earth. a. How will the weaker gravitational force on the Moon’s surface affect the path of the rock if it is thrown horizontally? b. If the thrower accidentally drops the rock on her toe, will it hurt more or less than it would on Earth? Explain. physicspp.com/self_check_quiz

7.2 Using the Law of Universal Gravitation

T

he planet Uranus was discovered in 1781. By 1830, it was clear that the law of gravitation didn’t correctly predict its orbit. Two astronomers proposed that Uranus was being attracted by the Sun and by an undiscovered planet. They calculated the orbit of such a planet in 1845, and, one year later, astronomers at the Berlin Observatory found the planet now called Neptune. How do planets, such as Neptune, orbit the Sun?

• Solve orbital motion problems. • Relate weightlessness to objects in free fall. • Describe gravitational fields. • Compare views on gravitation.

Orbits of Planets and Satellites Newton used a drawing similar to the one shown in Figure 7-9 to illustrate a thought experiment on the motion of satellites. Imagine a cannon, perched high atop a mountain, firing a cannonball horizontally with a given horizontal speed. The cannonball is a projectile, and its motion has both vertical and horizontal components. Like all projectiles on Earth, it would follow a parabolic trajectory and fall back to the ground. If the cannonball’s horizontal speed were increased, it would travel farther across the surface of Earth, and still fall back to the ground. If an extremely powerful cannon were used, however, the cannonball would travel all the way around Earth, and keep going. It would fall toward Earth at the same rate that Earth’s surface curves away. In other words, the curvature of the projectile would continue to just match the curvature of Earth, so that the cannonball would never get any closer or farther away from Earth’s curved surface. The cannonball would, therefore, be in orbit. Newton’s thought experiment ignored air resistance. For the cannonball to be free of air resistance, the mountain on which the cannon is perched would have to be more than 150 km above Earth’s surface. By way of comparison, the mountain would have to be much taller than the peak of Mount Everest, the world’s tallest mountain, which is only 8.85 km in height. A cannonball launched from a mountain that is 150 km above Earth’s surface would encounter little or no air resistance at an altitude of 150 km, because the mountain would be above most of the atmosphere. Thus, a cannonball or any object or satellite at or above this altitude could orbit Earth for a long time. v1

Objectives

Vocabulary gravitational field inertial mass gravitational mass

■ Figure 7-9 The horizontal speed v1 is not high enough and the cannonball falls to the ground. With a higher speed, v2, the cannonball travels farther. The cannonball travels all the way around Earth when the horizontal speed, v3, is high enough.

v2

v3

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Geosynchronous Orbit The GOES-12 weather satellite orbits Earth once a day at an altitude of 35,785 km. The orbital speed of the satellite matches Earth’s rate of rotation. Thus, to an observer on Earth, the satellite appears to remain above one spot on the equator. Satellite dishes on Earth can be directed to one point in the sky and not have to change position as the satellite orbits.

A satellite in an orbit that is always the same height above Earth moves in uniform circular motion. Recall that its centripetal acceleration is given by ac v2/r. Newton’s second law, Fnet mac, can thus be rewritten as Fnet mv2/r. If Earth’s mass is mE, then this expression combined with Newton’s law of universal gravitation produces the following equation: m m r

mv2 r

GE2 Solving for the speed of a satellite in circular orbit about Earth, v, yields the following. Speed of a Satellite Orbiting Earth v

r GmE

The speed of a satellite orbiting Earth is equal to the square root of the universal gravitational constant times the mass of Earth, divided by the radius of the orbit.

A satellite’s orbital period A satellite’s orbit around Earth is similar to a planet’s orbit about the Sun. Recall that the period of a planet orbiting the Sun is expressed by the following equation:

Gm 3

r T 2 S

Thus, the period for a satellite orbiting Earth is given by the following equation. Period of a Satellite Orbiting Earth T 2

r Gm 3

E

The period for a satellite orbiting Earth is equal to 2 times the square root of the radius of the orbit cubed, divided by the product of the universal gravitational constant and the mass of Earth.

■ Figure 7-10 Landsat 7, a remote sensing satellite, has a mass of about 2200 kg and orbits Earth at an altitude of about 705 km.

Geology Connection

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Russ Underwood, Lockheed Martin Space Systems/NASA

The equations for the speed and period of a satellite can be used for any object in orbit about another. The mass of the central body will replace mE in the equations, and r will be the distance between the centers of the orbiting body and the central body. If the mass of the central body is much greater than the mass of the orbiting body, then r is equal to the distance between the centers of the orbiting body and the central body. Orbital speed, v, and period, T, are independent of the mass of the satellite. Are there any factors that limit the mass of a satellite? A satellite’s mass Landsat 7, shown in Figure 7-10, is an artificial satellite that provides images of Earth’s continental surfaces. Landsat images have been used to create maps, study land use, and monitor resources and global changes. The Landsat 7 system enables researchers to monitor small-scale processes, such as deforestation, on a global scale. Satellites, such as Landsat 7, are accelerated to the speeds necessary for them to achieve orbit by large rockets, such as shuttle-booster rockets. Because the acceleration of any mass must follow Newton’s second law of motion, Fnet ma, more force is required to launch a more massive satellite into orbit. Thus, the mass of a satellite is limited by the capability of the rocket used to launch it.

Orbital Speed and Period Assume that a satellite orbits Earth 225 km above its surface. Given that the mass of Earth is 5.971024 kg and the radius of Earth is 6.38106 m, what are the satellite’s orbital speed and period? 1

Analyze and Sketch the Problem

Satellite

• Sketch the situation showing the height of the satellite’s orbit.

2

Known:

Unknown:

h 2.25105 m rE 6.38106 m mE 5.971024 kg G 6.671011 Nm2/kg2

v? T?

rE h

Earth

r

Solve for the Unknown Determine the orbital radius by adding the height of the satellite’s orbit to Earth’s radius. r h rE 2.25105 m + 6.38106 m 6.61106 m Substitute h 2.25105 m, rE 6.38106 m Solve for the speed. v

r GmE

11

(6.6710 Nm /kg )(5.9710 kg) 6.6110 m 2

2

24

6

Substitute G 6.671011 Nm2/kg2, mE 5.971024 kg, r 6.61106 m

7.76103 m/s Solve for the period.

r3 T 2 GmE

2

(6.61106 m)3 11 (6.6710 Nm2/kg2)(5.971024 kg)

5.35103 s This is approximately 89 min, or 1.5 h. 3

Math Handbook Square and Cube Roots pages 839–840 Substitute r 6.61106 m, G 6.671011 Nm2/kg2, mE 5.971024 kg

Evaluate the Answer • Are the units correct? The unit for speed is m/s and the unit for period is s.

For the following problems, assume a circular orbit for all calculations.

12. Suppose that the satellite in Example Problem 2 is moved to an orbit that is 24 km larger in radius than its previous orbit. What would its speed be? Is this faster or slower than its previous speed? 13. Use Newton’s thought experiment on the motion of satellites to solve the following. a. Calculate the speed that a satellite shot from a cannon must have to orbit Earth 150 km above its surface. b. How long, in seconds and minutes, would it take for the satellite to complete one orbit and return to the cannon? 14. Use the data for Mercury in Table 7-1 on page 173 to find the following. a. the speed of a satellite that is in orbit 260 km above Mercury's surface b. the period of the satellite

Section 7.2 Using the Law of Universal Gravitation

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Acceleration Due To Gravity Weightless Water This activity is best done outdoors. Use a pencil to poke two holes through a foam or paper cup: one on the bottom and the other on the side. Hold your fingers over the two holes to block them as your lab partner pours colored water into the cup until it is two-thirds full. 1. Predict what will happen as the cup is allowed to fall. 2. Test your prediction: drop the cup and watch closely. Analyze and Conclude 3. Describe your observations. 4. Explain your results.

■

Figure 7-11 Astronaut Chiaki Mukai experiences weightlessness on board the space shuttle Columbia, as the shuttle and everything in it falls freely toward Earth.

The acceleration of objects due to Earth’s gravity can be found by using Newton’s law of universal gravitation and his second law of motion. For a free-falling object, m, the following is true: m m r

m r

F GE2 ma, so a G2E Because a g and r rE on Earth’s surface, the following equation can be written: m gr 2 rE G mE You found above that a G2 for a free-falling object. Substituting the r

g G2E , thus, mE E

above expression for mE yields the following: 2

gr

GE

a G 2 r

2

r r

a g E

This shows that as you move farther from Earth’s center, that is, as r becomes larger, the acceleration due to gravity is reduced according to this inverse square relationship. What happens to your weight, mg , as you move farther and farther from Earth’s center? Weight and weightlessness You probably have seen photos similar to the one in Figure 7-11 in which astronauts are on the space shuttle in an environment often called “zero-g” or “weightlessness.” The shuttle orbits about 400 km above Earth’s surface. At that distance, g 8.7 m/s2, only slightly less than on Earth’s surface. Thus, Earth’s gravitational force is certainly not zero in the shuttle. In fact, gravity causes the shuttle to orbit Earth. Why, then, do the astronauts appear to have no weight? Remember that you sense weight when something, such as the floor or your chair, exerts a contact force on you. But if you, your chair, and the floor all are accelerating toward Earth together, then no contact forces are exerted on you. Thus, your apparent weight is zero and you experience weightlessness. Similarly, the astronauts experience weightlessness as the shuttle and everything in it falls freely toward Earth.

The Gravitational Field Recall from Chapter 6 that many common forces are contact forces. Friction is exerted where two objects touch, for example, when the floor and your chair or desk push on you. Gravity, however, is different. It acts on an apple falling from a tree and on the Moon in orbit. It even acts on you in midair as you jump up or skydive. In other words, gravity acts over a distance. It acts between objects that are not touching or that are not close together. Newton was puzzled by this concept. He wondered how the Sun could exert a force on planet Earth, which is hundreds of millions of kilometers away. 182 NASA

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The answer to the puzzle arose from a study of magnetism. In the 19th century, Michael Faraday developed the concept of a field to explain how a magnet attracts objects. Later, the field concept was applied to gravity. Any object with mass is surrounded by a gravitational field in which another object experiences a force due to the interaction between its mass and the gravitational field, g, at its location. This is expressed by the following equation. GM r The gravitational field is equal to the universal gravitational constant times the object’s mass, divided by the square of the distance from the object’s center. The direction is toward the mass’s center.

Gravitational Field g 2

Suppose the gravitational field is created by the Sun. Then a planet of mass m has a force exerted on it that depends on its mass and the magnitude of the gravitational field at its location. That is, F mg, toward the Sun. The force is caused by the interaction of the planet’s mass with the gravitational field at its location, not with the Sun millions of kilometers away. To find the gravitational field caused by more than one object you would calculate both gravitational fields and add them as vectors. The gravitational field can be measured by placing an object with a small mass, m, in the gravitational field and measuring the force, F, on it. The gravitational field can be calculated using g F/m. The gravitational field is measured in N/kg, which is also equal to m/s2. On Earth’s surface, the strength of the gravitational field is 9.80 N/kg, and its direction is toward Earth’s center. The field can be represented by a vector of length g pointing toward the center of the object producing the field. You can picture the gravitational field of Earth as a collection of vectors surrounding Earth and pointing toward it, as shown in Figure 7-12. The strength of the field varies inversely with the square of the distance from the center of Earth. The gravitational field depends on Earth’s mass, but not on the mass of the object experiencing it.

Two Kinds of Mass

■ Figure 7-12 Vectors representing Earth’s gravitational field all point toward Earth’s center. The field is weaker farther from Earth.

■

Figure 7-13 An inertial balance allows you to calculate the inertial mass of an object from the period (T ) of the back-and-forth motion of the object. Calibration masses, such as the cylindrical ones shown here, are used to create a graph of T 2 versus the mass. The period of the unknown mass is then measured, and the inertial mass is determined from the calibration graph.

Recall that when the concept of mass was discussed in Chapter 4, it was defined as the slope of a graph of force versus acceleration. That is, mass is equal to the ratio of the net force exerted on an object to its acceleration. This kind of mass, related to the inertia of an object, is called inertial mass and is represented by the following equation. F

et Inertial Mass minertial n

a Inertial mass is equal to the net force exerted on the object divided by the acceleration of the object.

The inertial mass of an object is measured by exerting a force on the object and measuring the object’s acceleration using an inertial balance, such as the one shown in Figure 7-13. The more inertial mass an object has, the less it is affected by any force—the less acceleration it undergoes. Thus, the inertial mass of an object is a measure of the object’s resistance to any type of force. Section 7.2 Using the Law of Universal Gravitation

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Newton’s law of universal gravitation, F = Gm1m2/r2, also involves mass, but a different kind of mass. Mass as used in the law of universal gravitation determines the size of the gravitational force between two objects and is called gravitational mass. It can be measured using a simple balance, such as the one shown in Figure 7-14. If you measure the attractive force exerted on an object by another object of mass, m, at a distance, r, then you can define the gravitational mass in the following way. r2Fgrav Gm The gravitational mass of an object is equal to the distance between the objects squared, times the gravitational force, divided by the product of the universal gravitational constant, times the mass of the other object.

Gravitational Mass mgrav

■ Figure 7-14 The platform balance shown here allows you to measure the force on a mass due to Earth’s gravity.

How different are these two kinds of mass? Suppose you have a watermelon in the trunk of your car. If you accelerate the car forward, the watermelon will roll backwards, relative to the trunk. This is a result of its inertial mass—its resistance to acceleration. Now, suppose your car climbs a steep hill at a constant speed. The watermelon will again roll backwards. But this time, it moves as a result of its gravitational mass. The watermelon is being attracted downward toward Earth. Newton made the claim that inertial mass and gravitational mass are equal in magnitude. This hypothesis is called the principle of equivalence. All experiments conducted so far have yielded data that support this principle. Albert Einstein also was intrigued by the principle of equivalence and made it a central point in his theory of gravity.

Einstein’s Theory of Gravity Newton’s law of universal gravitation allows us to calculate the gravitational force that exists between two objects because of their masses. The concept of a gravitational field allows us to picture the way gravity acts on objects that are far away. Einstein proposed that gravity is not a force, but rather, an effect of space itself. According to Einstein, mass changes the space around it. Mass causes space to be curved, and other bodies are accelerated because of the way they follow this curved space. ■ Figure 7-15 Matter causes space to curve just as an object on a rubber sheet curves the sheet around it. Moving objects near the mass follow the curvature of space. The red ball is moving clockwise around the center mass.

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(t)Horizons Companies, (b)Ted Kinsman/Photo Researchers

One way to picture how space is affected by mass is to compare space to a large, two-dimensional rubber sheet, as shown in Figure 7-15. The yellow ball on the sheet represents a massive object. It forms an indentation. A red ball rolling across the sheet simulates the motion of an object in space. If the red ball moves near the sagging region of the sheet, it will be accelerated. In the same way, Earth and the Sun are attracted to one another because of the way space is distorted by the two objects. Einstein’s theory, called the general theory of relativity, makes many predictions about how massive objects affect one another. In every test conducted to date, Einstein’s theory has been shown to give the correct results. Deflection of light Einstein’s theory predicts the deflection or bending of light by massive objects. Light follows the curvature of space around the massive object and is deflected, as shown in Figure 7-16. In 1919, during an eclipse of the Sun, astronomers found that light from distant stars that passed near the Sun was deflected in agreement with Einstein’s predictions. Another result of general relativity is the effect on light from very massive objects. If an object is massive and dense enough, the light leaving it will be totally bent back to the object. No light ever escapes the object. Objects such as these, called black holes, have been identified as a result of their effects on nearby stars. The radiation produced when matter is pulled into black holes has also been helpful in their detection. While Einstein’s theory provides very accurate predictions of gravity’s effects, it is still incomplete. It does not explain the origin of mass or how mass curves space. Physicists are working to understand the deeper meaning of gravity and the origin of mass itself.

7.2

■ Figure 7-16 The light from a distant star bends due to the Sun’s gravitational field, thereby changing the apparent position of the star. (Illustration not to scale)

Star Reference star

Actual position Apparent position

Sun

Moon Earth

Section Review

15. Gravitational Fields The Moon is 3.9105 km from Earth’s center and 1.5108 km from the Sun’s center. The masses of Earth and the Sun are 6.01024 kg and 2.01030 kg, respectively. a. Find the ratio of the gravitational fields due to Earth and the Sun at the center of the Moon. b. When the Moon is in its third quarter phase, as shown in Figure 7-17, its direction from Earth is at right angles to the Sun’s direction. What is the net gravitational field due to the Sun and Earth at the center of the Moon? ■

Figure 7-17 (Not to scale)

Moon

Sun Earth

16. Gravitational Field The mass of the Moon is 7.31022 kg and its radius is 1785 km. What is the strength of the gravitational field on the surface of the Moon? physicspp.com/self_check_quiz

17. A Satellite’s Mass When the first artificial satellite was launched into orbit by the former Soviet Union in 1957, U.S. president Dwight D. Eisenhower asked his scientific advisors to calculate the mass of the satellite. Would they have been able to make this calculation? Explain. 18. Orbital Period and Speed Two satellites are in circular orbits about Earth. One is 150 km above the surface, the other 160 km. a. Which satellite has the larger orbital period? b. Which one has the greater speed? 19. Theories and Laws Why is Einstein’s description of gravity called a “theory,” while Newton’s is a “law?” 20. Weightlessness Chairs in an orbiting spacecraft are weightless. If you were on board such a spacecraft and you were barefoot, would you stub your toe if you kicked a chair? Explain. 21. Critical Thinking It is easier to launch a satellite from Earth into an orbit that circles eastward than it is to launch one that circles westward. Explain. Section 7.2 Using the Law of Universal Gravitation

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Modeling the Orbits of Planets and Satellites In this experiment, you will analyze a model that will show how Kepler’s first and second laws of motion apply to orbits of objects in space. Kepler’s first law states that orbits of planets are ellipses, with the Sun at one focus. Kepler’s second law states that an imaginary line from the Sun to a planet sweeps out equal areas in equal time intervals. The shape of the elliptical orbit is defined by eccentricity, e, which is the ratio of the distance between the foci to the length of the major axis. When an object is at its farthest distance from the Sun along the major axis, it is at aphelion. When the object is at its closest distance from the Sun along the major axis, it is at perihelion.

QUESTION What is the shape of the orbits of planets and satellites in the solar system?

Objectives

Procedure

■ Formulate models to infer the shape of orbits

1. Place a piece of paper on a piece of cardboard using tape at the four corners.

of planets and satellites. ■ Collect and organize data for aphelion distances and perihelion distances of objects as they orbit the Sun. ■ Draw conclusions about Kepler’s first and second laws of motion.

3. Mark the center of the line and label it C. 4. Use the string to tie a loop, which, when stretched, has a length of 10 cm. For each object listed in the data table, calculate the distance between the foci, d, using the following equation:

Safety Precautions

■ Pins are sharp and can puncture the skin.

Materials piece of cardboard sheet of blank, white paper two push pins

2. Draw a line across the center of the paper, along the length of the paper. This line represents the major axis.

metric ruler sharp pencil or pen four small pieces of tape string (25 cm)

2e(10.0 cm) e1

d 5. For the circle, place a pin at C. Put the loop of string over the pin and pull it tight with your pencil. Move the pencil in a circular fashion around the center, letting the string guide it. 6. For the next object, place one pin a distance of d/2 from C along the major axis. 7. Place a second pin a distance of d/2 on the opposite side of C. The two pins represent the foci. One focus is the location of the Sun. 8. Put the loop of string over both pins and pull it tight with your pencil. Move the pencil in a circular fashion, letting the string guide it. 9. Using the same paper, repeat steps 6-8 for each of the listed objects. 10. After all of the orbits are plotted, label each orbit with the name and eccentricity of the object plotted.

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Data Table Object

Eccentricity (e)

Circle

Earth

0.017

Pluto

0.25

Comet

0.70

d (cm)

Measured A

Analyze 1. Measure the aphelion distance, A, by measuring the distance between one focus and the farthest point in the orbit along the major axis. Record your data in the data table. 2. Measure the perihelion distance, P, by measuring the closest distance between one focus and the closest point in the orbit along the major axis. Record the data in the data table. 3. Calculate the experimental eccentricity for each of the objects and record your data in the data table. Use the following equation: AP AP

e 4. Error Analysis Calculate the percent error for each object using the experimental eccentricities compared to the known eccentricities. Record your values in the data table. 5. Analyze Why is the shape of the orbit with e 0 a circle? 6. Compare How does Earth’s orbit compare to a circle?

Measured P

Experimental e

% Error

4. Kepler’s second law helps to determine the ratio between Pluto’s velocity at aphelion and perihelion (vA /vP). To determine this ratio, first calculate the area swept out by Pluto’s orbit. This area is approximately equal to the area of a triangle: Area 21 (distance to the Sun) current velocity time. If the area that the orbit sweeps out in a fixed amount of time, such as 30 days, is the same at aphelion and perihelion, this relationship can be written 1 1 P vPt AvAt 2 2

What is the ratio vP /vA for Pluto? 5. Pluto’s minimum orbital velocity is 3.7 km/s. What are the values for vP and vA?

Going Further 1. You used rough approximations to look at Kepler’s second law. Suggest an experiment to obtain precise results to confirm the second law. 2. Design an experiment to verify Kepler’s third law.

7. Observe Which of the orbits truly looks elliptical?

Real-World Physics Conclude and Apply 1. Does the orbit model you constructed obey Kepler’s first law? Explain. 2. Kepler studied the orbit data of Mars (e 0.093) and concluded that planets move about the Sun in elliptical orbits. What would Kepler have concluded if he had been on Mars and studied Earth’s orbit? 3. Where does a planet travel fastest: at aphelion or perihelion? Why?

Does a communications or weather satellite that is orbiting Earth follow Kepler’s laws? Collect data to verify your answer.

To find out more about gravitation, visit the Web site: physicspp.com

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Black Holes What would happen if you were to travel to a black hole? Your body would be stretched, flattened, and eventually pulled apart. What is a black hole? What is known about black holes? A black hole is one of the possible final stages in the evolution of a star. When fusion reactions stop in the core of a star that is at least 20 times more massive than the Sun, the core collapses forever, compacting matter into an increasingly smaller volume. The infinitely small, but infinitely dense, object that remains is called a singularity. The force of gravity is so immense in the region around the singularity that nothing, not even light, can escape it. This region is called a black hole.

Nothing Can Escape In 1917, German mathematician Karl Schwarzschild verified, mathematically, that black holes could exist. Schwarzschild used solutions to Einstein’s theory of general relativity to describe the properties of black holes. He derived an expression for a radius, called the Schwarzschild radius, within which neither light nor matter escapes the force of gravity of the singularity. The Schwarzschild radius is represented by the following equation: 2GM Rs 2 c

Hubble visible image of galaxy NGC 6240.

In this equation, G is Newton’s universal gravitational constant, M is the mass of the black hole, and c is the speed of light. The edge of the sphere defined by the Schwarzschild radius is called the event horizon. At the event horizon, the escape velocity equals the speed of light. Because nothing travels faster than the speed of light, objects that cross the event horizon can never escape.

Indirect and Direct Evidence Black holes have three physical properties that can theoretically be measured—mass, angular momentum, and electric charge. A black hole’s 188 NASA

Extreme Physics

mass can be determined by the gravitational field it generates. Mass is calculated by using a modified form of Kepler’s third law of planetary motion. Studies using NASA’s Rossi X-ray Timing Explorer have shown that black holes spin just as stars and planets do. A black hole spins because it retains the angular momentum of the star that formed it. Even though a black hole’s electric charge has not been measured, scientists hypothesize that a black hole may become charged when an excess of one type of electric charge falls into it. Super-heated gases in a black hole emit X rays, which can be detected by X-ray telescopes, such as the space-based Chandra X-ray Observatory. Although not everything is known about black holes, there is direct and indirect evidence of their existence. Continued research and special missions will provide a better understanding of black holes.

Chandra X-ray image of two black holes (blue) in NGC 6240.

Going Further Solve The escape velocity of an object leaving the event horizon can be represented by the following equation: v

2G M R s

In this equation, G is Newton’s universal gravitational constant, M is the mass of the black hole, and Rs is the radius of the black hole. Show that the escape velocity equals the speed of light.

7.1 Planetary Motion and Gravitation Vocabulary

Key Concepts

• • • • •

•

Kepler’s first law (p. 172) Kepler’s second law (p. 173) Kepler’s third law (p. 173) gravitational force (p. 175) law of universal gravitation (p. 175)

• •

Kepler’s first law states that planets move in elliptical orbits, with the Sun at one focus. Kepler’s second law states that an imaginary line from the Sun to a planet sweeps out equal areas in equal times. Kepler’s third law states that the square of the ratio of the periods of any two planets is equal to the cube of the ratio of their distances from the Sun.

T r TA 2

rA 3

B

•

B

Newton’s law of universal gravitation states that the gravitational force between any two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The force is attractive and along a line connecting their centers. m m r

F G 122

•

Newton’s law of universal gravitation can be used to rewrite Kepler’s third law to relate the radius and period of a planet to the mass of the Sun.

4 2

r3 T2 GmS

7.2 Using the Law of Universal Gravitation Vocabulary

Key Concepts

• gravitational field (p. 183) • inertial mass (p. 183) • gravitational mass (p. 184)

•

The speed of an object in circular orbit is given by the following expression. v

•

r

The period of a satellite in a circular orbit is given by the following expression. T 2

•

GmE

Gm r3

E

All objects have gravitational fields surrounding them. Gm r

g 2

•

Gravitational mass and inertial mass are two essentially different concepts. The gravitational and inertial masses of an object, however, are numerically equal. F

et m inertial n

a

•

r2Fgrav

mgrav Gm

Einstein’s general theory of relativity describes gravitational attraction as a property of space itself.

physicspp.com/vocabulary_puzzlemaker

189

Concept Mapping 22. Create a concept map using these terms: planets, stars, Newton’s law of universal gravitation, Kepler’s first law, Kepler’s second law, Kepler’s third law, Einstein’s general theory of relativity.

35. Show that the dimensions of g in the equation g F/m are in m/s2. (7.2)

36. If Earth were twice as massive but remained the same size, what would happen to the value of g? (7.2)

Mastering Concepts

Applying Concepts

23. In 1609, Galileo looked through his telescope at

37. Golf Ball The force of gravity acting on an object

Jupiter and saw four moons. The name of one of the moons that he saw is Io. Restate Kepler’s first law for Io and Jupiter. (7.1)

24. Earth moves more slowly in its orbit during summer

near Earth’s surface is proportional to the mass of the object. Figure 7-18 shows a tennis ball and golf ball in free fall. Why does a tennis ball not fall faster than a golf ball?

in the northern hemisphere than it does during winter. Is it closer to the Sun in summer or in winter? (7.1)

25. Is the area swept out per unit of time by Earth moving around the Sun equal to the area swept out per unit of time by Mars moving around the Sun? (7.1)

26. Why did Newton think that a force must act on the Moon? (7.1)

27. How did Cavendish demonstrate that a gravitational force of attraction exists between two small objects? (7.1)

28. What happens to the gravitational force between

■

two masses when the distance between the masses is doubled? (7.1)

29. According to Newton’s version of Kepler’s third law, how would the ratio T 2/r 3 change if the mass of the Sun were doubled? (7.1)

30. How do you answer the question, “What keeps a satellite up?” (7.2)

31. A satellite is orbiting Earth. On which of the

Figure 7-18

38. What information do you need to find the mass of Jupiter using Newton’s version of Kepler’s third law?

39. The mass of Pluto was not known until a satellite of the planet was discovered. Why?

40. Decide whether each of the orbits shown in Figure 7-19 is a possible orbit for a planet.

following does its speed depend? (7.2)

c c

a. mass of the satellite b. distance from Earth c. mass of Earth

Sun Sun

32. What provides the force that causes the centripetal acceleration of a satellite in orbit? (7.2)

33. During space flight, astronauts often refer to forces as multiples of the force of gravity on Earth’s surface. What does a force of 5g mean to an astronaut? (7.2)

34. Newton assumed that a gravitational force acts

c c

directly between Earth and the Moon. How does Einstein’s view of the attractive force between the two bodies differ from Newton’s view? (7.2)

190

Sun

Sun

■

Figure 7-19

Chapter 7 Gravitation For more problems, go to Additional Problems, Appendix B. Barry Runk/Grant Heilman

41. The Moon and Earth are attracted to each other by gravitational force. Does the more-massive Earth attract the Moon with a greater force than the Moon attracts Earth? Explain.

42. What would happen to the value of G if Earth were twice as massive, but remained the same size?

43. Figure 7-20 shows a satellite orbiting Earth. Examine the equation v

, relating the speed r GmE

of an orbiting satellite and its distance from the center of Earth. Does a satellite with a large or small orbital radius have the greater velocity?

48. If a mass in Earth’s gravitational field is doubled, what will happen to the force exerted by the field upon the mass?

49. Weight Suppose that yesterday your body had a mass of 50.0 kg. This morning you stepped on a scale and found that you had gained weight. a. What happened, if anything, to your mass? b. What happened, if anything, to the ratio of your weight to your mass?

50. As an astronaut in an orbiting space shuttle, how would you go about “dropping” an object down to Earth?

51. Weather Satellites The weather pictures that you see every day on TV come from a spacecraft in a stationary position relative to the surface of Earth, 35,700 km above Earth’s equator. Explain how it can stay in exactly the same position day after day. What would happen if it were closer? Farther out? Hint: Draw a pictorial model.

Satellite

r

Earth

Mastering Problems 7.1 Planetary Motion and Gravitation

■

52. Jupiter is 5.2 times farther from the Sun than Earth

Figure 7-20 (Not to scale)

is. Find Jupiter’s orbital period in Earth years.

44. Space Shuttle If a space shuttle goes into a higher orbit, what happens to the shuttle’s period?

45. Mars has about one-ninth the mass of Earth. Figure 7-21 shows satellite M, which orbits Mars with the same orbital radius as satellite E, which orbits Earth. Which satellite has a smaller period?

53. Figure 7-22 shows a Cavendish apparatus like the one used to find G. It has a large lead sphere that is 5.9 kg in mass and a small one with a mass of 0.047 kg. Their centers are separated by 0.055 m. Find the force of attraction between them. 5.9 kg 0.047 kg 0.055 m

Mars

Earth

rM

rE E

M ■

0.055 m 0.047 kg

5.9 kg ■

Figure 7-22

Figure 7-21 (Not to scale)

54. Use Table 7-1 on p. 173 to compute the gravitational 46. Jupiter has about 300 times the mass of Earth and about ten times Earth’s radius. Estimate the size of g on the surface of Jupiter.

47. A satellite is one Earth radius above the surface of Earth. How does the acceleration due to gravity at that location compare to acceleration at the surface of Earth? physicspp.com/chapter_test

force that the Sun exerts on Jupiter.

55. Tom has a mass of 70.0 kg and Sally has a mass of 50.0 kg. Tom and Sally are standing 20.0 m apart on the dance floor. Sally looks up and sees Tom. She feels an attraction. If the attraction is gravitational, find its size. Assume that both Tom and Sally can be replaced by spherical masses. Chapter 7 Assessment

191

56. Two balls have their centers 2.0 m apart, as shown in Figure 7-23. One ball has a mass of 8.0 kg. The other has a mass of 6.0 kg. What is the gravitational force between them?

67. Mimas, one of Saturn’s moons, has an orbital radius of 1.87108 m and an orbital period of about 23.0 h. Use Newton’s version of Kepler’s third law to find Saturn’s mass.

68. The Moon is 3.9108 m away from Earth and has a 2.0 m 8.0 kg ■

6.0 kg

Figure 7-23

69. Halley’s Comet Every 74 years, comet Halley is

57. Two bowling balls each have a mass of 6.8 kg. They are located next to each other with their centers 21.8 cm apart. What gravitational force do they exert on each other?

58. Assume that you have a mass of 50.0 kg. Earth has a mass of 5.971024 kg and a radius of 6.38106 m. a. What is the force of gravitational attraction between you and Earth? b. What is your weight?

59. The gravitational force between two electrons that

are 1.00 m apart is 5.541071 N. Find the mass of an electron.

60. A 1.0-kg mass weighs 9.8 N on Earth’s surface, and the radius of Earth is roughly

period of 27.33 days. Use Newton’s version of Kepler’s third law to find the mass of Earth. Compare this mass to the mass found in problem 60.

6.4106

m.

a. Calculate the mass of Earth. b. Calculate the average density of Earth.

61. Uranus Uranus requires 84 years to circle the Sun.

visible from Earth. Find the average distance of the comet from the Sun in astronomical units (AU).

70. Area is measured in m2, so the rate at which area is swept out by a planet or satellite is measured in m2/s. a. How quickly is an area swept out by Earth in its orbit about the Sun? b. How quickly is an area swept out by the Moon in its orbit about Earth? Use 3.9108 m as the average distance between Earth and the Moon, and 27.33 days as the period of the Moon.

7.2 Using the Law of Universal Gravitation 71. Satellite A geosynchronous satellite is one that appears to remain over one spot on Earth, as shown in Figure 7-24. Assume that a geosynchronous satellite has an orbital radius of 4.23107 m. a. Calculate its speed in orbit. b. Calculate its period.

Find Uranus’s orbital radius as a multiple of Earth’s orbital radius.

Satellite

62. Venus Venus has a period of revolution of 225 r

Earth days. Find the distance between the Sun and Venus as a multiple of Earth’s orbital radius. Earth

63. If a small planet, D, were located 8.0 times as far from the Sun as Earth is, how many years would it take the planet to orbit the Sun?

64. Two spheres are placed so that their centers are 2.6 m apart. The force between the two spheres is 2.751012 N. What is the mass of each sphere if one sphere is twice the mass of the other sphere?

65. The Moon is 3.9105 km from Earth’s center and 1.5108

km from the Sun’s center. If the masses of the Moon, Earth, and the Sun are 7.31022 kg, 6.01024 kg, and 2.01030 kg, respectively, find the ratio of the gravitational forces exerted by Earth and the Sun on the Moon.

66. Toy Boat A force of 40.0 N is required to pull a 10.0-kg wooden toy boat at a constant velocity across a smooth glass surface on Earth. What force would be required to pull the same wooden toy boat across the same glass surface on the planet Jupiter?

192

■

Figure 7-24 (Not to scale)

72. Asteroid The asteroid Ceres has a mass of 71020 kg and a radius of 500 km. a. What is g on the surface of Ceres? b. How much would a 90-kg astronaut weigh on Ceres?

73. Book A 1.25-kg book in space has a weight of 8.35 N. What is the value of the gravitational field at that location?

74. The Moon’s mass is 7.341022 kg, and it is 3.8108 m away from Earth. Earth’s mass is 5.971024 kg. a. Calculate the gravitational force of attraction between Earth and the Moon. b. Find Earth’s gravitational field at the Moon.

Chapter 7 Gravitation For more problems, go to Additional Problems, Appendix B.

75. Two 1.00-kg masses have their centers 1.00 m apart. What is the force of attraction between them?

76. The radius of Earth is about 6.38103 km. A 7.20103-N spacecraft travels away from Earth. What is the weight of the spacecraft at the following distances from Earth’s surface? a. 6.38103 km b. 1.28104 km

77. Rocket How high does a rocket have to go above Earth’s surface before its weight is half of what it is on Earth?

78. Two satellites of equal mass are put into orbit 30.0 m apart. The gravitational force between them is 2.0107 N. a. What is the mass of each satellite? b. What is the initial acceleration given to each satellite by gravitational force?

79. Two large spheres are suspended close to each other. Their centers are 4.0 m apart, as shown in Figure 7-25. One sphere weighs 9.8102 N. The other sphere has a weight of 1.96102 N. What is the gravitational force between them?

84. If you weigh 637 N on Earth’s surface, how much would you weigh on the planet Mars? Mars has a mass of 6.421023 kg and a radius of 3.40106 m.

85. Using Newton’s version of Kepler’s third law and information from Table 7-1 on page 173, calculate the period of Earth’s Moon if the orbital radius were twice the actual value of 3.9108 m.

86. Find the value of g, acceleration due to gravity, in the following situations. a. Earth’s mass is triple its actual value, but its radius remains the same. b. Earth’s radius is tripled, but its mass remains the same. c. Both the mass and radius of Earth are doubled.

87. Astronaut What would be the strength of Earth’s gravitational field at a point where an 80.0-kg astronaut would experience a 25.0 percent reduction in weight?

Mixed Review 88. Use the information for Earth in Table 7-1 on page 173 to calculate the mass of the Sun, using Newton’s version of Kepler’s third law.

89. Earth’s gravitational field is 7.83 N/kg at the altitude of the space shuttle. At this altitude, what is the size of the force of attraction between a student with a mass of 45.0 kg and Earth? 4.0 m

9.8102 N ■

1.96102 N

Figure 7-25

80. Suppose the centers of Earth and the Moon are 3.9108 m apart, and the gravitational force between them is about 1.91020 N. What is the approximate mass of the Moon?

90. Use the data from Table 7-1 on page 173 to find the speed and period of a satellite that orbits Mars 175 km above its surface.

91. Satellite A satellite is placed in orbit, as shown in Figure 7-26, with a radius that is half the radius of the Moon’s orbit. Find the period of the satellite in units of the period of the Moon. Moon

81. On the surface of the Moon, a 91.0-kg physics teacher weighs only 145.6 N. What is the value of the Moon’s gravitational field at its surface?

Earth

1 rM 2

82. The mass of an electron is 9.11031 kg. The mass

of a proton is 1.71027 kg. An electron and a proton are about 0.591010 m apart in a hydrogen atom. What gravitational force exists between the proton and the electron of a hydrogen atom?

83. Consider two spherical 8.0-kg objects that are 5.0 m apart. a. What is the gravitational force between the two objects? b. What is the gravitational force between them when they are 5.0101 m apart? physicspp.com/chapter_test

rM

Satellite ■

Figure 7-26

92. Cannonball The Moon’s mass is 7.31022 kg and its radius is 1785 km. If Newton’s thought experiment of firing a cannonball from a high mountain were attempted on the Moon, how fast would the cannonball have to be fired? How long would it take the cannonball to return to the cannon? Chapter 7 Assessment

193

93. The period of the Moon is one month. Answer the following questions assuming that the mass of Earth is doubled. a. What would the period of the Moon be? Express your results in months. b. Where would a satellite with an orbital period of one month be located? c. How would the length of a year on Earth be affected?

94. How fast would a planet of Earth’s mass and size have to spin so that an object at the equator would be weightless? Give the period of rotation of the planet in minutes.

95. Car Races Suppose that a Martian base has been established and car races are being considered. A flat, circular race track has been built for the race. If a car can achieve speeds of up to 12 m/s, what is the smallest radius of a track for which the coefficient of friction is 0.50?

96. Apollo 11 On July 19, 1969, Apollo 11’s revolution around the Moon was adjusted to an average orbit of 111 km. The radius of the Moon is 1785 km, and the mass of the Moon is 7.31022 kg. a. How many minutes did Apollo 11 take to orbit the Moon once? b. At what velocity did Apollo 11 orbit the Moon?

98. Make and Use Graphs Use Newton’s law of universal gravitation to find an equation where x is equal to an object’s distance from Earth’s center, and y is its acceleration due to gravity. Use a graphing calculator to graph this equation, using 6400–6600 km as the range for x and 9–10 m/s2 as the range for y. The equation should be of the form y c(1/x2). Trace along this graph and find y for the following locations. a. at sea level, 6400 km b. on top of Mt. Everest, 6410 km c. in a typical satellite orbit, 6500 km d. in a much higher orbit, 6600 km

Writing in Physics 99. Research and describe the historical development of the measurement of the distance between the Sun and Earth.

Thinking Critically 97. Analyze and Conclude Some people say that the tides on Earth are caused by the pull of the Moon. Is this statement true? a. Determine the forces that the Moon and the Sun exert on a mass, m, of water on Earth. Your answer will be in terms of m with units of N. b. Which celestial body, the Sun or the Moon, has a greater pull on the waters of Earth? c. Determine the difference in force exerted by the Moon on the water at the near surface and the water at the far surface (on the opposite side) of Earth, as illustrated in Figure 7-27. Again, your answer will be in terms of m with units of N. Far tidal bulge

d. Determine the difference in force exerted by the Sun on water at the near surface and on water at the far surface (on the opposite side) of Earth. e. Which celestial body has a greater difference in pull from one side of Earth to the other? f. Why is the statement that the tides result from the pull of the Moon misleading? Make a correct statement to explain how the Moon causes tides on Earth.

■

Figure 7-27 (Not to scale)

100. Explore the discovery of planets around other stars. What methods did the astronomers use? What measurements did they take? How did they use Kepler’s third law?

Cumulative Review 101. Airplanes A jet airplane took off from Pittsburgh at 2:20 P.M. and landed in Washington, DC, at 3:15 P.M. on the same day. If the jet’s average speed while in the air was 441.0 km/h, what is the distance between the cities? (Chapter 2)

102. Carolyn wants to know how much her brother Jared weighs. He agrees to stand on a scale for her, but only if they are riding in an elevator. If he steps on the scale while the elevator is accelerating upward at 1.75 m/s2 and the scale reads 716 N, what is Jared’s usual weight on Earth? (Chapter 4)

103. Potato Bug A 1.0-g potato bug is walking around the outer rim of an upside-down flying disk. If the disk has a diameter of 17.2 cm and the bug moves at a rate of 0.63 cm/s, what is the centripetal force acting on the bug? What agent provides this force?

Near tidal bulge Earth

194

Moon

(Chapter 6)

Chapter 7 Gravitation For more problems, go to Additional Problems, Appendix B.

Multiple TEXT TO COME Choice 1. Two satellites are in orbit around a planet. One satellite has an orbital radius of 8.0106 m. The period of rotation for this satellite is 1.0106 s. The other satellite has an orbital radius of 2.0107 m. What is this satellite’s period of rotation? 5.0105 s

4.0106 s

2.5106 s

1.3107 s

2. The illustration below shows a satellite in orbit around a small planet. The satellite’s orbital radius is 6.7104 km and its speed is 2.0105 m/s. What is the mass of the planet around which the satellite orbits? (G 6.71011 Nm2/kg2) 2.51018 kg

2.51023 kg

4.01020 kg

4.01028 kg Satellite

6.710 4 km

5. A moon in orbit around a planet experiences a gravitational force not only from the planet, but also from the Sun. The illustration below shows a moon during a solar eclipse, when the planet, the moon, and the Sun are aligned. The moon has a mass of about 3.91021 kg. The mass of the planet is 2.41026 kg, and the mass of the Sun is 2.01030 kg. The distance from the moon to the center of the planet is 6.0108 m, and the distance from the moon to the Sun is 1.51011 m. What is the ratio of the gravitational force on the moon due to the planet, compared to its gravitational force due to the Sun during the solar eclipse? 0.5 2.5

5.0 7.5

Sun 6.010 m 8

Planet 1.510 11 m (Not to scale)

Planet

(Not to scale)

3. Two satellites are in orbit around the same planet. Satellite A has a mass of 1.5102 kg, and satellite B has a mass of 4.5103 kg. The mass of the planet is 6.61024 kg. Both satellites have the same orbital radius of 6.8106 m. What is the difference in the orbital periods of the satellites? no difference

2.2102 s

1.5102 s

3.0102 s

4. A moon revolves around a planet with a speed of 9.0103 m/s. The distance from the moon to the center of the planet is 5.4106 m. What is the orbital period of the moon? 1.2102 s

1.2103 s

6.0102 s

1.2109 s

physicspp.com/standardized_test

Extended Answer 6. Two satellites are in orbit around a planet. Satellite S1 takes 20 days to orbit the planet at a distance of 2105 km from the center of the planet. Satellite S2 takes 160 days to orbit the planet. What is the distance of Satellite S2 from the center of the planet?

Plan Your Work and Work Your Plan Plan your workload so that you do a little work each day, rather than a lot of work all at once. The key to retaining information is repeated review and practice. You will retain more if you study one hour a night for five days in a row instead of cramming the night before a test.

Chapter 7 Standardized Test Practice

195

What You’ll Learn • You will learn how to describe and measure rotational motion. • You will learn how torque changes rotational velocity. • You will explore factors that determine the stability of an object. • You will learn the nature of centrifugal and Coriolis “forces.”

Why It’s Important You encounter many rotating objects in everyday life, such as CDs, wheels, and amusement-park rides. Spin Rides Amusementpark rides that spin are designed to thrill riders using the physics of rotational motion. The thrill is produced by a “force” that is present only when the ride spins.

Think About This Why do people who ride amusement-park rides that spin in circles, such as this one, experience such strong physical reactions?

physicspp.com 196 Paul L. Ruben

How do different objects rotate as they roll? Question Do different objects of similar size and mass roll at the same rate on an incline? Procedure 1. You will need a meterstick, a piece of foam board, a ball, a solid can, and a hollow can. 2. Position the foam board on a 20° incline. 3. Place the meterstick horizontally across the foam board, near the top of the incline, and hold it. 4. Place the ball, solid can, and hollow can against the meterstick. The solid can and hollow can should be placed sideways. 5. Simultaneously, release the three objects by lifting the meterstick. 6. As each object accelerates down the incline, due to gravity, observe the order in which each object reaches the bottom. 7. Repeat steps 2–5 two more times.

Critical Thinking Which of the objects’ properties may have contributed to their behavior? List the properties that were similar and those that were different for each object.

Analysis List the objects in order from the greatest to the least acceleration.

8.1 Describing Rotational Motion

Y

ou probably have observed a spinning object many times. How would you measure such an object’s rotation? Find a circular object, such as a CD. Mark one point on the edge of the CD so that you can keep track of its position. Rotate the CD to the left (counterclockwise), and as you do so, watch the location of the mark. When the mark returns to its original position, the CD has made one complete revolution. How can you measure a fraction of one revolution? It can be measured in several 1 1 of a revolution, whereas a degree is of different ways. A grad is 400 360 a revolution. In mathematics and physics, yet another form of measurement is used to describe fractions of revolutions. In one revolution, a point on the edge travels a distance equal to 2 times the radius of the object. For this reason, the radian is defined as 12 of a revolution. In other words, one complete revolution is equal to 2 radians. A radian is abbreviated “rad.”

Objectives • Describe angular displacement. • Calculate angular velocity. • Calculate angular acceleration. • Solve problems involving rotational motion.

Vocabulary radian angular displacement angular velocity angular acceleration

Section 8.1 Describing Rotational Motion

197

Horizons Companies

■ Figure 8-1 The pie chart shows the radian measure of the most common angles, measured in the counterclockwise direction. Each angle is measured from 0.

2 3

2

3

3 4

4

6

0 2

7 4

5 4

3 2

Angular Displacement The Greek letter theta, , is used to represent the angle of revolution. Figure 8-1 shows the angles in radians for several common fractions of a revolution. Note that counterclockwise rotation is designated as positive, while clockwise is negative. As an object rotates, the change in the angle is called angular displacement. As you know, Earth makes one complete revolution, or 2 rad, in 24 h. In 12 h, its rotation is through rad. Through what angle does Earth rotate in 6 h? Because 6 h is one-fourth of a day, Earth rotates through an angle of rad during that period. Earth’s rotation as seen from the north pole is 2 positive. Is it positive or negative when viewed from the south pole?

■ Figure 8-2 The dashed line shows the path of the point on the CD as the CD rotates counterclockwise about its center.

r

d

How far does a point on a rotating object move? You already found that a point on the edge of an object moves 2 times the radius in one revolution. In general, for rotation through an angle, , a point at a distance, r, from the center, as shown in Figure 8-2, moves a distance given by d r. If r is measured in meters, you might think that multiplying it by rad would result in d being measured in mrad. However, this is not the case. Radians indicate the ratio between d and r. Thus, d is measured in m.

Angular Velocity How fast does a CD spin? How do you determine its speed of rotation? Recall from Chapter 2 that velocity is displacement divided by the time taken to make the displacement. Likewise, the angular velocity of an object is angular displacement divided by the time taken to make the displacement. Thus, the angular velocity of an object is given by the following equation, where angular velocity is represented by the Greek letter omega, .

Angular Velocity of an Object t

The angular velocity is equal to the angular displacement divided by the time required to make the rotation.

198

Chapter 8 Rotational Motion

Recall that if the velocity changes over a time interval, the average velocity is not equal to the instantaneous velocity at any given instant. Similarly, the angular velocity calculated in this way is actually the average angular velocity over a time interval, t. Instantaneous angular velocity is equal to the slope of a graph of angular position versus time. Angular velocity is measured in rad/s. For Earth, E (2 rad)/ (24.0 h)(3600 s/h) 7.27105 rad/s. In the same way that counterclockwise rotation produces positive angular displacement, it also results in positive angular velocity. If an object’s angular velocity is , then the linear velocity of a point a distance, r, from the axis of rotation is given by v r. The speed at which an object on Earth’s equator moves as a result of Earth’s rotation is given by v r (6.38106 m)(7.27105 rad/s) 464 m/s. Earth is an example of a rotating, rigid body. Even though different points on Earth rotate different distances in each revolution, all points rotate through the same angle. All parts of a rigid body rotate at the same rate. The Sun, on the other hand, is not a rigid body. Different parts of the Sun rotate at different rates. Most objects that we will consider in this chapter are rigid bodies.

Astronomy Connection

Angular Acceleration What if angular velocity is changing? For example, if a car were accelerated from 0.0 m/s to 25 m/s in 15 s, then the angular velocity of the wheels also would change from 0.0 rad/s to 78 rad/s in the same 15 s. The wheels would undergo angular acceleration, which is defined as the change in angular velocity divided by the time required to make the change. Angular acceleration, , is represented by the following equation. t Angular acceleration is equal to the change in angular velocity divided by the time required to make that change.

Angular Acceleration of an Object

Angular acceleration is measured in rad/s2. If the change in angular velocity is positive, then the angular acceleration also is positive. Angular acceleration defined in this way is also the average angular acceleration over the time interval t. One way to find the instantaneous angular acceleration is to find the slope of a graph of angular velocity as a function of time. The linear acceleration of a point at a distance, r, from the axis of an object with angular acceleration, , is given by a r. Table 8-1 is a summary of linear and angular relationships.

Table 8-1 Linear and Angular Measures Quantity Displacement Velocity Acceleration

Linear

Angular

Relationship

d (m) v (m/s) a (m/s2)

(rad) (rad/s) (rad/s2)

d r v r a r

Section 8.1 Describing Rotational Motion

199

1. What is the angular displacement of each of the following hands of a clock in 1 h? State your answer in three significant digits. a. the second hand b. the minute hand c. the hour hand 2. If a truck has a linear acceleration of 1.85 m/s2 and the wheels have an angular acceleration of 5.23 rad/s2, what is the diameter of the truck’s wheels? 3. The truck in the previous problem is towing a trailer with wheels that have a diameter of 48 cm. a. How does the linear acceleration of the trailer compare with that of the truck? b. How do the angular accelerations of the wheels of the trailer and the wheels of the truck compare? 4. You want to replace the tires on your car with tires that have a larger diameter. After you change the tires, for trips at the same speed and over the same distance, how will the angular velocity and number of revolutions change?

Angular frequency A rotating object can make many revolutions in a given amount of time. For instance, a spinning wheel can go through several complete revolutions in 1 min. Thus, the number of complete revolutions made by the object in 1 s is called angular frequency. Angular frequency is f /2. In the next section, you will explore the factors that cause the angular frequency to change.

8.1 Section Review 5. Angular Displacement A movie lasts 2 h. During that time, what is the angular displacement of each of the following? a. the hour hand b. the minute hand 6. Angular Velocity The Moon rotates once on its axis in 27.3 days. Its radius is 1.74106 m. a. What is the period of the Moon’s rotation in seconds? b. What is the frequency of the Moon’s rotation in rad/s? c. What is the linear speed of a rock on the Moon’s equator due only to the Moon’s rotation? d. Compare this speed with the speed of a person on Earth’s equator due to Earth’s rotation. 7. Angular Displacement The ball in a computer mouse is 2.0 cm in diameter. If you move the mouse 12 cm, what is the angular displacement of the ball? 200

Chapter 8 Rotational Motion

8. Angular Displacement Do all parts of the minute hand on a watch have the same angular displacement? Do they move the same linear distance? Explain. 9. Angular Acceleration In the spin cycle of a clothes washer, the drum turns at 635 rev/min. If the lid of the washer is opened, the motor is turned off. If the drum requires 8.0 s to slow to a stop, what is the angular acceleration of the drum? 10. Critical Thinking A CD-ROM has a spiral track that starts 2.7 cm from the center of the disk and ends 5.5 cm from the center. The disk drive must turn the disk so that the linear velocity of the track is a constant 1.4 m/s. Find the following. a. the angular velocity of the disk (in rad/s and rev/min) for the start of the track b. the disk’s angular velocity at the end of the track c. the disk’s angular acceleration if the disk is played for 76 min physicspp.com/self_check_quiz

8.2 Rotational Dynamics

H

ow do you start the rotation of an object? That is, how do you change its angular velocity? Suppose you have a soup can that you want to spin. If you wrap a string around it and pull hard, you could make the can spin rapidly. Later in this chapter, you will learn why gravity, the force of Earth’s mass on the can, acts on the center of the can. The force of the string, on the other hand, is exerted at the outer edge of the can, and at right angles to the line from the center of the can, to the point where the string leaves the can’s surface. You have learned that a force changes the velocity of a point object. In the case of a soup can, a force that is exerted in a very specific way changes the angular velocity of an extended object, which is an object that has a definite shape and size. Consider how you open a door: you exert a force. How can you exert the force to open the door most easily? To get the most effect from the least force, you exert the force as far from the axis of rotation as possible, as shown in Figure 8-3. In this case, the axis of rotation is an imaginary vertical line through the hinges. The doorknob is near the outer edge of the door. You exert the force on the doorknob at right angles to the door, away from the hinges. Thus, the magnitude of the force, the distance from the axis to the point where the force is exerted, and the direction of the force determine the change in angular velocity.

Objectives • Describe torque and the factors that determine it. • Calculate net torque. • Calculate the moment of inertia.

Vocabulary lever arm torque moment of inertia Newton’s second law for rotational motion

Lever arm For a given applied force, the change in angular velocity depends on the lever arm, which is the perpendicular distance from the axis of rotation to the point where the force is exerted. If the force is ■ Figure 8-3 When opening a perpendicular to the radius of rotation, as it was with the soup can, then door that is free to rotate about the lever arm is the distance from the axis, r. For the door, it is the distance its hinges, the greatest torque from the hinges to the point where you exert the force, as illustrated is produced when the force is in Figure 8-4a, on the next page. If the force is not perpendicular, the applied farthest from the hinges perpendicular component of the force must be found. (a), at an angle perpendicular The force exerted by the string around to the door (b). the can is perpendicular to the radius. a No effect Little effect Maximum effect If a force is not exerted perpendicular to the radius, however, the lever arm is reduced. To find the lever arm, extend the line of the force until it forms a right angle with a line from the center of rotation. The distance between the intersection and the axis is the lever arm. Thus, using trigonometry, the lever b arm, L, can be calculated by the equation L r sin θ, as shown in Figure 8-4b. In this equation, r is the distance from the axis of rotation to the point where the force is exerted, and θ is the angle between the force and the radius from the axis of rotation to the point where No effect Some effect Maximum effect the force is applied. Section 8.2 Rotational Dynamics

201

■ Figure 8-4 The lever arm is along the width of the door, from the hinge to the point where the force is exerted (a). The lever arm is equal to r sin , when the angle, , between the force and the radius of rotation is not equal to 90° (b).

b

rm

Axis of rotation

Le

ra

ve

ve

ra

Le

rm

a

r sin r

F F

Torque is a measure of how effectively a force causes rotation. The magnitude of torque is the product of the force and the lever arm. Because force is measured in newtons, and distance is measured in meters, torque is measured in newton-meters (Nm). Torque is represented by the Greek letter tau, τ. The equation for torque is shown below. Torque

τ Fr sin θ

Torque is equal to the force times the lever arm.

Lever Arm A bolt on a car engine needs to be tightened with a torque of 35 Nm. You use a 25-cm-long wrench and pull on the end of the wrench at an angle of 60.0° from the perpendicular. How long is the lever arm, and how much force do you have to exert? 1

Analyze and Sketch the Problem 60.0°

• Sketch the situation. Find the lever arm by extending the force vector backwards until a line that is perpendicular to it intersects the axis of rotation. Known: r 0.25 m 35 Nm 60.0°

60.0°

L? F?

rm ra

Solve for the length of the lever arm. L r sin (0.25 m)(sin 60.0°) Substitute r 0.25 m, 60.0° 0.22 m

25 cm ve

Solve for the Unknown

Le

2

Unknown:

Solve for the force. Fr sin F

r sin 35 Nm (0.25 m)(sin 60.0°)

1.6102 N 3

Substitute 35 Nm, r 0.25 m, 60.0°

Math Handbook Trigonometric Ratios page 855

Evaluate the Answer • Are the units correct? Force is measured in newtons. • Does the sign make sense? Only the magnitude of the force needed to rotate the wrench clockwise is calculated.

202

Chapter 8 Rotational Motion

11. Consider the wrench in Example Problem 1. What force is needed if it is applied to the wrench at a point perpendicular to the wrench? 12. If a torque of 55.0 Nm is required and the largest force that can be exerted by you is 135 N, what is the length of the lever arm that must be used? 13. You have a 0.234-m-long wrench. A job requires a torque of 32.4 Nm, and you can exert a force of 232 N. What is the smallest angle, with respect to the vertical, at which the force can be exerted? 14. You stand on the pedal of a bicycle. If you have a mass of 65 kg, the pedal makes an angle of 35° above the horizontal, and the pedal is 18 cm from the center of the chain ring, how much torque would you exert? 15. If the pedal in problem 14 is horizontal, how much torque would you exert? How much torque would you exert when the pedal is vertical?

Finding Net Torque Try the following experiment. Get two pencils, some coins, and some transparent tape. Tape two identical coins to the ends of the pencil and balance it on the second pencil, as shown in Figure 8-5. Each coin exerts a torque that is equal to its weight, Fg, times the distance, r, from the balance point to the center of the coin, as follows:

Fgr But the torques are equal and opposite in direction. Thus, the net torque is zero:

1 2 0 or Fg1 r1 Fg2 r2 0 How can you make the pencil rotate? You could add a second coin on top of one of the two coins, thereby making the two forces different. You also could slide the balance point toward one end or the other of the pencil, thereby making the two distances different.

■ Figure 8-5 The torque exerted by the first coin, Fg1r1, is equal and opposite in direction to the torque exerted by the second coin, Fg2r2, when the pencil is balanced.

r1

Fg1

r2

Fg2

Section 8.2 Rotational Dynamics

203

Balancing Torques Kariann (56 kg) and Aysha (43 kg) want to balance on a 1.75-m-long seesaw. Where should they place the pivot point? 1

Analyze and Sketch the Problem • Sketch the situation. • Draw and label the vectors.

2

rK

Known:

Unknown:

mK 56 kg mA 43 kg rK rA 1.75 m

rK ? rA ?

Solve for the Unknown

rA

FgA

FgK

Find the two forces. Kariann: FgK mKg (56 kg)(9.80 m/s2)

Substitute mK 56 kg, g 9.80 m/s2

5.5102 N Aysha: FgA mAg (43 kg)(9.80 m/s2)

Substitute mA 43 kg, g 9.80 m/s2

4.2102 N Define Kariann’s distance in terms of the length of the seesaw and Aysha’s distance. rK 1.75 m rA When there is no rotation, the sum of the torques is zero. FgKrK FgArA FgKrK FgArA 0.0 Nm FgK(1.75 m rA) FgArA 0.0 Nm

Substitute rK 1.75 m rA

Solve for rA. FgK(1.75 m) FgK(rA) FgArA 0.0 Nm

Math Handbook

FgKrA FgArA FgK(1.75 m)

Isolating a Variable page 845

(FgK FgA) rA FgK(1.75 m) FgK(1.75 m)

rA (FgK FgA)

2

(5.510 N)(1.75 m) 2 2 (5.510 N 4.210 N)

Substitute FgK 5.5102 N, FgA 4.2102 N

0.99 m 3

Evaluate the Answer • Are the units correct? Distance is measured in meters. • Do the signs make sense? Distances are positive. • Is the magnitude realistic? Aysha is about 1 m from the center, so Kariann is about 0.75 m away from it. Because Kariann’s weight is greater than Aysha’s weight, the lever arm on Kariann’s side should be shorter. Aysha is farther from the pivot, as expected.

204

Chapter 8 Rotational Motion

■

4.5 cm 1.1 cm

16. Ashok, whose mass is 43 kg, sits 1.8 m from the center of a seesaw. Steve, whose mass is 52 kg, wants to balance Ashok. How far from the center of the seesaw should Steve sit?

Figure 8-6

(Not to scale)

17. A bicycle-chain wheel has a radius of 7.70 cm. If the chain exerts a 35.0-N force on the wheel in the clockwise direction, what torque is needed to keep the wheel from turning? 18. Two baskets of fruit hang from strings going around pulleys of different diameters, as shown in Figure 8-6. What is the mass of basket A?

A

19. Suppose the radius of the larger pulley in problem 18 was increased to 6.0 cm. What is the mass of basket A now?

0.23 kg 45.0°

20. A bicyclist, of mass 65.0 kg, stands on the pedal of a bicycle. The crank, which is 0.170 m long, makes a 45.0° angle with the vertical, as shown in Figure 8-7. The crank is attached to the chain wheel, which has a radius of 9.70 cm. What force must the chain exert to keep the wheel from turning?

9.70 cm 0.170 m

■

Figure 8-7

The Moment of Inertia If you exert a force on a point mass, its acceleration will be inversely proportional to its mass. How does an extended object rotate when a torque is exerted on it? To observe firsthand, recover the pencil, the coins, and the transparent tape that you used earlier in this chapter. First, tape the coins at the ends of the pencil. Hold the pencil between your thumb and forefinger, and wiggle it back and forth. Take note of the forces that your thumb and forefinger exert. These forces create torques that change the angular velocity of the pencil and coins. Now move the coins so that they are only 1 or 2 cm apart. Wiggle the pencil as before. Did the amount of torque and force need to be changed? The torque that was required was much less this time. Thus, the amount of mass is not the only factor that determines how much torque is needed to change angular velocity; the location of that mass also is relevant. The resistance to rotation is called the moment of inertia, which is represented by the symbol I and has units of mass times the square of the distance. For a point object located at a distance, r, from the axis of rotation, the moment of inertia is given by the following equation. Moment of Inertia of a Point Mass I mr2 The moment of inertia of a point mass is equal to the mass of the object times the square of the object’s distance from the axis of rotation.

Section 8.2 Rotational Dynamics

205

Table 8-2 Moments of Inertia for Various Objects Location of Axis

Object

Moment of Inertia

Diagram

Thin hoop of radius r

Through central diameter

r

Solid, uniform cylinder of radius r

Through center

r

Uniform sphere of radius r

Through center

Long, uniform rod of length l

Through center

Axis

mr 2

Axis

1 mr 2 2

Axis

2 mr 2 5

Axis

1 ml 2 12

Axis

1 ml 2 3

r

l

Long, uniform rod of length l

Through end

Thin, rectangular plate of length l and width w

Through center

l

■

Figure 8-8 The moment of inertia of a book depends on the axis of rotation. The moment of inertia of the book in (a) is larger than the moment of inertia of the book in (b) because the average distance of the book’s mass from the rotational axis is larger.

a

Axis of rotation

b

Axis of rotation

206

Chapter 8 Rotational Motion

Axis

1 m(l 2 w 2) 12

l w

As you have seen, the moment of inertia for complex objects, such as the pencil and coins, depends on how far the coins are from the axis of rotation. A bicycle wheel, for example, has almost all of its mass in the rim and tire. Its moment of inertia is almost exactly equal to mr 2, where r is the radius of the wheel. For most objects, however, the mass is distributed continuously and so the moment of inertia is less than mr 2. For example, as shown in Table 8-2, for a solid cylinder of radius r, 1 2 I mr 2, while for a solid sphere, I mr 2. 2

5

The moment of inertia also depends on the location of the rotational axis, as illustrated in Figure 8-8. To observe this firsthand, hold a book in the upright position, by placing your hands at the bottom of the book. Feel the torque needed to rock the book towards you, and then away from you. Now put your hands in the middle of the book and feel the torque needed to rock the book toward you and then away from you. Note that much less torque is needed when your hands are placed in the middle of the book because the average distance of the book’s mass from the rotational axis is much less in this case.

Moment of Inertia A simplified model of a twirling baton is a thin rod with two round objects at each end. The length of the baton is 0.65 m, and the mass of each object is 0.30 kg. Find the moment of inertia of the baton if it is rotated about the midpoint between the round objects. What is the moment of inertia of the baton when it is rotated around one end? Which is greater? Neglect the mass of the rod. 1

Analyze and Sketch the Problem • Sketch the situation. Show the baton with the two different axes of rotation and the distances from the axes of rotation to the masses. Known: m 0.30 kg l 0.65 m

2

0.33 m

0.30 kg

0.33 m

0.30 kg

Unknown: I? Axis

Solve for the Unknown

0.65 m

Calculate the moment of inertia of each mass separately. Rotating about the center of the rod: 1 2 1 (0.65 m) 2

r l

Substitute l = 0.65 m

0.33 m Isingle mass mr 2 (0.30 kg)(0.33 m)2 0.033

Substitute m 0.30 kg, r 0.33 m

kgm2 Math Handbook

Find the moment of inertia of the baton. I 2Isingle mass 2(0.033 kgm2) 0.066 kgm2

Operations with Significant Digits pages 835–836

Substitute Isingle mass 0.033 kgm2

Rotating about one end of the rod: Isingle mass mr 2 (0.30 kg)(0.65 m)2 0.13 kgm2

Substitute m 0.30 kg, r 0.65 m

Find the moment of inertia of the baton. I Isingle mass 0.13 kgm2 The moment of inertia is greater when the baton is swung around one end. 3

Evaluate the Answer • Are the units correct? Moment of inertia is measured in kgm2. • Is the magnitude realistic? Masses and distances are small, and so are the moments of inertia. Doubling the distance increases the moment of inertia by a factor of 4. Thus, doubling the distance overcomes having only one mass contributing.

Section 8.2 Rotational Dynamics

207

21. Two children of equal masses sit 0.3 m from the center of a seesaw. Assuming that their masses are much greater than that of the seesaw, by how much is the moment of inertia increased when they sit 0.6 m from the center? 22. Suppose there are two balls with equal diameters and masses. One is solid, and the other is hollow, with all its mass distributed at its surface. Are the moments of inertia of the balls equal? If not, which is greater? 23. Figure 8-9 shows three massive spheres on a rod of very small mass. Consider the moment of inertia of the system, first when it is rotated about sphere A, and then when it is rotated about sphere C. Are the moments of inertia the A C same or different? Explain. If the moments of inertia are different, in which case is the moment of inertia greater? 24. Each sphere in the previous problem has a mass of 0.10 kg. The distance between spheres A and C is 0.20 m. Find the moment of inertia in the following instances: rotation about sphere A, rotation about sphere C.

■

Figure 8-9

Newton’s Second Law for Rotational Motion Newton's second law for linear motion is expressed as a Fnet/m. If you rewrite this equation to represent rotational motion, acceleration is replaced by angular acceleration, , force is replaced by net torque, net, and mass is replaced by moment of inertia, I. Thus, Newton’s second law for rotational motion states that angular acceleration is directly proportional to the net torque and inversely proportional to the moment of inertia. This law is expressed by the following equation. Newton’s Second Law for Rotational Motion

et n

I The angular acceleration of an object is equal to the net torque on the object, divided by the moment of inertia.

Recall the coins taped on the pencil. To change the direction of rotation of the pencil—to give it angular acceleration—you had to apply torque to the pencil. The greater the moment of inertia, the more torque needed to produce the same angular acceleration.

Rank the objects shown in the diagram according to their moments of inertia about the indicated axes. All spheres have equal masses and all separations are the same.

A B

C

D

208

Chapter 8 Rotational Motion

Torque A solid steel wheel has a mass of 15 kg and a diameter of 0.44 m. It starts at rest. You want to make it rotate at 8.0 rev/s in 15 s. a. What torque must be applied to the wheel? F b. If you apply the torque by wrapping a strap around the outside of the wheel, how much force should you exert on the strap? 1

τ

Analyze and Sketch the Problem • Sketch the situation. The torque must be applied in a counterclockwise direction; force must be exerted as shown. Known:

Unknown:

m 15 kg

?

1 2

2

r (0.44 m) = 0.22 m

I ?

i 0.0 rad/s f 2(8.0 rev/s) t 15 s

? F ?

0.44 m

Solve for the Unknown a. Solve for angular acceleration. t

2(8.0 rev/s) (0.0 rad/s) 15 s

Substitute f 2(8.0 rev/s), i 0.0 rad/s

3.4 rad/s2 Solve for the moment of inertia. 1 2 1 (15 kg)(0.22 m)2 2

I mr 2

Substitute m 15 kg, r 0.22 m

0.36 kgm2 Solve for torque. I (0.36 kgm2)(3.4 rad/s2) 1.2 kgm2/s2 1.2 Nm

Substitute I 0.36 kgm2, 3.4 rad/s2

Math Handbook

b. Solve for force. Fr r 1.2 Nm 0.22 m

Operations with Significant Digits pages 835–836

F

Substitute 1.2 Nm, r 0.22 m

5.5 N 3

Evaluate the Answer • Are the units correct? Torque is measured in Nm and force is measured in N. • Is the magnitude realistic? Despite its large mass, the small size of the wheel makes it relatively easy to spin.

Section 8.2 Rotational Dynamics

209

25. Consider the wheel in Example Problem 4. If the force on the strap were twice as great, what would be the speed of rotation of the wheel after 15 s? 26. A solid wheel accelerates at 3.25 rad/s2 when a force of 4.5 N exerts a torque on it. If the wheel is replaced by a wheel with all of its mass on the rim, the moment of inertia is given by I mr 2. If the same angular velocity were desired, what force would have to be exerted on the strap? 27. A bicycle wheel can be accelerated either by pulling on the chain that is on the gear or by pulling on a string wrapped around the tire. The wheel’s radius is 0.38 m, while the radius of the gear is 0.14 m. If you obtained the needed acceleration with a force of 15 N on the chain, what force would you need to exert on the string? 28. The bicycle wheel in problem 27 is used with a smaller gear whose radius is 0.11 m. The wheel can be accelerated either by pulling on the chain that is on the gear or by pulling string that is wrapped around the tire. If you obtained the needed acceleration with a force of 15 N on the chain, what force would you need to exert on the string? 29. A disk with a moment of inertia of 0.26 kgm2 is attached to a smaller disk mounted on the same axle. The smaller disk has a diameter of 0.180 m and a mass of 2.5 kg. A strap is wrapped around the smaller disk, as shown in Figure 8-10. Find the force needed to give this system an angular acceleration of 2.57 rad/s2.

F

■

Figure 8-10

In summary, changes in the amount of torque applied to an object, or changes in the moment of inertia, affect the rate of rotation. In this section, you learned how Newton’s second law of motion applies to rotational motion. In the next section, you will learn how to keep objects from rotating.

8.2 Section Review 30. Torque Vijesh enters a revolving door that is not moving. Explain where and how Vijesh should push to produce a torque with the least amount of force. 31. Lever Arm You try to open a door, but you are unable to push at a right angle to the door. So, you push the door at an angle of 55° from the perpendicular. How much harder would you have to push to open the door just as fast as if you were to push it at 90°? 32. Net Torque Two people are pulling on ropes wrapped around the edge of a large wheel. The wheel has a mass of 12 kg and a diameter of 2.4 m. One person pulls in a clockwise direction with a 43-N force, while the other pulls in a counterclockwise direction with a 67-N force. What is the net torque on the wheel? 210

Chapter 8 Rotational Motion

33. Moment of Inertia Refer to Table 8-2 on page 206 and rank the moments of inertia from least to greatest of the following objects: a sphere, a wheel with almost all of its mass at the rim, and a solid disk. All have equal masses and diameters. Explain the advantage of using the one with the least moment of inertia. 34. Newton’s Second Law for Rotational Motion A rope is wrapped around a pulley and pulled with a force of 13.0 N. The pulley’s radius is 0.150 m. The pulley’s rotational speed goes from 0.0 to 14.0 rev/min in 4.50 s. What is the moment of inertia of the pulley? 35. Critical Thinking A ball on an extremely lowfriction, tilted surface, will slide downhill without rotating. If the surface is rough, however, the ball will roll. Explain why, using a free-body diagram. physicspp.com/self_check_quiz

8.3 Equilibrium

W

hy are some vehicles more likely than others to roll over when involved in an accident? What causes a vehicle to roll over? The answer lies in the design of the vehicle. In this section, you will learn some of the factors that cause an object to tip over.

• Define center of mass. • Explain how the location of the center of mass affects the stability of an object.

The Center of Mass How does an object rotate around its center of mass? A wrench may spin about its handle or end-over-end. Does any single point on the wrench follow a straight path? Figure 8-11a shows the path of the wrench. You can see that there is a single point whose path traces a straight line, as if the wrench could be replaced by a point particle at that location. The center of mass of an object is the point on the object that moves in the same way that a point particle would move.

Objectives

• Define the conditions for equilibrium. • Describe how rotating frames of reference give rise to apparent forces.

Vocabulary center of mass centrifugal “force” Coriolis “force”

Locating the center of mass How can you locate the center of mass of an object? First, suspend the object from any point. When the object stops swinging, the center of mass is along the vertical line drawn from the suspension point as shown in Figure 8-11b. Draw the line. Then, suspend the object from another point. Again, the center of mass must be below this point. Draw a second vertical line. The center of mass is at the point where the two lines cross, as shown in Figure 8-11c. The wrench, racket, and all other freely rotating objects rotate about an axis that goes through their center of mass. Where is the center of mass of a person located?

a

b

c

■

Center of mass

Figure 8-11 The path of the center of mass of a wrench is a straight line (a). The center of mass of an object such as a tennis racket can be found by first suspending it from any point (b). The point where the strings intersect is the location of the racket’s center of mass (c). Section 8.3 Equilibrium

211

(t)Richard Megna/Fundamental Photographs, (others)Hutchings Photography

■ Figure 8-12 The upward motion of the ballet dancer’s head is less than the upward motion of the center of mass. Thus, the head and torso move in a nearly horizontal path. This creates an illusion of floating.

The Fosbury-Flop In high jumping, a technique called the Fosbury-Flop allows a high jumper to clear the bar when it is placed at the highest position. This is possible because the athlete’s center of mass passes below the bar as he or she somersaults over the bar, with his or her back toward it.

, Path of the dancer s head

, f the dancer s center of mass Path o

The center of mass of a human body For a person who is standing with his or her arms hanging straight down, the center of mass is a few centimeters below the navel, midway between the front and back of the person’s body. It is slightly higher in young children, because of their relatively larger heads. Because the human body is flexible, however, its center of mass is not fixed. If you raise your hands above your head, your center of mass rises 6 to 10 cm. A ballet dancer, for example, can appear to be floating on air by changing her center of mass in a leap. By raising her arms and legs while in the air, as shown in Figure 8-12, the dancer moves her center of mass closer to her head. The path of the center of mass is a parabola, so the dancer’s head stays at almost the same height for a surprisingly long time.

Center of Mass and Stability What factors determine whether a vehicle is stable or prone to roll over in an accident? To understand the problem, think about tipping over a box. A tall, narrow box, standing on end, tips more easily than a low, broad box. Why? To tip a box, as shown in Figure 8-13, you must rotate it about a corner. You pull at the top with a force, F, applying a torque, F. The weight of the box, acting on the center of mass, Fg, applies an opposing torque, w. When the center of mass is directly above the point of support, w is zero. The only torque is the one applied by you. As the box rotates farther, its center of mass is no longer above its base of support, and both torques act in the same direction. At this point, the box tips over rapidly.

τF

■

Figure 8-13 The bent arrows show the direction of the torque produced by the force exerted to tip over a box.

τF

F

τw Fg

212

Chapter 8 Rotational Motion

τF

F

τw

F Fg

Fg

Stability An object is said to be stable if an external force is required to tip it. The box in Figure 8-13 is stable as long as the direction of the torque due to its weight, w tends to keep it upright. This occurs as long as the box’s center of mass lies above its base. To tip the box over, you must rotate its center of mass around the axis of rotation until it is no longer above the base of the box. To rotate the box, you must lift its center of mass. The broader the base, the more stable the object is. For this reason, if you are standing on a bus that is weaving through traffic and you want to avoid falling down, you need to stand with your feet spread apart. Why do vehicles roll over? Figure 8-14 shows two vehicles rolling over. Note that the one with the higher center of mass does not have to be tilted very far for its center of mass to be outside its base—its center of mass does not have to be raised as much as the other vehicle’s. The lower the location of an object’s center of mass, the greater its stability. You are stable when you stand flat on your feet. When you stand on tiptoe, however, your center of mass moves forward directly above the balls of your feet, and you have very little stability. A small person can use torque, rather than force, to defend himself or herself against a stronger person. In judo, aikido, and other martial arts, the fighter uses torque to rotate the opponent into an unstable position, where the opponent’s center of mass does not lie above his or her feet. In summary, if the center of mass is outside the base of an object, it is unstable and will roll over without additional torque. If the center of mass is above the base of the object, it is stable. If the base of the object is very narrow and the center of mass is high, then the object is stable, but the slightest force will cause it to tip over.

Spinning Tops 1. Cut out two cardboard disks of 10-cm and 15-cm diameter. 2. Use a pencil with an eraser that has rounded edges from use. If it is new, rub it on paper to round it. 3. Spin the pencil and try to make it stand on the eraser. Repeat several times and record your observations. 4. Carefully push the pencil through the center of the 10-cm disk. 5. Spin the pencil with the disk and try to make it stand on the eraser. 6. Move the disk to different points on the pencil. Spin and record your observations. 7. Repeat steps 4-6 with the 15-cm disk. Analyze and Conclude 8. Sequence the three trials in order from least to most stable. 9. Describe the location of the pencil’s center of mass. 10. Analyze the placement of the disk and its effect on stability.

Conditions for Equilibrium If your pen is at rest, what is needed to keep it at rest? You could either hold it up or place it on a desk or some other surface. An upward force must be exerted on the pen to balance the downward force of gravity. You must also hold the pen so that it will not rotate. An object is said to be in static equilibrium if both its velocity and angular velocity are zero or constant. Thus, for an object to be in static equilibrium, it must meet two conditions. First, it must be in translational equilibrium; that is, the net force exerted on the object must be zero. Second, it must be in rotational equilibrium; that is, the net torque exerted on the object must be zero.

■ Figure 8-14 Larger vehicles have a higher center of mass than smaller ones. The higher the center of mass, the smaller the tilt needed to cause the vehicle’s center of mass to move outside its base and cause the vehicle to roll over.

1

2

Section 8.3 Equilibrium

213

Static Equilibrium A 5.8-kg ladder, 1.80 m long, rests on two sawhorses. Sawhorse A is 0.60 m from one end of the ladder, and sawhorse B is 0.15 m from the other end of the ladder. What force does each sawhorse exert on the ladder? 1

0.60 m

0.30 m

0.75 m

Analyze and Sketch the Problem • Sketch the situation. • Choose the axis of rotation at the point where FA acts on the ladder. Thus, the torque due to FA is zero. Known: m 5.8 kg l 1.80 m lA 0.60 m lB 0.15 m

2

0.15 m

1.80 m

A

B

FA

FB

Unknown: FA ? FB ?

Fg rB rg

FB

Solve for the Unknown For a ladder that has a constant density, the center of mass is at the center rung.

Axis of rotation

Fg

The net force is the sum of all forces on the ladder. Fnet FA FB (Fg) The ladder is in translational equilibrium, so the net force exerted on it is zero. 0.0 N FA FB Fg Solve for FA. FA Fg FB Find the torques due to Fg and FB. g rgFg g is in the clockwise direction. B rBFB B is in the counterclockwise direction. The net torque is the sum of all torques on the object. net B g 0.0 Nm B g The ladder is in rotational equilibrium, so net 0.0 Nm. B g rBFB rgFg Substitute B rBFB, g rgFg Solve for FB.

Math Handbook

rgFg FB rB rgmg rB

Isolating a Variable page 845 Substitute Fg mg

Using the expression FA Fg FB, substitute in the expressions for FB and Fg. FA Fg FB rgmg

Fg

rB rgmg mg rB rg mg (1 ) rB

214

Chapter 8 Rotational Motion

rgmg

Substitute FB rB

Substitute Fg mg

Solve for rg. 1 2

rg lA

For a ladder, which has a constant density, the center of mass is at the center rung.

0.90 m 0.60 m 0.30 m

l 2

Substitute 0.90 m, lA 0.60 m

Solve for rB. rB (0.90 m l B) (0.90 m l A) (0.90 m 0.15 m) (0.90 m 0.60 m) 0.75 m 0.30 m 1.05 m

Substitute lB 0.15 m, lA 0.60 m

Calculate FB. rgmg

FB rB

(0.30 m)(5.8 kg)(9.80 m/s2) (1.05 m)

Substitute rg 0.30 m, m 5.8 kg, g 9.80 m/s2, rB 1.05 m

16 N Calculate FA.

rg

FA mg 1

rB

(0.30 m) (1.05 m)

1 (5.8 kg)(9.80 m/s2) 41 N 3

Substitute rg 0.30 m, m 5.8 kg, g 9.80 m/s2, rB 1.05 m

Evaluate the Answer • Are the units correct? Forces are measured in newtons. • Do the signs make sense? Both forces are upward. • Is the magnitude realistic? The forces add up to the weight of the ladder, and the force exerted by the sawhorse closer to the center of mass is greater, which is correct.

36. What would be the forces exerted by the two sawhorses if the ladder in Example Problem 5 had a mass of 11.4 kg? 37. A 7.3-kg ladder, 1.92 m long, rests on two sawhorses, as shown in Figure 8-15. Sawhorse A, on the left, is located 0.30 m from the end, and sawhorse B, on the right, is located 0.45 m from the other end. Choose the axis 1.92 m of rotation to be the center of mass of the ladder. a. What are the torques acting on the ladder?

0.30 m

0.66 m

0.45 m

0.51 m

b. Write the equation for rotational equilibrium. c. Solve the equation for FA in terms of Fg. d. How would the forces exerted by the two sawhorses change if A were moved very close to, but not directly under, the center of mass?

■

A Figure 8-15

B

38. A 4.5-m-long wooden plank with a 24-kg mass is supported in two places. One support is directly under the center of the board, and the other is at one end. What are the forces exerted by the two supports? 39. A 85-kg diver walks to the end of a diving board. The board, which is 3.5 m long with a mass of 14 kg, is supported at the center of mass of the board and at one end. What are the forces on the two supports?

Section 8.3 Equilibrium

215

Rotating Frames of Reference When you are on a on a rapidly spinning amusement-park ride, it feels like a strong force is pushing you to the outside. A pebble on the floor of the ride would accelerate outward without a horizontal force being exerted on it in the same direction. The pebble would not move in a straight line. In other words, Newton’s laws would not apply. This is because rotating frames of reference are accelerated frames. Newton’s laws are valid only in inertial or nonaccelerated frames. Motion in a rotating reference frame is important to us because Earth rotates. The effects of the rotation of Earth are too small to be noticed in the classroom or lab, but they are significant influences on the motion of the atmosphere and therefore, on climate and weather.

Centrifugal “Force” Suppose you fasten one end of a spring to the center of a rotating platform. An object lies on the platform and is attached to the other end of the spring. As the platform rotates, an observer on the platform sees the object stretch the spring. The observer might think that some force toward the outside of the platform is pulling on the object. This apparent force is called centrifugal “force.” It is not a real force because there is no physical outward push on the object. Still, this “force” seems real, as anyone who has ever been on an amusement-park ride can attest. As the platform rotates, an observer on the ground sees things differently. This observer sees the object moving in a circle. The object accelerates toward the center because of the force of the spring. As you know, the acceleration is centripetal acceleration and is given by ac v 2/r. It also can be written in terms of angular velocity, as ac 2/r. Centripetal acceleration is proportional to the distance from the axis of rotation and depends on the square of the angular velocity. Thus, if you double the rotational frequency, the acceleration increases by a factor of 4.

The Coriolis “Force” A second effect of rotation is shown in Figure 8-16. Suppose a person standing at the center of a rotating disk throws a ball toward the edge of the disk. Consider the horizontal motion of the ball as seen by two observers and ignore the vertical motion of the ball as it falls. ■ Figure 8-16 The Coriolis “force” exists only in rotating reference frames.

a

Viewed from fixed frame

Observed positions of ball

Chapter 8 Rotational Motion

Coriolis "force" viewed from rotating frame

Center

216

b

Observed positions of ball

An observer standing outside the disk, as shown in Figure 8-16a, sees the ball travel in a straight line at a constant speed toward the edge of the disk. However, the other observer, who is stationed on the disk and rotating with it, as shown in Figure 8-16b, sees the ball follow a curved path at a constant speed. A force seems to be acting to deflect the ball. This apparent force is called the Coriolis “force.” Like the centrifugal “force,” the Coriolis “force” is not a real force. It seems to exist because we observe a deflection in horizontal motion when we are in a rotating frame of reference. Coriolis “force” due to Earth Suppose a cannon is fired from a point on the equator toward a target due north of it. If the projectile were fired directly northward, it would also have an eastward velocity component because of the rotation of Earth. This eastward speed is greater at the equator than at any other latitude. Thus, as the projectile moves northward, it also moves eastward faster than points on Earth below it do. The result is that the projectile lands east of the target as shown in Figure 8-17. While an observer in space would see Earth’s rotation, an observer on Earth could claim that the projectile missed the target because of the Coriolis “force” on the rocket. Note that for objects moving toward the equator, the direction of the apparent force is westward. A projectile will land west of the target when fired due south. The direction of winds around high- and low-pressure areas results from the Coriolis “force.” Winds flow from areas of high to low pressure. Because of the Coriolis “force” in the northern hemisphere, winds from the south go to the east of low-pressure areas. Winds from the north, however, end up west of low-pressure areas. Therefore, winds rotate counterclockwise around low-pressure areas in the northern hemisphere. In the southern hemisphere however, winds rotate clockwise around low-pressure areas. Most amusement-park rides thrill the riders because they are in accelerated reference frames while on the ride. The “forces” felt by roller-coaster riders at the tops and bottoms of hills, and when moving almost vertically downward, are mostly related to linear acceleration. On Ferris wheels, rotors, other circular rides, and on the curves of roller coasters, centrifugal “forces” provide most of the excitement.

Actual path Expected path

Equator

■ Figure 8-17 An observer on Earth sees the Coriolis “force” cause a projectile fired due north to deflect to the right of the intended target.

Meteorology Connection

8.3 Section Review 40. Center of Mass Can the center of mass of an object be located in an area where the object has no mass? Explain. 41. Stability of an Object Why is a modified vehicle with its body raised high on risers less stable than a similar vehicle with its body at normal height? 42. Conditions for Equilibrium Give an example of an object for each of the following conditions. a. rotational equilibrium, but not translational equilibrium b. translational equilibrium, but not rotational equilibrium physicspp.com/self_check_quiz

43. Center of Mass Where is the center of mass of a roll of masking tape? 44. Locating the Center of Mass Describe how you would find the center of mass of this textbook. 45. Rotating Frames of Reference A penny is placed on a rotating, old-fashioned record turntable. At the highest speed, the penny starts sliding outward. What are the forces acting on the penny? 46. Critical Thinking You have learned why the winds around a low-pressure area move in a counterclockwise direction. Would the winds move in the same or opposite direction in the southern hemisphere? Explain. Section 8.3 Equilibrium

217

For maintenance on large buildings, scaffolding can be hung on the outside. In order for the scaffolding to support workers, it must be in translational and rotational equilibrium. If two or more forces act on the scaffolding, each can produce a rotation about either end. Scaffolding with uniform mass distribution acts as though all of the mass is concentrated at its center. In translational equilibrium the object is not accelerating; thus, the upward and downward forces are equal. In order to achieve rotational equilibrium, the sum of all the clockwise torques must equal the sum of all the counterclockwise torques as measured from a pivot point. That is, the net torque must be zero. In this lab you will model scaffolding hung from two ropes using a meterstick and spring scales, and use numbers to measure the forces on the scaffolding.

QUESTION What conditions are required for equilibrium when parallel forces act on an object?

Objectives

Procedure

■ Collect and organize data about the forces

The left spring scale will be considered a pivot point for the purposes of this lab. Therefore, the lever arm will be measured from this point.

acting on the scaffolding. ■ Describe clockwise and counterclockwise torque. ■ Compare and contrast translational and rotational equilibrium.

2. Attach a Buret clamp to each of the ring stands. 3. Verify that the scales are set to zero before use. If the scales need to be adjusted, ask your teacher for assistance.

Safety Precautions

■ Use care to avoid dropping masses.

Materials meterstick two 0-5 N spring scales two ring stands

1. Place the ring stands 80 cm apart.

two Buret clamps 500-g hooked mass 200-g hooked mass

4. Hang a spring scale from each Buret clamp attached to a ring stand. 5. Hook the meterstick onto the spring scale in such a manner that the 10-cm mark is supported by one hook and the 90-cm mark is supported by the other hook. 6. Read each spring scale and record the force in Data Table 1. 7. Hang a 500-g mass on the meterstick at the 30-cm mark. This point should be 20-cm from the left scale. 8. Read each spring scale and record the force in Data Table 1. 9. Hang a 200-g mass on the meterstick at the 70-cm mark. This point should be 60 cm from the left scale. 10. Read each spring scale and record the force in Data Table 1.

218

Horizons Companies

Translational and Rotational Equilibrium Alternate CBL instructions can be found on the Web site. physicspp.com

Data Table 1 Object Added

Distance From Left Scale (m)

Meterstick

0.4

500-g mass

0.2

200-g mass

0.6

Left Scale Reading (N)

Right Scale Reading (N)

Data Table 2 Object Added

c

cc

Lever Arm (m)

Force (N)

Meterstick 500-g mass 200-g mass Right scale

Data Table 3 Object Added

c (Nm)

cc (Nm)

Meterstick 500-g mass 200-g mass Right scale

Analyze 1. Calculate Find the mass of the meterstick. 2. Calculate Find the force, or weight, that results from each object and record it in Data Table 2. For the right scale, read the force it exerts and record it in Data Table 2. 3. Using the point where the left scale is attached as a pivot point, identify the forces located elsewhere that cause the scaffold to rotate clockwise or counterclockwise. Mark these in Data Table 2 with an x. 4. Record the lever arm distance of each force from the pivot point in Data Table 2. 5. Use Numbers Calculate the torque for each object by multiplying the force and lever arm distance. Record these values in Data Table 3.

3. Compare and contrast the sum of the clockwise torques, c, and the counterclockwise torques, cc. 4. What is the percent difference between c and cc?

Going Further Use additional masses at locations of your choice with your teacher’s permission and record your data.

Real-World Physics Research the safety requirements in your area for putting up, using, and dismantling scaffolding.

Conclude and Apply 1. Is the system in translational equilibrium? How do you know? 2. Draw a free-body diagram of your system, showing all the forces.

To find out more about rotational motion, visit the Web site: physicspp.com

219

The Stability of Sport-Utility Vehicles Why are sport-utility vehicles more flippable? Many believe that the large size of the sport-utility vehicle makes it more stable and secure. But, a sport-utility vehicle, as well as other tall vehicles such as vans, is much more likely to roll over than a car.

ESC hydraulic unit

Engine control unit

Wheel speed sensors

The Problem A sport-utility vehicle has a Steering high center of mass which makes it more angle likely to topple. Another factor that affects sensor rollover is the static stability factor, which Yaw is the ratio of the track width to the center sensor of mass. Track width is defined as half the distance between the two front wheels. The higher the static stability factor, the more likely a vehicle will stay upright. Many sport-utility vehicles have a center of Wheel mass 13 or 15 cm higher than passenger cars. speed sensors Their track width, however, is about the same as that of passenger cars. Suppose the stability factor for a sport-utility vehicle is 1.06 An ESC system processes information from the sensors and automatically applies the brakes to individual and 1.43 for a car. Statistics show that in a wheels when instability is detected. single-vehicle crash, the sport-utility vehicle has a 37 percent chance of rolling over, while A promising new technology called Electronic the car has a 10.6 percent chance of rolling over. Stability Control (ESC) can be used to prevent However, the static stability factor oversimplirollover accidents. An ESC system has elecfies the issue. Weather and driver behavior are tronic sensors that detect when a vehicle begins also contributers to rollover crashes. Vehicle to spin due to oversteering, and also when it factors, such as tires, suspension systems, begins to slide in a plowlike manner because inertial properties, and advanced handling of understeering. In these instances, an ESC systems all play a role as well. system automatically applies the brakes at one It is true that most rollover crashes occur or more wheels, thereby reorienting the vehicle when a vehicle swerves off the road and hits a in the right direction. rut, soft soil, or other surface irregularity. This Safe driving is the key to preventing many usually occurs when a driver is not paying automobile accidents. Knowledge of the physics proper attention or is speeding. Safe drivers behind rollover accidents and the factors that greatly reduce their chances of being involved affect rollover accidents may help make you an in a rollover accident by paying attention and informed, safe driver. driving at the correct speed. Still, weather and driver behavior being equal, the laws of physics indicate that sport-utility vehicles carry an Going Further increased risk. What Is Being Done? Some models are being built with wider track widths or stronger roofs. Optional side-curtain air bags have sensors to keep the bags inflated for up to 6 s, rather than the usual fraction of a second. This will cushion passengers if the vehicle should flip several times. 220

Technology and Society

1. Hypothesize In a multi-vehicle accident, sport-utility vehicles generally fare better than the passenger cars involved in the accident. Why is this so? 2. Debate the Issue ESC is a life-saving technology. Should it be mandatory in all sport-utility vehicles? Why or why not?

8.1 Describing Rotational Motion Vocabulary

Key Concepts

• radian (p. 197) • angular displacement

• •

(p. 198)

Angular position and its changes are measured in radians. One complete revolution is 2 rad. Angular velocity is given by the following equation.

• angular velocity (p. 198) • angular acceleration

t

(p. 199)

•

Angular acceleration is given by the following equation. t

•

For a rotating, rigid object, the angular displacement, velocity, and acceleration can be related to the linear displacement, velocity, and acceleration for any point on the object. d r

v r

a r

8.2 Rotational Dynamics Vocabulary

Key Concepts

• • • •

• •

lever arm (p. 201) torque (p. 202) moment of inertia (p. 205) Newton’s second law for rotational motion (p. 208)

When torque is exerted on an object, its angular velocity changes. Torque depends on the magnitude of the force, the distance from the axis of rotation at which it is applied, and the angle between the force and the radius from the axis of rotation to the point where the force is applied. Fr sin

•

The moment of inertia of an object depends on the way the object’s mass is distributed about the rotational axis. For a point object: I mr2

•

Newton’s second law for rotational motion states that angular acceleration is directly proportional to the net torque and inversely proportional to the moment of inertia. net

I

8.3 Equilibrium Vocabulary

Key Concepts

• center of mass (p. 211) • centrifugal “force” (p. 216) • Coriolis “force” (p. 217)

• • • •

The center of mass of an object is the point on the object that moves in the same way that a point particle would move. An object is stable against rollover if its center of mass is above its base. An object is in equilibrium if there are no net forces exerted on it and if there are no net torques acting on it. Centrifugal “force” and the Coriolis “force” are two apparent forces that appear when a rotating object is analyzed from a coordinate system that rotates with it.

physicspp.com/vocabulary_puzzlemaker

221

Concept Mapping 47. Complete the following concept map using the following terms: angular acceleration, radius, tangential acceleration, centripetal acceleration.

57. A stunt driver maneuvers a monster truck so that it is traveling on only two wheels. Where is the center of mass of the truck? (8.3)

58. Suppose you stand flat-footed, then you rise and

Angular velocity

balance on tiptoe. If you stand with your toes touching a wall, you cannot balance on tiptoe. Explain. (8.3)

59. Why does a gymnast appear to be floating on air when she raises her arms above her head in a leap? (8.3)

60. Why is a vehicle with wheels that have a large

Mastering Concepts 48. A bicycle wheel rotates at a constant 25 rev/min. Is its angular velocity decreasing, increasing, or constant? (8.1)

49. A toy rotates at a constant 5 rev/min. Is its angular acceleration positive, negative, or zero? (8.1)

50. Do all parts of Earth rotate at the same rate? Explain. (8.1)

diameter more likely to roll over than a vehicle with wheels that have a smaller diameter? (8.3)

Applying Concepts 61. Two gears are in contact and rotating. One is larger than the other, as shown in Figure 8-19. Compare their angular velocities. Also compare the linear velocities of two teeth that are in contact.

51. A unicycle wheel rotates at a constant 14 rev/min. Is the total acceleration of a point on the tire inward, outward, tangential, or zero? (8.1)

52. Think about some possible rotations of your textbook. Are the moments of inertia about these three axes the same or different? Explain. (8.2)

53. Torque is important when tightening bolts. Why is force not important? (8.2)

54. Rank the torques on the five doors shown in Figure 8-18 from least to greatest. Note that the magnitude of all the forces is the same. (8.2) B

A

■

Figure 8-19

62. Videotape When a videotape is rewound, why does it wind up fastest towards the end?

63. Spin Cycle What does a spin cycle of a washing machine do? Explain in terms of the forces on the clothes and water. C

D

64. How can you experimentally find the moment of inertia of an object?

65. Bicycle Wheels Three bicycle wheels have masses E ■

Figure 8-18

55. Explain how you can change an object’s angular frequency. (8.2)

56. To balance a car’s wheel, it is placed on a vertical shaft and weights are added to make the wheel horizontal. Why is this equivalent to moving the center of mass until it is at the center of the wheel? (8.3)

222

that are distributed in three different ways: mostly at the rim, uniformly, and mostly at the hub. The wheels all have the same mass. If equal torques are applied to them, which one will have the greatest angular acceleration? Which one will have the least?

66. Bowling Ball When a bowling ball leaves a bowler’s hand, it does not spin. After it has gone about half the length of the lane, however, it does spin. Explain how its rotation rate increased and why it does not continue to increase.

Chapter 8 Rotational Motion For more problems, go to Additional Problems, Appendix B.

67. Flat Tire Suppose your car has a flat tire. You get out your tools and find a lug wrench to remove the nuts off the bolt studs. You find it impossible to turn the nuts. Your friend suggests ways you might produce enough torque to turn them. What three ways might your friend suggest?

73. The outer edge of a truck tire that has a radius of 45 cm has a velocity of 23 m/s. What is the angular velocity of the tire in rad/s?

74. A steering wheel is rotated through 128°, as shown in Figure 8-22. Its radius is 22 cm. How far would a point on the steering wheel’s edge move?

68. Tightrope Walkers Tightrope walkers often carry long poles that sag so that the ends are lower than the center as shown in Figure 8-20. How does such a pole increase the tightrope walker’s stability? Hint: Consider both center of mass and moment of inertia.

22 cm

128°

■

Figure 8-22

75. Propeller A propeller spins at 1880 rev/min. ■

a. What is its angular velocity in rad/s? b. What is the angular displacement of the propeller in 2.50 s?

Figure 8-20

69. Merry-Go-Round While riding a merry-go-round, you toss a key to a friend standing on the ground. For your friend to be able to catch the key, should you toss it a second or two before you reach the spot where your friend is standing or wait until your friend is directly behind you? Explain.

70. Why can you ignore forces that act on the axis of

76. The propeller in the previous problem slows from 475 rev/min to 187 rev/min in 4.00 s. What is its angular acceleration?

77. An automobile wheel with a 9.00 cm radius, as shown in Figure 8-23, rotates at 2.50 rad/s. How fast does a point 7.00 cm from the center travel?

0 cm

rotation of an object in static equilibrium when determining the net torque?

71. In solving problems about static equilibrium, why

9.0

is the axis of rotation often placed at a point where one or more forces are acting on the object?

7.00 cm

Mastering Problems 8.1 Describing Rotational Motion 72. A wheel is rotated so that a point on the edge moves through 1.50 m. The radius of the wheel is 2.50 m, as shown in Figure 8-21. Through what angle (in radians) is the wheel rotated?

■

Figure 8-23

2.

50

m

78. Washing Machine A washing machine’s two spin

1.50 m

cycles are 328 rev/min and 542 rev/min. The diameter of the drum is 0.43 m. a. What is the ratio of the centripetal accelerations for the fast and slow spin cycles? Recall that v2 ac and v rw r

b. What is the ratio of the linear velocity of an object at the surface of the drum for the fast and slow spin cycles?

79. Find the maximum centripetal acceleration in terms ■

Figure 8-21 physicspp.com/chapter_test

of g for the washing machine in problem 78. Chapter 8 Assessment

223

Mark D. Phillips/AFP/CORBIS

80. A laboratory ultracentrifuge is designed to produce a centripetal acceleration of 0.35106 g at a distance of 2.50 cm from the axis. What angular velocity in rev/min is required?

8.2 Rotational Dynamics 81. Wrench A bolt is to be tightened with a torque of 8.0 Nm. If you have a wrench that is 0.35 m long, what is the least amount of force you must exert?

82. What is the torque on a bolt produced by a 15-N force exerted perpendicular to a wrench that is 25 cm long, as shown in Figure 8-24? F

8.3 Equilibrium 86. A 12.5-kg board, 4.00 m long, is being held up on one end by Ahmed. He calls for help, and Judi responds. a. What is the least force that Judi could exert to lift the board to the horizontal position? What part of the board should she lift to exert this force? b. What is the greatest force that Judi could exert to lift the board to the horizontal position? What part of the board should she lift to exert this force?

87. Two people are holding up the ends of a 4.25-kg wooden board that is 1.75 m long. A 6.00-kg box sits on the board, 0.50 m from one end, as shown in Figure 8-26. What forces do the two people exert?

90°

6.00 kg

25 cm 1.25 m

0.50 m

■

Figure 8-26

88. A car’s specifications state that its weight distribution

■

is 53 percent on the front tires and 47 percent on the rear tires. The wheel base is 2.46 m. Where is the car’s center of mass?

Figure 8-24

83. A toy consisting of two balls, each 0.45 kg, at the ends of a 0.46-m-long, thin, lightweight rod is shown in Figure 8-25. Find the moment of inertia of the toy. The moment of inertia is to be found about the center of the rod. 0.45 kg

0.45 kg

Mixed Review 89. A wooden door of mass, m, and length, l, is held horizontally by Dan and Ajit. Dan suddenly drops his end. a. What is the angular acceleration of the door just after Dan lets go? b. Is the acceleration constant? Explain.

90. Topsoil Ten bags of topsoil, each weighing 175 N,

0.46 m ■

are placed on a 2.43-m-long sheet of wood. They are stacked 0.50 m from one end of the sheet of wood, as shown in Figure 8-27. Two people lift the sheet of wood, one at each end. Ignoring the weight of the wood, how much force must each person exert?

Figure 8-25

84. A bicycle wheel with a radius of 38 cm is given an angular acceleration of 2.67 rad/s2 by applying a force of 0.35 N on the edge of the wheel. What is the wheel’s moment of inertia?

85. Toy Top A toy top consists of a rod with a diameter of 8.0-mm and a disk of mass 0.0125 kg and a diameter of 3.5 cm. The moment of inertia of the rod can be neglected. The top is spun by wrapping a string around the rod and pulling it with a velocity that increases from zero to 3.0 m/s over 0.50 s.

1.93 m

a. What is the resulting angular velocity of the top? b. What force was exerted on the string?

224

Chapter 8 Rotational Motion For more problems, go to Additional Problems, Appendix B.

0.50 m ■

Figure 8-27

91. Basketball A basketball is rolled down the court. A regulation basketball has a diameter of 24.1 cm, a mass of 0.60 kg, and a moment of inertia of 5.8103 kgm2. The basketball’s initial velocity is 2.5 m/s. a. What is its initial angular velocity? b. The ball rolls a total of 12 m. How many revolutions does it make? c. What is its total angular displacement?

92. The basketball in the previous problem stops rolling after traveling 12 m. a. If its acceleration was constant, what was its angular acceleration? b. What torque was acting on it as it was slowing down?

97. The second hand on a watch is 12 mm long. What is the velocity of its tip?

98. Lumber You buy a 2.44-m-long piece of 10 cm 10 cm lumber. Your friend buys a piece of the same size and cuts it into two lengths, each 1.22 m long, as shown in Figure 8-29. You each carry your lumber on your shoulders. a. Which load is easier to lift? Why? b. Both you and your friend apply a torque with your hands to keep the lumber from rotating. Which load is easier to keep from rotating? Why?

2.44 m

93. A cylinder with a 50 m diameter, as shown in Figure 8-28, is at rest on a surface. A rope is wrapped around the cylinder and pulled. The cylinder rolls without slipping. a. After the rope has been pulled a distance of 2.50 m at a constant speed, how far has the center of mass of the cylinder moved? b. If the rope was pulled a distance of 2.50 m in 1.25 s, how fast was the center of mass of the cylinder moving? c. What is the angular velocity of the cylinder?

1.22 m 1.22 m ■

Figure 8-29

99. Surfboard Harris and Paul carry a surfboard that is 2.43 m long and weighs 143 N. Paul lifts one end with a force of 57 N. a. What force must Harris exert? b. What part of the board should Harris lift?

100. A steel beam that is 6.50 m long weighs 325 N. 50 m

■

Figure 8-28

94. Hard Drive A hard drive on a modern computer spins at 7200 rpm (revolutions per minute). If the drive is designed to start from rest and reach operating speed in 1.5 s, what is the angular acceleration of the disk?

95. Speedometers Most speedometers in automobiles measure the angular velocity of the transmission and convert it to speed. How will increasing the diameter of the tires affect the reading of the speedometer?

96. A box is dragged across the floor using a rope that is a distance h above the floor. The coefficient of friction is 0.35. The box is 0.50 m high and 0.25 m wide. Find the force that just tips the box. physicspp.com/chapter_test

It rests on two supports, 3.00 m apart, with equal amounts of the beam extending from each end. Suki, who weighs 575 N, stands on the beam in the center and then walks toward one end. How close to the end can she come before the beam begins to tip?

Thinking Critically 101. Apply Concepts Consider a point on the edge of a rotating wheel. a. Under what conditions can the centripetal acceleration be zero? b. Under what conditions can the tangential (linear) acceleration be zero? c. Can the tangential acceleration be nonzero while the centripetal acceleration is zero? Explain. d. Can the centripetal acceleration be nonzero while the tangential acceleration is zero? Explain.

102. Apply Concepts When you apply the brakes in a car, the front end dips. Why? Chapter 8 Assessment

225

103. Analyze and Conclude A banner is suspended from a horizontal, pivoted pole, as shown in Figure 8-30. The pole is 2.10 m long and weighs 175 N. The banner, which weighs 105 N, is suspended 1.80 m from the pivot point or axis of rotation. What is the tension in the cable supporting the pole?

105. Analyze and Conclude Gerald and Evelyn carry the following objects up a flight of stairs: a large mirror, a dresser, and a television. Evelyn is at the front end, and Gerald is at the bottom end. Assume that both Evelyn and Gerald exert only upward forces. a. Draw a free-body diagram showing Gerald and Evelyn exerting the same force on the mirror. b. Draw a free-body diagram showing Gerald exerting more force on the bottom of the dresser.

Cable

Axis of rotation

Center of mass

c. Where would the center of mass of the television have to be so that Gerald carries all the weight? 25.0°

1.05 m

Writing in Physics

1.80 m

106. Astronomers know that if a satellite is too close to

2.10 m

a planet, it will be torn apart by tidal forces. That is, the difference in the gravitational force on the part of the satellite nearest the planet and the part farthest from the planet is stronger than the forces holding the satellite together. Do research on the Roche limit and determine how close the Moon would have to orbit Earth to be at the Roche limit.

107. Automobile engines are rated by the torque that ■

they produce. Research and explain why torque is an important quantity to measure.

Figure 8-30

104. Analyze and Conclude A pivoted lamp pole is shown in Figure 8-31. The pole weighs 27 N, and the lamp weighs 64 N. a. What is the torque caused by each force? b. Determine the tension in the rope supporting the lamp pole.

Cumulative Review 108. Two blocks, one of mass 2.0 kg and the other of mass 3.0 kg, are tied together with a massless rope. This rope is strung over a massless, resistance-free pulley. The blocks are released from rest. Find the following. (Chapter 4) a. the tension in the rope b. the acceleration of the blocks.

Rope

y

109. Eric sits on a see-saw. At what angle, relative to the vertical, will the component of his weight parallel to the plane be equal to one-third the perpendicular component of his weight? (Chapter 5)

105.0° Axis of rotation

x

110. The pilot of a plane wants to reach an airport

0.33 m

325 km due north in 2.75 hours. A wind is blowing from the west at 30.0 km/h. What heading and airspeed should be chosen to reach the destination on time? (Chapter 6)

0.44 m

Lamp

■

226

Figure 8-31

111. A 60.0-kg speed skater with a velocity of 18.0 m/s comes into a curve of 20.0-m radius. How much friction must be exerted between the skates and ice to negotiate the curve? (Chapter 6)

Chapter 8 Rotational Motion For more problems, go to Additional Problems, Appendix B.

Multiple Choice 1. The illustration below shows two boxes on opposite ends of a board that is 3.0 m long. The board is supported in the middle by a fulcrum. The box on the left has a mass, m1, of 25 kg, and the box on the right has a mass, m2, of 15 kg. How far should the fulcrum be positioned from the left side of the board in order to balance the masses horizontally? 0.38 m 0.60 m

1.1 m 1.9 m

1.3 rad/s 2.5 rad/s

5.0 rad/s 6.3 rad/s

6. Two of the tires on a farmer’s tractor have diameters of 1.5 m. If the farmer drives the tractor at a linear velocity of 3.0 m/s, what is the angular velocity of each tire? m2

m1

2. A force of 60 N is exerted on one end of a 1.0-m-long lever. The other end of the lever is attached to a rotating rod that is perpendicular to the lever. By pushing down on the end of the lever, you can rotate the rod. If the force on the lever is exerted at an angle of 30°, what torque is exerted on the lever? (sin 30° 0.5; cos 30° 0.87; tan 30° 0.58) 30 N 52 N

5. A thin hoop with a mass of 5.0 kg rotates about a perpendicular axis through its center. A force of 25 N is exerted tangentially to the hoop. If the hoop’s radius is 2.0 m, what is its angular acceleration?

2.0 rad/s 2.3 rad/s

4.0 rad/s 4.5 rad/s

Extended Answer 7. You use a 25-cm long wrench to remove the lug nuts on a car wheel, as shown in the illustration below. If you pull up on the end of the wrench with a force of 2.0102 N at an angle of 30°, what is the torque on the wrench? (sin 30° 0.5, cos 30° 0.87) F 30°

60 N 69 N 25

3. A child attempts to use a wrench to remove a nut on a bicycle. Removing the nut requires a torque of 10 N·m. The maximum force the child is capable of exerting at a 90° angle is 50 N. What is the length of the wrench the child must use to remove the nut? 0.1 m 0.15 m

cm

0.2 m 0.25 m

4. A car moves a distance of 420 m. Each tire on the car has a diameter of 42 cm. Which of the following shows how many revolutions each tire makes as they move that distance? 5.0101 rev 1.0102 rev

1.5102 rev 1.0103 rev

physicspp.com/standardized_test

When Eliminating, Cross It Out Consider each answer choice individually and cross out the ones you have eliminated. If you cannot write in the test booklet, use the scratch paper to list and cross off the answer choices. You will save time and stop yourself from choosing an answer you have mentally eliminated.

Chapter 8 Standardized Test Practice

227

What You’ll Learn • You will describe momentum and impulse and apply them to the interactions between objects. • You will relate Newton’s third law of motion to conservation of momentum. • You will explore the momentum of rotating objects.

Why It’s Important Momentum is the key to success in many sporting events, including baseball, football, ice hockey, and tennis. Baseball Every baseball player dreams of hitting a home run. When a player hits the ball, at the moment of collision, the ball and the bat are deformed by the collision. The resulting change in momentum determines the batter’s success.

Think About This What is the force on a baseball bat when a home run is hit out of the park?

physicspp.com 228 fotobankyokohama/firstlight.ca

What happens when a hollow plastic ball strikes a bocce ball? Question What direction will a hollow plastic ball and a bocce ball move after a head-on collision? Procedure 1. Roll a bocce ball and a hollow plastic ball toward each other on a smooth surface. 2. Observe the direction each one moves after the collision. 3. Repeat the experiment, this time keeping the bocce ball stationary, while rolling the hollow plastic ball toward it. 4. Observe the direction each one moves after the collision. 5. Repeat the experiment one more time, but keep the hollow plastic ball stationary, while rolling the bocce ball toward it. 6. Observe the direction each one moves after the collision.

Critical Thinking What factor(s) would cause the bocce ball to move backward after colliding with the hollow plastic ball?

Analysis What factors affect how fast the balls move after the collision? What factors determine the direction each one moves after the collision?

9.1 Impulse and Momentum

I

t is always exciting to watch a baseball player hit a home run. The pitcher fires the baseball toward the plate. The batter swings at the baseball and the baseball recoils from the impact of the bat at high speed. Rather than concentrating on the force between the baseball and bat and their resulting accelerations, as in previous chapters, you will approach this collision in a different way in this chapter. The first step in analyzing this type of interaction is to describe what happens before, during, and after the collision between the baseball and bat. You can simplify the collision between the baseball and the bat by making the assumption that all motion is in the horizontal direction. Before the collision, the baseball moves toward the bat. During the collision, the baseball is squashed against the bat. After the collision, however, the baseball moves at a higher velocity away from the bat, and the bat continues in its path, but at a slower velocity.

Objectives • Define the momentum of an object. • Determine the impulse given to an object. • Define the angular momentum of an object.

Vocabulary impulse momentum impulse-momentum theorem angular momentum angular impulse-angular momentum theorem

Section 9.1 Impulse and Momentum

229

Horizons Companies

1.5104

Force on a Baseball

Impulse and Momentum

Force (N)

How are the velocities of the ball, before and after the collision, related to the force acting on it? Newton’s second law of motion describes how the 1.0104 velocity of an object is changed by a net force acting on it. The change in velocity of the ball must have been caused by the force exerted by the bat on 5.0103 the ball. The force changes over time, as shown in Figure 9-1. Just after contact is made, the ball is squeezed, and the force increases. After the force 0.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 reaches its maximum, which is more than 10,000 times the weight of the ball, the ball recovers its Time (ms) shape and snaps away from the bat. The force rapidly returns to zero. This whole event takes place within about 3.0 ms. ■ Figure 9-1 The force acting on a baseball increases, then rapidly How can you calculate the change in velocity of the baseball? decreases during a collision, as shown in this force-time graph.

Impulse Newton’s second law of motion, F ma, can be rewritten by using the definition of acceleration as the change in velocity divided by the time needed to make that change. It can be represented by the following equation:

t

v F ma m

Multiplying both sides of the equation by the time interval, t, results in the following equation: Ft mv Impulse, or Ft, is the product of the average force on an object and the time interval over which it acts. Impulse is measured in newton-seconds. For instances in which the force varies with time, the magnitude of an impulse is found by determining the area under the curve of a force-time graph, such as the one shown in Figure 9-1. The right side of the equation, mv, involves the change in velocity: v vf vi. Therefore, mv mvf mvi. The product of the object’s mass, m, and the object’s velocity, v, is defined as the momentum of the object. Momentum is measured in kgm/s. An object’s momentum, also known as linear momentum, is represented by the following equation. Momentum p mv The momentum of an object is equal to the mass of the object times the object’s velocity.

• Momentum and impulse vectors are orange. • Force vectors are blue. • Acceleration vectors are violet. • Velocity vectors are red. • Displacement vectors are green.

230

Recall the equation Ft mv mvf mvi. Because mvf pf and mvi pi, this equation can be rewritten as follows: Ft mv pf pi. The right side of this equation, pf pi, describes the change in momentum of an object. Thus, the impulse on an object is equal to the change in its momentum, which is called the impulse-momentum theorem. The impulse-momentum theorem is represented by the following equation. Impulse-Momentum Theorem Ft pf pi The impulse on an object is equal to the object’s final momentum minus the object’s initial momentum.

Chapter 9 Momentum and Its Conservation

If the force on an object is constant, the impulse is the product of the force multiplied by the time interval over which it acts. Generally, the force is not constant, however, and the impulse is found by using an average force multiplied by the time interval over which it acts, or by finding the area under a force-time graph. Because velocity is a vector, momentum also is a vector. Similarly, impulse is a vector because force is a vector. This means that signs will be important for motion in one dimension.

Using the Impulse-Momentum Theorem What is the change in momentum of a baseball? From the impulsemomentum theorem, you know that the change in momentum is equal to the impulse acting on it. The impulse on a baseball can be calculated by using a force-time graph. In Figure 9-1, the area under the curve is approximately 13.1 Ns. The direction of the impulse is in the direction of the force. Therefore, the change in momentum of the ball also is 13.1 Ns. Because 1 Ns is equal to 1 kgm/s, the momentum gained by the ball is 13.1 kgm/s in the direction of the force acting on it. Assume that a batter hits a fastball. Before the collision of the ball and bat, the ball, with a mass of 0.145 kg, has a velocity of 38 m/s. Assume that the positive direction is toward the pitcher. Therefore, the baseball’s momentum is pi (0.145 kg)(38 m/s) 5.5 kgm/s. What is the momentum of the ball after the collision? Solve the impulse-momentum theorem for the final momentum: pf pi Ft. The ball’s final momentum is the sum of the initial momentum and the impulse. Thus, the ball’s final momentum is calculated as follows.

Running Shoes Running is hard on the feet. When a runner’s foot strikes the ground, the force exerted by the ground on it is as much as four times the runner’s weight. The cushioning in an athletic shoe is designed to reduce this force by lengthening the time interval over which the force is exerted.

pf pi 13.1 kgm/s 5.5 kgm/s 13.1 kgm/s 7.6 kgm/s What is the baseball’s final velocity? Because pf mvf, solving for vf yields the following: p m

7.6 kgm/s 0.145 kgm/s

vf f 52 m/s A speed of 52 m/s is fast enough to clear most outfield fences if the baseball is hit in the correct direction.

■ Figure 9-2 An air bag is inflated during a collision when the force due to the impact triggers the sensor. The chemicals in the air bag’s inflation system react and produce a gas that rapidly inflates the air bag.

Using the Impulse-Momentum Theorem to Save Lives A large change in momentum occurs only when there is a large impulse. A large impulse can result either from a large force acting over a short period of time or from a smaller force acting over a long period of time. What happens to the driver when a crash suddenly stops a car? An impulse is needed to bring the driver’s momentum to zero. According to the impulse-momentum equation, Ft pf pi. The final momentum, pf , is zero. The initial momentum, pi, is the same with or without an air bag. Thus, the impulse, Ft, also is the same. An air bag, such as the one shown in Figure 9-2, reduces the force by increasing the time interval during which it acts. It also exerts the force over a larger area of the person’s body, thereby reducing the likelihood of injuries. Section 9.1 Impulse and Momentum

231

Rick Fischer/Masterfile

Average Force A 2200-kg vehicle traveling at 94 km/h (26 m/s) can be stopped in 21s by gently applying the brakes. It can be stopped in 3.8 s if the driver slams on the brakes, or in 0.22 s if it hits a concrete wall. What average force is exerted on the vehicle in each of these stops? 1

2200 kg

Analyze and Sketch the Problem • Sketch the system. • Include a coordinate axis and select the positive direction to be the direction of the velocity of the car. • Draw a vector diagram for momentum and impulse. Known: m 2200 kg vi +26 m/s vf +0.0 m/s

2

x

tgentle braking 21 s t hard braking 3.8 s t hitting a wall 0.22 s

94 km/h

Unknown:

Vector diagram

Fgentle braking ? F hard braking ? F hitting a wall ?

pi

pf

Impulse

Solve for the Unknown Determine the initial momentum, pi. pi mvi (2200 kg)(26 m/s) Substitute m 2200 kg, vi 26 m/s 5.7104 kgm/s Determine the final momentum, pf. pf mvf (2200 kg)(0.0 m/s) Substitute m 2200 kg, vf 0.0 m/s 0.0 kgm/s

Apply the impulse-momentum theorem to obtain the force needed to stop the vehicle. Ft pf pi Ft (0.0 kgm/s) (5.7104 kgm/s) Substitute pf 0.0 kgm/s, pi 5.7104 kgm/s 5.7104 kgm/s 5.7104 kgm/s

F t 5.7104 kgm/s 21 s

Fgentle braking

Substitute tgentle braking 21 s

2.7103 N 5.7104

kgm/s 3.8 s

F hard braking

Math Handbook Substitute thard braking 3.8 s

1.5104 N 5.7104 kgm/s 0.22 s

F hitting a wall

Substitute thitting a wall 0.22 s

2.6105 N 3

Evaluate the Answer • Are the units correct? Force is measured in newtons. • Does the direction make sense? Force is exerted in the direction opposite to the velocity of the car and thus, is negative. • Is the magnitude realistic? People weigh hundreds of newtons, so it is reasonable that the force needed to stop a car would be in the thousands of newtons. The impulse is the same for all three stops. Thus, as the stopping time is shortened by more than a factor of 10, the force is increased by more than a factor of 10.

232

Chapter 9 Momentum and Its Conservation

Operations with Significant Digits pages 835–836

1. A compact car, with mass 725 kg, is moving at 115 km/h toward the east. Sketch the moving car. a. Find the magnitude and direction of its momentum. Draw an arrow on your sketch showing the momentum.

Force (N)

a

b. A second car, with a mass of 2175 kg, has the same momentum. What is its velocity? 2. The driver of the compact car in the previous problem suddenly applies the brakes hard for 2.0 s. As a result, an average force of 5.0103 N is exerted on the car to slow it down.

1 2 Time (s)

a. What is the change in momentum; that is, the magnitude and direction of the impulse, on the car?

Force (N)

b

b. Complete the “before” and “after” sketches, and determine the momentum and the velocity of the car now. 3. A 7.0-kg bowling ball is rolling down the alley with a velocity of 2.0 m/s. For each impulse, shown in Figures 9-3a and 9-3b, find the resulting speed and direction of motion of the bowling ball. 4. The driver accelerates a 240.0-kg snowmobile, which results in a force being exerted that speeds up the snowmobile from 6.00 m/s to 28.0 m/s over a time interval of 60.0 s.

5 0 5

5 0 5

1 2

Time (s) ■

Figure 9-3

a. Sketch the event, showing the initial and final situations. b. What is the snowmobile’s change in momentum? What is the impulse on the snowmobile? c. What is the magnitude of the average force that is exerted on the snowmobile? 5. Suppose a 60.0-kg person was in the vehicle that hit the concrete wall in Example Problem 1. The velocity of the person equals that of the car both before and after the crash, and the velocity changes in 0.20 s. Sketch the problem. a. What is the average force exerted on the person? b. Some people think that they can stop their bodies from lurching forward in a vehicle that is suddenly braking by putting their hands on the dashboard. Find the mass of an object that has a weight equal to the force you just calculated. Could you lift such a mass? Are you strong enough to stop your body with your arms?

Angular Momentum As you learned in Chapter 8, the angular velocity of a rotating object changes only if torque is applied to it. This is a statement of Newton’s law for rotational motion, I/t. This equation can be rearranged in the same way as Newton’s second law of motion was, to produce t I. The left side of this equation, t, is the angular impulse of the rotating object. The right side can be rewritten as f i. The product of a rotating object’s moment of inertia and angular velocity is called angular momentum, which is represented by the symbol L. The angular momentum of an object can be represented by the following equation. Angular Momentum L = I The angular momentum of an object is equal to the product of the object’s moment of inertia and the object’s angular velocity.

Section 9.1 Impulse and Momentum

233

Angular momentum is measured in kgm2/s. Just as the linear momentum of an object changes when an impulse acts on it, the angular momentum of an object changes when an angular impulse acts on it. Thus, the angular impulse on the object is equal to the change in the object’s angular momentum, which is called the angular impulse-angular momentum theorem. The angular impulse-angular momentum theorem is represented by the following equation. Angular Impulse-Angular Momentum Theorem t Lf Li The angular impulse on an object is equal to the object’s final angular momentum minus the object’s initial angular momentum.

Astronomy Connection

If there are no forces acting on an object, its linear momentum is constant. If there are no torques acting on an object, its angular momentum is also constant. Because an object’s mass cannot be changed, if its momentum is constant, then its velocity is also constant. In the case of angular momentum, however, the object’s angular velocity does not remain constant. This is because the moment of inertia depends on the object’s mass and the way it is distributed about the axis of rotation or revolution. Thus, the angular velocity of an object can change even if no torques are acting on it. Consider, for example, a planet orbiting the Sun. The torque on the planet is zero because the gravitational force acts directly toward the Sun. Therefore, the planet’s angular momentum is constant. When the distance between the planet and the Sun decreases, however, the planet’s moment of inertia of revolution in orbit about the Sun also decreases. Thus, the planet’s angular velocity increases and it moves faster. This is an explanation of Kepler’s second law of planetary motion, based on Newton’s laws of motion. ■ Figure 9-4 The diver’s center of mass is in front of her feet as she gets ready to dive (a). As the diver changes her moment of inertia by moving her arms and legs to increase her angular momentum, the location of the center of mass changes, but the path of the center of mass remains a parabola (b).

a

b

Center of mass

Fg

F

234 Tim Fuller

Chapter 9 Momentum and Its Conservation

Consider the diver in Figure 9-4. How does she start rotating her body? She uses the diving board to apply an external torque to her body. Then, she moves her center of mass in front of her feet and uses the board to give a final upward push to her feet. This torque acts over time, t, and thus increases the angular momentum of the diver. Before the diver reaches the water, she can change her angular velocity by changing her moment of inertia. She may go into a tuck position, grabbing her knees with her hands. By moving her mass closer to the axis of rotation, the diver decreases her moment of inertia and increases her angular velocity. When she nears the water, she stretches her body straight, thereby increasing the moment of inertia and reducing the angular velocity. As a result, she goes straight into the water. An ice-skater uses a similar method to spin. To begin rotating on one foot, the ice-skater applies an external torque to her body by pushing a portion of the other skate into the ice, as shown in Figure 9-5. If she pushes on the ice in one direction, the ice will exert a force on her in the opposite direction. The force results in a torque if the force is exerted some distance away from the pivot point, and in a direction that is not toward it. The greatest torque for a given force will result if the push is perpendicular to the lever arm. The ice-skater then can control his angular velocity by changing her moment of inertia. Both arms and one leg can be extended from the body to slow the rotation, or pulled in close to the axis of rotation to speed it up. To stop spinning, another torque must be exerted by using the second skate to create a way for the ice to exert the needed force.

■ Figure 9-5 To spin on one foot, an ice-skater extends one leg and pushes on the ice. The ice exerts an equal and opposite force on her body and produces an external torque.

9.1 Section Review 6. Momentum Is the momentum of a car traveling south different from that of the same car when it travels north at the same speed? Draw the momentum vectors to support your answer. 7. Impulse and Momentum When you jump from a height to the ground, you let your legs bend at the knees as your feet hit the floor. Explain why you do this in terms of the physics concepts introduced in this chapter. 8. Momentum Which has more momentum, a supertanker tied to a dock or a falling raindrop? 9. Impulse and Momentum A 0.174-kg softball is pitched horizontally at 26.0 m/s. The ball moves in the opposite direction at 38.0 m/s after it is hit by the bat. a. Draw arrows showing the ball’s momentum before and after the bat hits it. b. What is the change in momentum of the ball? c. What is the impulse delivered by the bat? d. If the bat and softball are in contact for 0.80 ms, what is the average force that the bat exerts on the ball? physicspp.com/self_check_quiz

10. Momentum The speed of a basketball as it is dribbled is the same when the ball is going toward the floor as it is when the ball rises from the floor. Is the basketball’s change in momentum equal to zero when it hits the floor? If not, in which direction is the change in momentum? Draw the basketball’s momentum vectors before and after it hits the floor. 11. Angular Momentum An ice-skater spins with his arms outstretched. When he pulls his arms in and raises them above his head, he spins much faster than before. Did a torque act on the iceskater? If not, how could his angular velocity have increased? 12. Critical Thinking An archer shoots arrows at a target. Some of the arrows stick in the target, while others bounce off. Assuming that the masses of the arrows and the velocities of the arrows are the same, which arrows produce a bigger impulse on the target? Hint: Draw a diagram to show the momentum of the arrows before and after hitting the target for the two instances. Section 9.1 Impulse and Momentum

235

Rick Stewart/Getty Images

9.2 Conservation of Momentum

Objectives • Relate Newton’s third law to conservation of momentum. • Recognize the conditions under which momentum is conserved. • Solve conservation of momentum problems.

Two-Particle Collisions

Vocabulary closed system isolated system law of conservation of momentum law of conservation of angular momentum

Before Collision (initial)

C

I

n the first section of this chapter, you learned how a force applied during a time interval changes the momentum of a baseball. In the discussion of Newton’s third law of motion, you learned that forces are the result of interactions between two objects. The force of a bat on a ball is accompanied by an equal and opposite force of the ball on the bat. Does the momentum of the bat, therefore, also change?

D

The bat, the hand and arm of the batter, and the ground on which the batter is standing are all objects that interact when a batter hits the ball. Thus, the bat cannot be considered a single object. In contrast to this complex system, examine for a moment the much simpler system shown in Figure 9-6, the collision of two balls. During the collision of the two balls, each one briefly exerts a force on the other. Despite the differences in sizes and velocities of the balls, the forces that they exert on each other are equal and opposite, according to Newton’s third law of motion. These forces are represented by the following equation: FD on C FC on D How do the impulses imparted by both balls compare? Because the time intervals over which the forces are exerted are the same, the impulses must be equal in magnitude but opposite in direction. How did the momenta of the balls change as a result of the collision? According to the impulse-momentum theorem, the change in momentum is equal to the impulse. Compare the changes in the momenta of the two balls. For ball C: pCf pCi FD on C t

pDi

pCi

During Collision

C

For ball D: pDf pDi FC on D t Because the time interval over which the forces were exerted is the same, the impulses are equal in magnitude, but opposite in direction. According to Newton’s third law of motion, FC on D FD on C. Thus,

D

pCf pCi (pDf pDi), or pCf pDf pCi pDi. FD on C

FC on D

After Collision (final)

C

This equation states that the sum of the momenta of the balls is the same before and after the collision. That is, the momentum gained by ball D is equal to the momentum lost by ball C. If the system is defined as the two balls, the momentum of the system is constant, and therefore, momentum is conserved for the system.

D

Momentum in a Closed, Isolated System pCf

■ Figure 9-6 When two balls collide, they exert forces on each other that change their momenta.

236

Under what conditions is the momentum of the system of two balls conserved? The first and most obvious condition is that no balls are lost and no balls are gained. Such a system, which does not gain or lose mass, is said to be a closed system. The second condition required to conserve the momentum of a system is that the forces involved are internal forces; that is, there are no forces acting on the system by objects outside of it.

Chapter 9 Momentum and Its Conservation

When the net external force on a closed system is zero, the system is described as an isolated system. No system on Earth can be said to be absolutely isolated, however, because there will always be some interactions between a system and its surroundings. Often, these interactions are small enough to be ignored when solving physics problems. Systems can contain any number of objects, and the objects can stick together or come apart in a collision. Under these conditions, the law of conservation of momentum states that the momentum of any closed, isolated system does not change. This law will enable you to make a connection between conditions, before and after an interaction, without knowing any of the details of the interaction.

Speed A 1875-kg car going 23 m/s rear-ends a 1025-kg compact car going 17 m/s on ice in the same direction. The two cars stick together. How fast do the two cars move together immediately after the collision? 1

Analyze and Sketch the Problem • Define the system. • Establish a coordinate system. • Sketch the situation showing the “before” and “after” states. • Draw a vector diagram for the momentum. Known: mC = 1875 kg vCi = +23 m/s mD = 1025 kg vDi = +17 m/s

2

Unknown: vf = ?

Solve for the Unknown Momentum is conserved because the ice makes the total external force on the cars nearly zero. pi pf pCi pDi pCf pDf mCvCi mDvDi mCvCf mDvDf

x

Before (initial)

After (final)

C

D

C

vCi

vDi

vCf vDf vf

Vector diagram pCi

(m v m v ) (mC mD) (1875 kg)(23 m/s) (1025 kg)(17 m/s) (1875 kg 1025 kg)

pDi

pf

pi pCi pDi

Because the two cars stick together, their velocities after the collision, denoted as vf, are equal. vCf vDf vf mCvCi mDvDi (mC mD)vf Solve for vf.

D

Math Handbook Order of Operations page 843

C Ci D Di vf

21 m/s 3

Substitute mC 1875 kg, vCi 23 m/s, mD 1025 kg, vDi 17 m/s

Evaluate the Answer • Are the units correct? Velocity is measured in m/s. • Does the direction make sense? vi and vf are in the positive direction; therefore, vf should be positive. • Is the magnitude realistic? The magnitude of vf is between the initial speeds of the two cars, but closer to the speed of the more massive one, so it is reasonable.

Section 9.2 Conservation of Momentum

237

13. Two freight cars, each with a mass of 3.0105 kg, collide and stick together. One was initially moving at 2.2 m/s, and the other was at rest. What is their final speed? 14. A 0.105-kg hockey puck moving at 24 m/s is caught and held by a 75-kg goalie at rest. With what speed does the goalie slide on the ice? 15. A 35.0-g bullet strikes a 5.0-kg stationary piece of lumber and embeds itself in the wood. The piece of lumber and bullet fly off together at 8.6 m/s. What was the original speed of the bullet? 16. A 35.0-g bullet moving at 475 m/s strikes a 2.5-kg bag of flour that is on ice, at rest. The bullet passes through the bag, as shown in Figure 9-7, and exits it at 275 m/s. How fast is the bag moving when the bullet exits?

275 m/s

17. The bullet in the previous problem strikes a 2.5-kg steel ball that is at rest. The bullet bounces backward after its collision at a speed of 5.0 m/s. How fast is the ball moving when the bullet bounces backward? 18. A 0.50-kg ball that is traveling at 6.0 m/s collides head-on with a 1.00-kg ball moving in the opposite direction at a speed of 12.0 m/s. The 0.50-kg ball bounces backward at 14 m/s after the collision. Find the speed of the second ball after the collision.

■

Figure 9-7

Recoil It is very important to define a system carefully. The momentum of a baseball changes when the external force of a bat is exerted on it. The baseball, therefore, is not an isolated system. On the other hand, the total momentum of two colliding balls within an isolated system does not change because all forces are between the objects within the system. Can you find the final velocities of the two in-line skaters in Figure 9-8? Assume that they are skating on a smooth surface with no external forces. They both start at rest, one behind the other.

a

■ Figure 9-8 The internal forces exerted by Skater C, the boy, and Skater D, the girl, cannot change the total momentum of the system.

238 Laura Sifferlin

Chapter 9 Momentum and Its Conservation

b

Skater C, the boy, gives skater D, the girl, a push. Now, both skaters are moving, making this situation similar to that of an explosion. Because the push was an internal force, you can use the law of conservation of momentum to find the skaters’ relative velocities. The total momentum of the system was zero before the push. Therefore, it must be zero after the push. Before

After

pCi pDi

pCf pDf

pCf pDf

pCf

mCvCf

mDvDf

The coordinate system was chosen so that the positive direction is to the left. The momenta of the skaters after the push are equal in magnitude but opposite in direction. The backward motion of skater C is an example of recoil. Are the skaters’ velocities equal and opposite? The last equation shown above, for the velocity of skater C, can be rewritten as follows:

■ Figure 9-9 The xenon atoms in the ion engine are ionized by bombarding them with electrons. Then, the positively charged xenon ions are accelerated to high speeds.

m mC

D vCf vDf

The velocities depend on the skaters’ relative masses. If skater C has a mass of 68.0 kg and skater D’s mass is 45.4 kg, then the ratio of their velocities will be 68.0 : 45.4, or 1.50. The less massive skater moves at the greater velocity. Without more information about how hard skater C pushed skater D, however, you cannot find the velocity of each skater.

Propulsion in Space How does a rocket in space change its velocity? The rocket carries both fuel and oxidizer. When the fuel and oxidizer combine in the rocket motor, the resulting hot gases leave the exhaust nozzle at high speed. If the rocket and chemicals are the system, then the system is a closed system. The forces that expel the gases are internal forces, so the system is also an isolated system. Thus, objects in space can accelerate by using the law of conservation of momentum and Newton’s third law of motion. A NASA space probe, called Deep Space 1, performed a flyby of an asteroid a few years ago. The most unusual of the 11 new technologies on board was an ion engine that exerts as much force as a sheet of paper resting on a person’s hand. The ion engine shown in Figure 9-9, operates differently from a traditional rocket engine. In a traditional rocket engine, the products of the chemical reaction taking place in the combustion chamber are released at high speed from the rear. In the ion engine, however, xenon atoms are expelled at a speed of 30 km/s, producing a force of only 0.092 N. How can such a small force create a significant change in the momentum of the probe? Instead of operating for only a few minutes, as the traditional chemical rockets do, the ion engine can run continuously for days, weeks, or months. Therefore, the impulse delivered by the engine is large enough to increase the momentum of the 490-kg spacecraft until it reaches the speed needed to complete its mission.

Rebound Height An object’s momentum is the product of its mass and velocity. 1. Drop a large rubber ball from about 15 cm above a table. 2. Measure and record the ball’s rebound height. 3. Repeat steps 1–2 with a small rubber ball. 4. Hold the small rubber ball on top of, and in contact with, the large rubber ball. 5. Release the two rubber balls from the same height, so that they fall together. 6. Measure the rebound heights of both rubber balls. Analyze and Conclude 7. Describe the rebound height of each rubber ball dropped by itself. 8. Compare and contrast the rebound heights from number 7 with those from number 6. 9. Explain your observations.

Section 9.2 Conservation of Momentum

239 NASA

Speed An astronaut at rest in space fires a thruster pistol that expels 35 g of hot gas at 875 m/s. The combined mass of the astronaut and pistol is 84 kg. How fast and in what direction is the astronaut moving after firing the pistol? 1

Before (initial)

Analyze and Sketch the Problem • • • •

2

+x

vDf

Define the system. Establish a coordinate axis. Sketch the “before” and “after” conditions. Draw a vector diagram showing momenta. Known:

Unknown:

mC 84 kg mD 0.035 kg vCi vDi +0.0 m/s vDf 875 m/s

vCf ?

After (final)

vi 0.0 m/s • pi

vCf

D

Vector Diagram pDf

pCf • pf pCf pDf

Solve for the Unknown

The system is the astronaut, the gun, and the chemicals that produce the gas. pi pCi pDi = 0.0 kgm/s Before the pistol is fired, all parts of the system are at rest; thus, the initial momentum is zero.

Use the law of conservation of momentum to find pf. pi pf 0.0 kgm/s pCf pDf The momentum of the astronaut is equal in magnitude, but opposite pCf pDf in direction to the momentum of the gas leaving the pistol. Solve for the final velocity of the astronaut, vCf. Math Handbook mCvCf mDvDf vCf

m v mC (0.035 kg)(875 m/s) 84 kg

(

Isolating a Variable page 845

)

Df D

Substitute mD 0.035 kg, vDf 875 m/s, mC 84 kg

+0.36 m/s 3

Evaluate the Answer • Are the units correct? The velocity is measured in m/s. • Does the direction make sense? The velocity of the astronaut is in the opposite direction to that of the expelled gas. • Is the magnitude realistic? The astronaut’s mass is much larger than that of the gas, so the velocity of the astronaut is much less than that of the expelled gas.

19. A 4.00-kg model rocket is launched, expelling 50.0 g of burned fuel from its exhaust at a speed of 625 m/s. What is the velocity of the rocket after the fuel has burned? Hint: Ignore the external forces of gravity and air resistance. 20. A thread holds a 1.5-kg cart and a 4.5-kg cart together. After the thread is burned, a compressed spring pushes the carts apart, giving the 1.5-kg cart a speed of 27 cm/s to the left. What is the velocity of the 4.5-kg cart? 21. Carmen and Judi dock a canoe. 80.0-kg Carmen moves forward at 4.0 m/s as she leaves the canoe. At what speed and in what direction do the canoe and Judi move if their combined mass is 115 kg?

240

Chapter 9 Momentum and Its Conservation

Two-Dimensional Collisions Up until now, you have looked at momentum in only one dimension. The law of conservation of momentum holds for all closed systems with no external forces. It is valid regardless of the directions of the particles before or after they interact. But what happens in two or three dimensions? Figure 9-10 shows the result of billiard ball C striking stationary billiard ball D. Consider the two billiard balls to be the system. The original momentum of the moving ball is pCi and the momentum of the stationary ball is zero. Therefore, the momentum of the system before the collision is equal to pCi. After the collision, both billiard balls are moving and have momenta. As long as the friction with the tabletop can be ignored, the system is closed and isolated. Thus, the law of conservation of momentum can be used. The initial momentum equals the vector sum of the final momenta, so pCi pCf pDf. The equality of the momenta before and after the collision also means that the sum of the components of the vectors before and after the collision must be equal. Suppose the x-axis is defined to be in the direction of the initial momentum, then the y-component of the initial momentum is equal to zero. Therefore, the sum of the final y-components also must be zero: pCf, y pDf, y 0 The y-components are equal in magnitude but are in the opposite direction and, thus, have opposite signs. The sum of the horizontal components also is equal: pCi pCf, x pDf, x

■

Figure 9-10 The law of conservation of momentum holds for all isolated, closed systems, regardless of the directions of objects before and after a collision. C

pCf C

C pCi

D

D

Section 9.2 Conservation of Momentum

241

Speed A 1325-kg car, C, moving north at 27.0 m/s, collides with a 2165-kg car, D, moving east at 11.0 m/s. The two cars are stuck together. In what direction and with what speed do they move after the collision? • Define the system. • Sketch the “before” and “after” states. • Establish the coordinate axis with the y-axis north and the x-axis east. • Draw a momentum-vector diagram. Unknown:

mC 1325 kg mD 2165 kg vCi, y 27.0 m/s vDi, x 11.0 m/s

vf, x ? vf, y ? ?

vf x

D

D

C

vDi 11.0 m/s Vector Diagram

Solve for the Unknown

vCi

2

Known:

After (final)

Before (initial) y

Analyze and Sketch the Problem

27.0 m/s

1

pDi C

Determine the initial momenta of the cars and the momentum of the system. pCi mCvCi, y (1325 kg)(27.0 m/s) Substitute mC = 1325 kg, vCi, y 27.0 m/s 4 3.5810 kgm/s (north) pDi mDvDi, x (2165 kg)(11.0 m/s) Substitute mD 2165 kg, vDi, x 11.0 m/s 2.38104 kgm/s (east)

90° pCi

pf

Use the law of conservation of momentum to find pf. pf, x pi, x 2.38104 kgm/s Substitute pi, x pDi 2.38104 kgm/s pf, y pi, y 3.58104 kgm/s Substitute pi, y pCi 3.58104 kgm/s Use the diagram to set up equations for pf, x and pf, y . 2 pf (pf, x )2 (pf, y)

(2.38 104 kg m/s)2 (3.5 8104 kgm/ s)2 4 4.3010 kgm/s Solve for .

tan1

3.58104 kgm/s 2.3810 kgm/s

tan1 4

Math Handbook

Inverses of Sine, Cosine, and Tangent page 856

pf, y pf, x

( ) (

Substitute pf, x 2.38104 kgm/s, pf, y 3.58104 kgm/s

)

Substitute pf, y 3.58104 kgm/s, pf, x 2.38104 kgm/s

56.4° Determine the final speed. p (mC mD)

f vf

4.30104 kgm/s (1325 kg 2165 kg)

Substitute pf 4.30104 kgm/s, mC 1325 kg, mD 2165 kg

12.3 m/s 3

Evaluate the Answer • Are the units correct? The correct unit for speed is m/s. • Do the signs make sense? Answers are both positive and at the appropriate angles. • Is the magnitude realistic? The cars stick together, so vf must be smaller than vCi.

242

Chapter 9 Momentum and Its Conservation

22. A 925-kg car moving north at 20.1 m/s collides with a 1865-kg car moving west at 13.4 m/s. The two cars are stuck together. In what direction and at what speed do they move after the collision? 23. A 1383-kg car moving south at 11.2 m/s is struck by a 1732-kg car moving east at 31.3 m/s. The cars are stuck together. How fast and in what direction do they move immediately after the collision? 24. A stationary billiard ball, with a mass of 0.17 kg, is struck by an identical ball moving at 4.0 m/s. After the collision, the second ball moves 60.0° to the left of its original direction. The stationary ball moves 30.0° to the right of the moving ball’s original direction. What is the velocity of each ball after the collision? 25. A 1345-kg car moving east at 15.7 m/s is struck by a 1923-kg car moving north. They are stuck together and move with an initial velocity of 14.5 m/s at = 63.5°. Was the north-moving car exceeding the 20.1 m/s speed limit?

Conservation of Angular Momentum Like linear momentum, angular momentum can be conserved. The law of conservation of angular momentum states that if no net external torque acts on an object, then its angular momentum does not change. This is represented by the following equation. Law of Conservation of Angular Momentum L1 L2 An object’s initial angular momentum is equal to its final angular momentum.

For example, Earth spins on its axis with no external torques. Its angular momentum is constant. Thus, Earth’s angular momentum is conserved. As a result, the length of a day does not change. A spinning ice-skater also demonstrates conservation of angular momentum. Figure 9-11a shows an ice-skater spinning with his arms extended. When he pulls in his arms, as shown in Figure 9-11b, he begins spinning faster. Without an external torque, his angular momentum does not change; that is, L I is constant. Thus, the ice-skater’s increased angular velocity must be accompanied by a decreased moment of inertia. By pulling his arms close to his body, the ice-skater brings more mass closer to the axis of rotation, thereby decreasing the radius of rotation and decreasing his moment of inertia. You can calculate changes in angular velocity using the law of conservation of angular momentum. Li Lf

■

Figure 9-11 When the iceskater’s arms are extended, the moment of inertia increases and his angular velocity decreases (a). When his arms are closer to his body the moment of inertia decreases and results in an increased angular velocity (b).

a

b

thus, Iii If f I f i i If

Because frequency is f /2, the above equation can be rewritten as follows: 2 ( ff) I i If 2 ( fi) f I thus, f i fi If

Section 9.2 Conservation of Momentum

243

F. Scott Grant/IMAGE Communications

Rotational axis

Notice that because f, , and I appear as ratios in these equations, any units may be used, as long as the same unit is used for both values of the quantity. If a torque-free object starts with no angular momentum, it must continue to have no angular momentum. Precession Thus, if part of an object rotates in one direction, due to torque another part must rotate in the opposite direction. For example, if you switch on a loosely held electric drill, the drill body will rotate in the direction opposite to the rotation of the motor and bit. Angle of Consider a ball thrown at a weather vane. The ball, rotational axis moving in a straight line, can start the vane rotating. with the Consider the ball and vane to be a system. With no vertical external torques, angular momentum is conserved. The Pivot point vane spins faster if the ball has a large mass, m, a large velocity, v, and hits at right angles as far as possible from the pivot of the vane. The angular momentum of a moving object, such as the ball, is given by L mvr, where r is the perpendicular distance from the axis of rotation.

Vertical Spin

Center of mass Gravitational force

■ Figure 9-12 The upper end of the top precesses due to the torque acting on the top.

Tops and Gyroscopes Because of the conservation of angular momentum, the direction of rotation of a spinning object can be changed only by applying a torque. If you played with a top as a child, you may have spun it by pulling the string wrapped around its axle. When a top is vertical, there is no torque on it, and the direction of its rotation does not change. If the top is tipped, as shown in Figure 9-12, a torque tries to rotate it downward. Rather than tipping over, however, the upper end of the top revolves, or precesses slowly about the vertical axis. Because Earth is not a perfect sphere, the Sun exerts a torque on it, causing it to precess. It takes about 26,000 years for Earth’s rotational axis to go through one cycle of precession.

Your friend was driving her 1265-kg car north on Oak Street when she was hit by a 925-kg compact car going west on Maple Street. The cars stuck together and slid 23.1 m at 42° north of west. The speed limit on both streets is 22 m/s (50 mph). Assume that momentum was conserved during the collision and that acceleration was constant during the skid. The coefficient of kinetic friction between the tires and the pavement is 0.65.

y x

42°

1. Your friend claims that she wasn’t speeding, but that the driver of other car was. How fast was your friend driving before the crash?

2. How fast was the other car moving before the crash? Can you support your friend’s case in court?

244

Chapter 9 Momentum and Its Conservation

1265 kg

925 kg

file photo

A gyroscope, such as the one shown in Figure 9-13, is a wheel or disk that spins rapidly around one axis while being free to rotate around one or two other axes. The direction of its large angular momentum can be changed only by applying an appropriate torque. Without such a torque, the direction of the axis of rotation does not change. Gyroscopes are used in airplanes, submarines, and spacecraft to keep an unchanging reference direction. Giant gyroscopes are used in cruise ships to reduce their motion in rough water. Gyroscopic compasses, unlike magnetic compasses, maintain direction even when they are not on a level surface. A football quarterback uses the gyroscope effect to make an accurate forward pass. As he throws, he spins, or spirals the ball. If the quarterback throws the ball in the direction of its spin axis of rotation, the ball keeps its pointed end forward, thereby reducing air resistance. Thus, the ball can be thrown far and accurately. If its spin direction is slightly off, the ball wobbles. If the ball is not spun, it tumbles end over end. The flight of a plastic disk also is stabilized by spin. A well-spun plastic disk can fly many meters through the air without wobbling. You are able to perform tricks with a yo-yo because its fast rotational speed keeps it rotating in one plane.

■

Figure 9-13 Because the orientation of the spin axis of the gyroscope does not change even when it is moved, the gyroscope can be used to fix direction.

9.2 Section Review 26. Angular Momentum The outer rim of a plastic disk is thick and heavy. Besides making it easier to catch, how does this affect the rotational properties of the plastic disk? 27. Speed A cart, weighing 24.5 N, is released from rest on a 1.00-m ramp, inclined at an angle of 30.0° as shown in Figure 9-14. The cart rolls down the incline and strikes a second cart weighing 36.8 N. a. Calculate the speed of the first cart at the bottom of the incline. b. If the two carts stick together, with what initial speed will they move along? 24 .5

N

1.0

0m

36.8 N 30.0°

■

Figure 9-14 physicspp.com/self_check_quiz

28. Conservation of Momentum During a tennis serve, the racket of a tennis player continues forward after it hits the ball. Is momentum conserved in the collision? Explain, making sure that you define the system. 29. Momentum A pole-vaulter runs toward the launch point with horizontal momentum. Where does the vertical momentum come from as the athlete vaults over the crossbar? 30. Initial Momentum During a soccer game, two players come from opposite directions and collide when trying to head the ball. They come to rest in midair and fall to the ground. Describe their initial momenta. 31. Critical Thinking You catch a heavy ball while you are standing on a skateboard, and then you roll backward. If you were standing on the ground, however, you would be able to avoid moving while catching the ball. Explain both situations using the law of conservation of momentum. Explain which system you use in each case. Section 9.2 Conservation of Momentum

245 file photo

Horizons Companies

Sticky Collisions Alternate CBL instructions can be found on the Web site. physicspp.com

In this activity, one moving cart will strike a stationary cart. During the collision, the two carts will stick together. You will measure mass and velocity, both before and after the collision. You then will calculate the momentum both before and after the collision.

QUESTION How is the momentum of a system affected by a sticky collision?

Objectives

Procedure

■ Describe how momentum is transferred during

1. View Chapter 9 lab video clip 1 at physicspp.com/internet_lab to determine the mass of the carts.

a collision. ■ Calculate the momenta involved. ■ Interpret data from a collision. ■ Draw conclusions that support the law of

conservation of momentum.

Safety Precautions

2. Record the mass of each cart. 3. Watch video clip 2: Cart 1 strikes Cart 2. 4. In the video, three frames represent 0.1 s. Record in the data table the distance Cart 1 travels in 0.1 s before the collision. 5. Observe the collision. Record in the data table the distance the Cart 1-Cart 2 system travels in 0.1 s after the collision. 6. Repeat steps 3–5 for video clip 3: Carts 1 and 3 strike Cart 2.

Materials Internet access required

7. Repeat steps 3–5 for video clip 4: Carts 1, 3, and 4 strike Cart 2. 8. Repeat steps 3-5 for video clip 5: Carts 1 and 3 strike Carts 2 and 4. 9. Repeat steps 3–5 for video clip 6: Cart 1 strikes Carts 2, 3, and 4.

246

Data Tables Cart

Mass (kg)

1 2 3 4 Time of Approach (s)

Distance Covered in Approach (cm)

Initial Velocity (cm/s)

Mass of Approaching Cart(s) (g)

Initial Momentum (gcm/s)

Time of Departure (s)

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

Distance Covered in Departure (cm)

Final Velocity (cm/s)

Mass of Departing Cart(s) (g)

Final Momentum (gcm/s)

Analyze

Real-World Physics

1. Calculate the initial and final velocities for each of the cart systems. 2. Calculate the initial and final momentum for each of the cart systems.

1. Suppose a linebacker collides with a stationary quarterback and they become entangled. What will happen to the velocity of the linebackerquarterback system if momentum is conserved?

3. Make and Use Graphs Make a graph showing final momentum versus initial momentum for all the video clips.

2. If a car rear-ends a stationary car so that the two cars become attached, what will happen to the velocity of the first car? The second car?

Conclude and Apply 1. What is the relationship between the initial momentum and the final momentum of the cart systems in a sticky collision? 2. In theory, what should be the slope of the line in your graph? 3. The initial and final data numbers may not be the same due to the precision of the instruments, friction, and other variables. Is the initial momentum typically greater or less than the final momentum? Explain.

Interpret Data Visit physicspp.com/ internet_lab to post your findings from the experiment testing the impact of friction during the collisions of the cart systems. Examine the data and graph of the final momentum versus initial momentum on the Web site. Notice how close to or far off the slope is from 1.00.

Going Further 1. Describe what the velocity and momentum data might look like if the carts did not stick together, but rather, bounced off of each other? 2. Design an experiment to test the impact of friction during the collision of the cart systems. Predict how the slope of the line in your graph will change with your experiment, and then try your experiment.

To find out more about momentum, visit the Web site: physicspp.com

247

Solar Sailing Nearly 400 years ago, Johannes Kepler observed that comet tails appeared to be blown by a solar breeze. He suggested that ships would be able to travel in space with sails designed to catch this breeze. Thus, the idea for solar sails was born.

but solar sails only require photons from the Sun. Thus, solar sails may be a superior way to move large masses over great distances in outer space.

Future Journeys The Cosmos 1, the first solar-sail prototype, is scheduled for a launch in the very near future. The Cosmos-1 mission How Does a Solar Sail Work? A solar is an international, privately funded venture. sail is a spacecraft without an engine. A solar The spacecraft looks like a flower with eight sail works like a giant fabric mirror that is huge, solar-sail petals. free to move. Solar sails Being the first solar sail, usually are made of goals are modest. The 5-micron-thick alumission will be considminized polyester film ered successful if the or polyimide film with Cosmos 1 operates for just a 100-nm-thick alua few days, accelerating minum layer deposited under sunlight pressure. on one side to form Solar sails are importhe reflective surface. tant, not only for travel, Reflected sunlight, but also for creating new rather than rocket fuel, types of space and Earth provides the force. weather monitoring Sunlight is made up stations. These stations of individual particles would be able to provide called photons. Photons greater coverage of Earth have momentum, and and more advanced when a photon bounces This artist’s rendering shows Cosmos 1, warning of solar storms off a solar sail, it transthe first solar sail scheduled for launch in that cause problems to fers its momentum to the near future. communication and electhe sail, which propels tric power grids. It is the spacecraft along. hoped that in the next few decades, solar sails The force of impacting photons is small in will be used as interplanetary shuttles because comparison to the force rocket fuel can supply. of their ability to travel great distances in So, small sails experience only a small amount convenient time frames. Vast distances could of force from sunlight, while larger sails experisomeday be traversed by vehicles that do not ence a greater force. Thus, solar sails may be a consume any fuel. kilometer or so across. What speeds can a solar sail achieve? This depends on the momentum transferred to the sail by photons, as well as the sail’s mass. To Going Further travel quickly through the solar system, a sail and the spacecraft should be lightweight. 1. Research how solar sails can help proPhotons supplied by the Sun are constant. vide advanced warning of solar storms. They impact the sail every second of every hour 2. Critical Thinking A certain solar-sail of every day during a space flight. The Sun’s model is predicted to take more time continuous supply of photons over time allows to reach Mars than a rocket-propelled the sail to build up huge velocities and enables spacecraft would, but less time to the spacecraft to travel great distances within a go to Pluto than a rocket-propelled convenient time frame. Rockets require enorspacecraft would. Explain why this is so. mous amounts of fuel to move large masses,

248

Future Technology

9.1 Impulse and Momentum Vocabulary

Key Concepts

• impulse (p. 230) • momentum (p. 230) • impulse-momentum theorem (p. 230) • angular momentum

• •

When doing a momentum problem, first examine the system before and after the event. The momentum of an object is the product of its mass and velocity and is a vector quantity. p mv

(p. 233)

• angular impulse-angular momentum theorem

•

The impulse on an object is the average net force exerted on the object multiplied by the time interval over which the force acts.

(p. 234)

Impluse Ft

•

The impulse on an object is equal to the change in momentum of the object. Ft pf pi

•

The angular momentum of a rotating object is the product of its moment of inertia and its angular velocity. L I

•

The angular impulse-angular momentum theorem states that the angular impulse on an object is equal to the change in the object’s angular momentum.

t Lf Li

9.2 Conservation of Momentum Vocabulary

Key Concepts

• closed system (p. 236) • isolated system (p. 237) • law of conservation of momentum (p. 237) • law of conservation of angular momentum

•

(p. 243)

•

According to Newton’s third law of motion and the law of conservation of momentum, the forces exerted by colliding objects on each other are equal in magnitude and opposite in direction. Momentum is conserved in a closed, isolated system. pf pi

• • •

The law of conservation of momentum can be used to explain the propulsion of rockets. Vector analysis is used to solve momentum-conservation problems in two dimensions. The law of conservation of angular momentum states that if there are no external torques acting on a system, then the angular momentum is conserved. Lf Li

•

Because angular momentum is conserved, the direction of rotation of a spinning object can be changed only by applying a torque.

physicspp.com/vocabulary_puzzlemaker

249

Concept Mapping 32. Complete the following concept map using the following terms: mass, momentum, average force, time over which the force is exerted.

Product is the

41. Consider a ball falling toward Earth. (9.2) a. Why is the momentum of the ball not conserved? b. In what system that includes the falling ball is the momentum conserved?

42. A falling basketball hits the floor. Just before it hits, the momentum is in the downward direction, and after it hits the floor, the momentum is in the upward direction. (9.2)

impulse

a. Why isn’t the momentum of the basketball conserved even though the bounce is a collision? b. In what system is the momentum conserved?

Produces a change in

43. Only an external force can change the momentum Product is the velocity

of a system. Explain how the internal force of a car’s brakes brings the car to a stop. (9.2)

44. Children’s playgrounds often have circular-motion

Mastering Concepts

rides. How could a child change the angular momentum of such a ride as it is turning? (9.2)

33. Can a bullet have the same momentum as a truck? 34. A pitcher throws a curve ball to the catcher. Assume that the speed of the ball doesn’t change in flight. (9.1) a. Which player exerts the larger impulse on the ball? b. Which player exerts the larger force on the ball?

35. Newton’s second law of motion states that if no net force is exerted on a system, no acceleration is possible. Does it follow that no change in momentum can occur? (9.1)

36. Why are cars made with bumpers that can be pushed in during a crash? (9.1)

37. An ice-skater is doing a spin. (9.1) a. How can the skater’s angular momentum be changed? b. How can the skater’s angular velocity be changed without changing the angular momentum?

38. What is meant by “an isolated system?” (9.2) 39. A spacecraft in outer space increases its velocity by firing its rockets. How can hot gases escaping from its rocket engine change the velocity of the craft when there is nothing in space for the gases to push against? (9.2)

40. A cue ball travels across a pool table and collides with the stationary eight ball. The two balls have equal masses. After the collision, the cue ball is at rest. What must be true regarding the speed of the eight ball? (9.2)

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Applying Concepts 45. Explain the concept of impulse using physical ideas rather than mathematics.

46. Is it possible for an object to obtain a larger impulse from a smaller force than it does from a larger force? Explain.

47. Foul Ball You are sitting at a baseball game when a foul ball comes in your direction. You prepare to catch it bare-handed. To catch it safely, should you move your hands toward the ball, hold them still, or move them in the same direction as the moving ball? Explain.

48. A 0.11-g bullet leaves a pistol at 323 m/s, while a similar bullet leaves a rifle at 396 m/s. Explain the difference in exit speeds of the two bullets, assuming that the forces exerted on the bullets by the expanding gases have the same magnitude.

49. An object initially at rest experiences the impulses described by the graph in Figure 9-15. Describe the object’s motion after impulses A, B, and C. 2 Force (N)

Explain. (9.1)

A 0

2

6 4

8

10 11

B

C

2 Time (s) ■

Figure 9-15

Chapter 9 Momentum and Its Conservation For more problems, go to Additional Problems, Appendix B.

50. During a space walk, the tether connecting an astronaut to the spaceship breaks. Using a gas pistol, the astronaut manages to get back to the ship. Use the language of the impulse-momentum theorem and a diagram to explain why this method was effective.

51. Tennis Ball As a tennis ball bounces off a wall, its momentum is reversed. Explain this action in terms of the law of conservation of momentum. Define the system and draw a diagram as a part of your explanation.

52. Imagine that you command spaceship Zeldon, which is moving through interplanetary space at high speed. How could you slow your ship by applying the law of conservation of momentum?

53. Two trucks that appear to be identical collide on an icy road. One was originally at rest. The trucks are stuck together and move at more than half the original speed of the moving truck. What can you conclude about the contents of the two trucks?

54. Explain, in terms of impulse and momentum, why it is advisable to place the butt of a rifle against your shoulder when first learning to shoot.

55. Bullets Two bullets of equal mass are shot at equal speeds at blocks of wood on a smooth ice rink. One bullet, made of rubber, bounces off of the wood. The other bullet, made of aluminum, burrows into the wood. In which case does the block of wood move faster? Explain.

60. In a ballistics test at the police department, Officer Rios fires a 6.0-g bullet at 350 m/s into a container that stops it in 1.8 ms. What is the average force that stops the bullet?

61. Volleyball A 0.24-kg volleyball approaches Tina with a velocity of 3.8 m/s. Tina bumps the ball, giving it a speed of 2.4 m/s but in the opposite direction. What average force did she apply if the interaction time between her hands and the ball was 0.025 s?

62. Hockey A hockey player makes a slap shot, exerting a constant force of 30.0 N on the hockey puck for 0.16 s. What is the magnitude of the impulse given to the puck?

63. Skateboarding Your brother’s mass is 35.6 kg, and he has a 1.3-kg skateboard. What is the combined momentum of your brother and his skateboard if they are moving at 9.50 m/s?

64. A hockey puck has a mass of 0.115 kg and is at rest. A hockey player makes a shot, exerting a constant force of 30.0 N on the puck for 0.16 s. With what speed does it head toward the goal?

65. Before a collision, a 25-kg object was moving at 12 m/s. Find the impulse that acted on the object if, after the collision, it moved at the following velocities. a. 8.0 m/s b. 8.0 m/s

66. A 0.150-kg ball, moving in the positive direction at 12 m/s, is acted on by the impulse shown in the graph in Figure 9-16. What is the ball’s speed at 4.0 s?

Mastering Problems 56. Golf Rocío strikes a 0.058-kg golf ball with a force of 272 N and gives it a velocity of 62.0 m/s. How long was Rocío’s club in contact with the ball?

2 Force (N)

9.1 Impulse and Momentum

1 2 3 4

2

57. A 0.145-kg baseball is pitched at 42 m/s. The batter hits it horizontally to the pitcher at 58 m/s. a. Find the change in momentum of the ball. b. If the ball and bat are in contact for 4.6104 s, what is the average force during contact?

58. Bowling A force of 186 N acts on a 7.3-kg bowling ball for 0.40 s. What is the bowling ball’s change in momentum? What is its change in velocity?

59. A 5500-kg freight truck accelerates from 4.2 m/s to 7.8 m/s in 15.0 s by the application of a constant force. a. What change in momentum occurs? b. How large of a force is exerted? physicspp.com/chapter_test

Time (s) ■

Figure 9-16

67. Baseball A 0.145-kg baseball is moving at 35 m/s when it is caught by a player. a. Find the change in momentum of the ball. b. If the ball is caught with the mitt held in a stationary position so that the ball stops in 0.050 s, what is the average force exerted on the ball? c. If, instead, the mitt is moving backward so that the ball takes 0.500 s to stop, what is the average force exerted by the mitt on the ball? Chapter 9 Assessment

251

68. Hockey A hockey puck has a mass of 0.115 kg and strikes the pole of the net at 37 m/s. It bounces off in the opposite direction at 25 m/s, as shown in Figure 9-17. a. What is the impulse on the puck? b. If the collision takes 5.0104 s, what is the average force on the puck?

72. Accident A car moving at 10.0 m/s crashes into a barrier and stops in 0.050 s. There is a 20.0-kg child in the car. Assume that the child’s velocity is changed by the same amount as that of the car, and in the same time period. a. What is the impulse needed to stop the child? b. What is the average force on the child? c. What is the approximate mass of an object whose weight equals the force in part b? d. Could you lift such a weight with your arm? e. Why is it advisable to use a proper restraining seat rather than hold a child on your lap?

9.2 Conservation of Momentum

0.115 kg

73. Football A 95-kg fullback, running at 8.2 m/s, 25 m/s ■

Figure 9-17

69. A nitrogen molecule with a mass of 4.71026 kg, moving at 550 m/s, strikes the wall of a container and bounces back at the same speed. a. What is the impulse the molecule delivers to the wall? b. If there are 1.51023 collisions each second, what is the average force on the wall?

70. Rockets Small rockets are used to make tiny adjustments in the speeds of satellites. One such rocket has a thrust of 35 N. If it is fired to change the velocity of a 72,000-kg spacecraft by 63 cm/s, how long should it be fired?

71. An animal rescue plane flying due east at 36.0 m/s drops a bale of hay from an altitude of 60.0 m, as shown in Figure 9-18. If the bale of hay weighs 175 N, what is the momentum of the bale the moment before it strikes the ground? Give both magnitude and direction. 36.0 m/s

175 N

60.0 m

collides in midair with a 128-kg defensive tackle moving in the opposite direction. Both players end up with zero speed. a. Identify the “before” and “after” situations and draw a diagram of both. b. What was the fullback’s momentum before the collision? c. What was the change in the fullback’s momentum? d. What was the change in the defensive tackle’s momentum? e. What was the defensive tackle’s original momentum? f. How fast was the defensive tackle moving originally?

74. Marble C, with mass 5.0 g, moves at a speed of 20.0 cm/s. It collides with a second marble, D, with mass 10.0 g, moving at 10.0 cm/s in the same direction. After the collision, marble C continues with a speed of 8.0 cm/s in the same direction. a. Sketch the situation and identify the system. Identify the “before” and “after” situations and set up a coordinate system. b. Calculate the marbles’ momenta before the collision. c. Calculate the momentum of marble C after the collision. d. Calculate the momentum of marble D after the collision. e. What is the speed of marble D after the collision?

75. Two lab carts are pushed together with a spring

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252

Figure 9-18

mechanism compressed between them. Upon release, the 5.0-kg cart repels one way with a velocity of 0.12 m/s, while the 2.0-kg cart goes in the opposite direction. What is the velocity of the 2.0-kg cart?

Chapter 9 Momentum and Its Conservation For more problems, go to Additional Problems, Appendix B.

76. A 50.0-g projectile is launched with a horizontal velocity of 647 m/s from a 4.65-kg launcher moving in the same direction at 2.00 m/s. What is the launcher’s velocity after the launch?

77. A 12.0-g rubber bullet travels at a velocity of 150 m/s, hits a stationary 8.5-kg concrete block resting on a frictionless surface, and ricochets in the opposite direction with a velocity of 1.0102 m/s, as shown in Figure 9-19. How fast will the concrete block be moving?

1.0102 m/s

compact car at rest. They move off together at 8.5 m/s. Assuming that the friction with the road is negligible, calculate the initial speed of the van.

81. In-line Skating Diego and Keshia are on in-line skates and stand face-to-face, then push each other away with their hands. Diego has a mass of 90.0 kg and Keshia has a mass of 60.0 kg. a. Sketch the event, identifying the “before” and “after” situations, and set up a coordinate axis. b. Find the ratio of the skaters’ velocities just after their hands lose contact. c. Which skater has the greater speed? d. Which skater pushed harder?

82. A 0.200-kg plastic ball moves with a velocity of 8.5 kg

12.0 g ■

80. A 2575-kg van runs into the back of an 825-kg

Figure 9-19

78. Skateboarding Kofi, with mass 42.00 kg, is riding a skateboard with a mass of 2.00 kg and traveling at 1.20 m/s. Kofi jumps off and the skateboard stops dead in its tracks. In what direction and with what velocity did he jump?

79. Billiards A cue ball, with mass 0.16 kg, rolling at 4.0 m/s, hits a stationary eight ball of similar mass. If the cue ball travels 45° above its original path and the eight ball travels 45° below the horizontal, as shown in Figure 9-20, what is the velocity of each ball after the collision?

0.30 m/s. It collides with a second plastic ball of mass 0.100 kg, which is moving along the same line at a speed of 0.10 m/s. After the collision, both balls continue moving in the same, original direction. The speed of the 0.100-kg ball is 0.26 m/s. What is the new velocity of the 0.200-kg ball?

Mixed Review 83. A constant force of 6.00 N acts on a 3.00-kg object for 10.0 s. What are the changes in the object’s momentum and velocity?

84. The velocity of a 625-kg car is changed from 10.0 m/s to 44.0 m/s in 68.0 s by an external, constant force. a. What is the resulting change in momentum of the car? b. What is the magnitude of the force?

85. Dragster An 845-kg dragster accelerates on a race track from rest to 100.0 km/h in 0.90 s. a. What is the change in momentum of the dragster? b. What is the average force exerted on the dragster? c. What exerts that force? 45° 45°

86. Ice Hockey A 0.115-kg hockey puck, moving at 35.0 m/s, strikes a 0.365-kg jacket that is thrown onto the ice by a fan of a certain hockey team. The puck and jacket slide off together. Find their velocity.

87. A 50.0-kg woman, riding on a 10.0-kg cart, is

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Figure 9-20 physicspp.com/chapter_test

moving east at 5.0 m/s. The woman jumps off the front of the cart and lands on the ground at 7.0 m/s eastward, relative to the ground. a. Sketch the “before” and “after” situations and assign a coordinate axis to them. b. Find the cart’s velocity after the woman jumps off. Chapter 9 Assessment

253

88. Gymnastics Figure 9-21 shows a gymnast performing a routine. First, she does giant swings on the high bar, holding her body straight and pivoting around her hands. Then, she lets go of the high bar and grabs her knees with her hands in the tuck position. Finally, she straightens up and lands on her feet. a. In the second and final parts of the gymnast’s routine, around what axis does she spin? b. Rank in order, from greatest to least, her moments of inertia for the three positions. c. Rank in order, from greatest to least, her angular velocities in the three positions.

92. Analyze and Conclude Two balls during a collision are shown in Figure 9-22, which is drawn to scale. The balls enter from the left, collide, and then bounce away. The heavier ball, at the bottom of the diagram, has a mass of 0.600 kg, and the other has a mass of 0.400 kg. Using a vector diagram, determine whether momentum is conserved in this collision. Explain any difference in the momentum of the system before and after the collision.

■

Figure 9-22

Writing in Physics 93. How can highway barriers be designed to be more ■

Figure 9-21

89. A 60.0-kg male dancer leaps 0.32 m high. a. With what momentum does he reach the ground? b. What impulse is needed to stop the dancer? c. As the dancer lands, his knees bend, lengthening the stopping time to 0.050 s. Find the average force exerted on the dancer’s body. d. Compare the stopping force with his weight.

Thinking Critically 90. Apply Concepts A 92-kg fullback, running at 5.0 m/s, attempts to dive directly across the goal line for a touchdown. Just as he reaches the line, he is met head-on in midair by two 75-kg linebackers, both moving in the direction opposite the fullback. One is moving at 2.0 m/s and the other at 4.0 m/s. They all become entangled as one mass. a. Sketch the event, identifying the “before” and “after” situations. b. What is the velocity of the football players after the collision? c. Does the fullback score a touchdown?

91. Analyze and Conclude A student, holding a bicycle wheel with its axis vertical, sits on a stool that can rotate without friction. She uses her hand to get the wheel spinning. Would you expect the student and stool to turn? If so, in which direction? Explain.

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effective in saving people’s lives? Research this issue and describe how impulse and change in momentum can be used to analyze barrier designs.

94. While air bags save many lives, they also have caused injuries and even death. Research the arguments and responses of automobile makers to this statement. Determine whether the problems involve impulse and momentum or other issues.

Cumulative Review 95. A 0.72-kg ball is swung vertically from a 0.60-m string in uniform circular motion at a speed of 3.3 m/s. What is the tension in the cord at the top of the ball’s motion? (Chapter 6)

96. You wish to launch a satellite that will remain above the same spot on Earth’s surface. This means the satellite must have a period of exactly one day. Calculate the radius of the circular orbit this satellite must have. Hint: The Moon also circles Earth and both the Moon and the satellite will obey Kepler’s third law. The Moon is 3.910 8 m from Earth and its period is 27.33 days. (Chapter 7)

97. A rope is wrapped around a drum that is 0.600 m in diameter. A machine pulls with a constant 40.0 N force for a total of 2.00 s. In that time, 5.00 m of rope is unwound. Find , at 2.00 s, and I. (Chapter 8)

Chapter 9 Momentum and Its Conservation For more problems, go to Additional Problems, Appendix B.

Multiple Choice 1. When a star that is much larger than the Sun nears the end of its lifetime, it begins to collapse, but continues to rotate. Which of the following describes the conditions of the collapsing star’s moment of inertia (I), angular momentum (L), and angular velocity (ω)? I increases, L stays constant, ω decreases I decreases, L stays constant, ω increases I increases, L increases, ω increases I increases, L increases, ω stays constant

6. When the large gear in the diagram rotates, it turns the small gear in the opposite direction at the same linear speed. The larger gear has twice the radius and four times the mass of the smaller gear. What is the angular momentum of the larger gear as a function of the angular momentum of the smaller gear? Hint: The moment of inertia for a disk is 12 mr 2, where m is mass and r is the radius of the disk. 2Lsmall 4Lsmall

8Lsmall 16Lsmall

2. A 40.0-kg ice-skater glides with a speed of 2.0 m/s toward a 10.0-kg sled at rest on the ice. The iceskater reaches the sled and holds on to it. The ice-skater and the sled then continue sliding in the same direction in which the ice-skater was originally skating. What is the speed of the ice-skater and the sled after they collide? 0.4 m/s 0.8 m/s

1.6 m/s 3.2 m/s

3. A bicyclist applies the brakes and slows the motion of the wheels. The angular momentum of each wheel then decreases from 7.0 kgm2/s to 3.5 kgm2/s over a period of 5.0 s. What is the angular impulse on each wheel? 0.7 1.4 kgm2/s 2.1 kgm2/s 3.5 kgm2/s kgm2/s

4. A 45.0-kg ice-skater stands at rest on the ice. A friend tosses the skater a 5.0-kg ball. The skater and the ball then move backwards across the ice with a speed of 0.50 m/s. What was the speed of the ball at the moment just before the skater caught it? 2.5 m/s 3.0 m/s

4.0 m/s 5.0 m/s

5. What is the difference in momentum between a 50.0-kg runner moving at a speed of 3.00 m/s and a 3.00103-kg truck moving at a speed of only 1.00 m/s? 1275 kgm/s 2550 kgm/s

2850 kgm/s 2950 kgm/s

physicspp.com/standardized_test

7. A force of 16 N exerted against a rock with an impulse of 0.8 kgm/s causes the rock to fly off the ground with a speed of 4.0 m/s. What is the mass of the rock? 0.2 kg 0.8 kg 1.6 kg 4.0 kg

Extended Answer 8. A 12.0-kg rock falls to the ground. What is the impulse on the rock if its velocity at the moment it strikes the ground is 20.0 m/s?

If It Looks Too Good To Be True Beware of answer choices in multiple-choice questions that seem ready-made and obvious. Remember that only one answer choice for each question is correct. The rest are made up by testmakers to distract you. This means that they might look very appealing. Check each answer choice carefully before making your final selection.

Chapter 9 Standardized Test Practice

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What You’ll Learn • You will recognize that work and power describe how the external world changes the energy of a system. • You will relate force to work and explain how machines ease the load.

Why It’s Important Simple machines and the compound machines formed from them make many everyday tasks easier to perform. Mountain Bikes A multispeed mountain bicycle with shock absorbers allows you to match the ability of your body to exert forces, to do work, and to deliver power climbing steep hills, traversing flat terrain at high speeds, and safely descending hills.

Think About This How does a multispeed mountain bicycle enable a cyclist to ride over any kind of terrain with the least effort?

physicspp.com 256 Colorstock/Getty Images

What factors affect energy? Question What factors affect the energy of falling objects and their ability to do work? Procedure

Analysis

1. Place about 2 cm of fine sand in the bottom of a pie plate or baking pan. 2. Obtain a variety of metal balls or glass marbles of different sizes. 3. Hold a meterstick vertically in one hand, with one end just touching the surface of the sand. With the other hand, drop one of the balls into the sand. Record the height from which you dropped the ball. 4. Carefully remove the ball from the sand, so as not to disturb the impact crater it made. Measure the depth of the crater and how far sand was thrown from the crater. 5. Record the mass of the ball. 6. Smooth out the sand in the pie plate and perform steps 3–5 with different sizes of balls and drop them from varying heights. Be sure to drop different sizes of balls from the same height, as well as the same ball from different heights.

Compare your data for the different craters. Is there an overall trend to your data? Explain. Critical Thinking As the balls are dropped into the sand, they do work on the sand. Energy can be defined as the ability of an object to do work on itself or its surroundings. Relate the trend(s) you found in this lab to the energy of the balls. How can the energy of a ball be increased?

10.1 Energy and Work

I

n Chapter 9, you learned about the conservation of momentum. You learned that you could examine the state of a system before and after an impulse acted on it without knowing the details about the impulse. The law of conservation of momentum was especially useful when considering collisions, during which forces sometimes changed dramatically. Recall the discussion in Chapter 9 of the two skaters who push each other away. While momentum is conserved in this situation, the skaters continue to move after pushing each other away; whereas before the collision, they were at rest. When two cars crash into each other, momentum is conserved. Unlike the skaters, however, the cars, which were moving prior to the collision, became stationary after the crash. The collision probably resulted in a lot of twisted metal and broken glass. In these types of situations, some other quantity must have been changed as a result of the force acting on each system.

Objectives • Describe the relationship between work and energy. • Calculate work. • Calculate the power used.

Vocabulary work energy kinetic energy work-energy theorem joule power watt

Section 10.1 Energy and Work

257

Work and Energy Recall that change in momentum is the result of an impulse, which is the product of the average force exerted on an object and the time of the interaction. Consider a force exerted on an object while the object moves a certain distance. Because there is a net force, the object will be accelerated, a F/m, and its velocity will increase. Examine Table 3-3 in Chapter 3, on page 68, which lists equations describing the relationships among position, velocity, and time for motion under constant acceleration. Consider the equation involving acceleration, velocity, and distance: 2ad vf2 vi2. If you use Newton’s second law to replace a with F/m and multiply both sides by m/2, you obtain Fd 12 mvf2 12 mvi2.

d

F

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Figure 10-1 Work is done when a constant force, F, is exerted on the backpack in the direction of motion and the backpack moves a distance, d.

Work The left side of the equation describes something that was done to the system by the external world (the environment). A force, F, was exerted on an object while the object moved a distance, d, as shown in Figure 10-1. If F is a constant force, exerted in the direction in which the object is moving, then work, W, is the product of the force and the object’s displacement. Work

W Fd

Work is equal to a constant force exerted on an object in the direction of motion, times the object’s displacement.

You probably have used the word work in many other ways. For example, a computer might work well, learning physics can be hard work, and you might work at an after-school job. To physicists, however, work has a very precise meaning. Recall that Fd 12 mvf2 12 mvi2. Rewriting the equation W Fd results

in W 12 mvf2 12 mvi2. The right side of the equation involves the object’s mass and its velocities after and before the force was exerted. The quantity 1 mvi2 describes a property of the system. 2

v F

Sun Planet

Kinetic energy What property of a system does 12 mvi2 describe? A massive, fast-moving vehicle can do damage to objects around it, and a baseball hit at high speed can rise high into the air. That is, an object with this property can produce a change in itself or the world around it. This property, the ability of an object to produce a change in itself or the world around it, is called energy. The fast-moving vehicle and the baseball possess energy that is associated with their motion. This energy resulting from motion is called kinetic energy and is represented by the symbol KE. Kinetic Energy

1 2

KE mv2 1

The kinetic energy of an object is equal to times the mass of the object 2 multiplied by the speed of the object squared. ■

Figure 10-2 If a planet is in a circular orbit, then the force is perpendicular to the direction of motion. Consequently, the gravitational force does no work on the planet.

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Substituting KE into the equation W 12 mvf2 12 mvi2 results in W KEf KEi. The right side is the difference, or change, in kinetic energy. The work-energy theorem states that when work is done on an object, the result is a change in kinetic energy. The work-energy theorem can be represented by the following equation.

Chapter 10 Energy, Work, and Simple Machines

Work-Energy Theorem W KE Work is equal to the change in kinetic energy.

The relationship between work done and the change in energy that results was established by nineteenth-century physicist James Prescott Joule. To honor his work, a unit of energy is called a joule (J). For example, if a 2-kg object moves at 1 m/s, it has a kinetic energy of 1 kgm2/s2, or 1 J. Recall that a system is the object of interest and the external world is everything else. For example, one system might be a box in a warehouse and the external world might consist of yourself, Earth’s mass, and anything else external to the box. Through the process of doing work, energy can move between the external world and the system. Notice that the direction of energy transfer can go both ways. If the external world does work on a system, then W is positive and the energy of the system increases. If, however, a system does work on the external world, then W is negative and the energy of the system decreases. In summary, work is the transfer of energy by mechanical means.

Calculating Work The first equation used to calculate work is W Fd. This equation, however, holds only for constant forces exerted in the direction of motion. What happens if the force is exerted perpendicular to the direction of motion? An everyday example of this is the motion of a planet around the Sun, as shown in Figure 10-2. If the orbit is circular, then the force is always perpendicular to the direction of motion. Recall from Chapter 6 that a perpendicular force does not change the speed of an object, only its direction. Consequently, the speed of the planet doesn’t change. Therefore, its kinetic energy also is constant. Using the equation W KE, you can see that when KE is constant, KE 0 and thus, W 0. This means that if F and d are at right angles, then W 0. Because the work done on an object equals the change in energy, work also is measured in joules. One joule of work is done when a force of 1 N acts on an object over a displacement of 1 m. An apple weighs about 1 N. Thus, when you lift an apple a distance of 1 m, you do 1 J of work on it. Constant force exerted at an angle You’ve learned that a force exerted in the direction of motion does an amount of work given by W Fd. A force exerted perpendicular to the motion does no work. What work does a force exerted at an angle do? For example, what work does the person pushing the car in Figure 10-3a do? You know that any force can be replaced by its components. If the coordinate system shown in Figure 10-3b is used, the 125-N force, F, exerted in the direction of the person’s arm, has two components. The magnitude of the horizontal component, Fx, is related to the magnitude of the force, F, by a cosine function: cos 25.0° Fx/F. By solving for Fx , you obtain Fx F cos 25.0° (125 N)(cos 25.0°) 113 N. Using the same method, the vertical component Fy F sin 25.0° (125 N)(sin 25.0°) 52.8 N, where the negative sign shows that the force is downward. Because the displacement is in the x direction, only the x-component does work. The y-component does no work.

a

b y

Fx x

25.0°

Fy

F 125 N

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Figure 10-3 If a force is applied to a car at an angle, the net force doing the work is the component that acts in the direction of the displacement.

Section 10.1 Energy and Work

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Hutchings Photography

The work you do when you exert a force on an object, at an angle to the direction of motion, is equal to the component of the force in the direction of the displacement, multiplied by the distance moved. The magnitude of the component force acting in the direction of displacement is found by multiplying the magnitude of force, F, by the cosine of the angle between F and the direction of the displacement: Fx F cos . Thus, the work done is represented by the following equation. W Fd cos

Work (Angle Between Force and Displacement)

Work is equal to the product of force and displacement, times the cosine of the angle between the force and the direction of the displacement.

Other agents exert forces on the pushed car as well. Which of these agents do work? Earth’s gravity acts downward, the ground exerts a normal force upward, and friction exerts a horizontal force opposite the direction of motion. The upward and downward forces are perpendicular to the direction of motion and do no work. For these forces, 90°, which makes cos 0, and thus, W 0. The work done by friction acts in the direction opposite that of motion—at an angle of 180°. Because cos 180° 1, the work done by friction is negative. Negative work done by a force exerted by something in the external world reduces the kinetic energy of the system. If the person in Figure 10-3a were to stop pushing, the car would quickly stop moving— its energy of motion would be reduced. Positive work done by a force increases the energy, while negative work decreases it. Use the problemsolving strategies below when you solve problems related to work.

Work When solving work-related problems, use the following strategies.

Work Diagram d

1. Sketch the system and show the force that is doing the work.

FN

2. Draw the force and displacement vectors of the system.

Fg

3. Find the angle, , between each force and displacement. 4. Calculate the work done by each force using W Fd cos . 5. Calculate the net work done. Check the sign of the work using the direction of energy transfer. If the energy of the system has increased, the work done by that force is positive. If the energy has decreased, then the work done by that force is negative.

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Chapter 10 Energy, Work, and Simple Machines

FA

d

A

d

FA

N

FN

d g

Fg

Work and Energy A 105-g hockey puck is sliding across the ice. A player exerts a constant 4.50-N force over a distance of 0.150 m. How much work does the player do on the puck? What is the change in the puck’s energy? 1

Analyze and Sketch the Problem

x

• Sketch the situation showing initial conditions. • Establish a coordinate system with x to the right. • Draw a vector diagram.

2

Known:

Unknown:

m 105 g F 4.50 N d 0.150 m

W ? KE ?

F d

F

Solve for the Unknown Use the equation for work when a constant force is exerted in the same direction as the object’s displacement. Math Handbook W Fd Operations with (4.50 N)(0.150 m) Substitute F 4.50 N, d 0.150 m Significant Digits 0.675 Nm pages 835—836 0.675 J 1 J 1 Nm Use the work-energy theorem to determine the change in energy of the system. W KE KE 0.675 J Substitute W 0.675 J

3

Evaluate the Answer • Are the units correct? Work is measured in joules. • Does the sign make sense? The player (external world) does work on the puck (the system). So the sign of work should be positive.

1. Refer to Example Problem 1 to solve the following problem. a. If the hockey player exerted twice as much force, 9.00 N, on the puck, how would the puck’s change in kinetic energy be affected? b. If the player exerted a 9.00 N-force, but the stick was in contact with the puck for only half the distance, 0.075 m, what would be the change in kinetic energy? 2. Together, two students exert a force of 825 N in pushing a car a distance of 35 m. a. How much work do the students do on the car? b. If the force was doubled, how much work would they do pushing the car the same distance? 3. A rock climber wears a 7.5-kg backpack while scaling a cliff. After 30.0 min, the climber is 8.2 m above the starting point. a. How much work does the climber do on the backpack? b. If the climber weighs 645 N, how much work does she do lifting herself and the backpack? c. What is the average power developed by the climber?

Section 10.1 Energy and Work

261

Force and Displacement at an Angle A sailor pulls a boat a distance of 30.0 m along a dock using a rope that makes a 25.0° angle with the horizontal. How much work does the sailor do on the boat if he exerts a force of 255 N on the rope? 1

Analyze and Sketch the Problem • Establish coordinate axes. • Sketch the situation showing the boat with initial conditions. • Draw a vector diagram showing the force and its component in the direction of the displacement.

2

Known:

Unknown:

F 255 N d 30.0 m 25.0°

W?

y x

F 25.0° F

d

F cos 25.0°

Solve for the Unknown Use the equation for work done when there is an angle between the force and displacement. W Fd cos (255 N)(30.0 m)(cos 25.0°) Substitute F 255 N, d 30.0 m, 25.0° 6.93103 J

3

Evaluate the Answer

Math Handbook

• Are the units correct? Work is measured in joules. • Does the sign make sense? The sailor does work on the boat, which agrees with a positive sign for work.

Trigonometric Ratios page 855

4. If the sailor in Example Problem 2 pulled with the same force, and along the same distance, but at an angle of 50.0°, how much work would he do? 5. Two people lift a heavy box a distance of 15 m. They use ropes, each of which makes an angle of 15° with the vertical. Each person exerts a force of 225 N. How much work do they do? 6. An airplane passenger carries a 215-N suitcase up the stairs, a displacement of 4.20 m vertically, and 4.60 m horizontally. a. How much work does the passenger do? b. The same passenger carries the same suitcase back down the same set of stairs. How much work does the passenger do now? 7. A rope is used to pull a metal box a distance of 15.0 m across the floor. The rope is held at an angle of 46.0° with the floor, and a force of 628 N is applied to the rope. How much work does the force on the rope do? 8. A bicycle rider pushes a bicycle that has a mass of 13 kg up a steep hill. The incline is 25° and the road is 275 m long, as shown in Figure 10-4. The rider pushes the bike parallel to the road with a force of 25 N.

275

25

m

N 25°

a. How much work does the rider do on the bike? b. How much work is done by the force of gravity on the bike?

262

Chapter 10 Energy, Work, and Simple Machines

■

Figure 10-4 (Not to scale)

Work done by many forces Newton’s second law of motion relates the net force on an object to its acceleration. In the same way, the work-energy theorem relates the net work done on a system to its energy change. If several forces are exerted on a system, calculate the work done by each force, and then add the results.

Power Until now, none of the discussions of work has mentioned the time it takes to move an object. The work done by a person lifting a box of books is the same whether the box is lifted onto a shelf in 2 s or each book is lifted separately so that it takes 20 min to put them all on the shelf. Although the work done is the same, the rate at which it is done is different. Power is the work done, divided by the time taken to do the work. In other words, power is the rate at which the external force changes the energy of the system. It is represented by the following equation. W t Power is equal to the work done, divided by the time taken to do the work.

Power P

■

Figure 10-5 Work can be obtained graphically by finding the area under a force-displacement graph.

a

Force (N)

20.0

0.0

1.50 Displacement (m)

b 20.0 Force (N)

Finding work done when forces change A graph of force versus displacement lets you determine the work done by a force. This graphical method can be used to solve problems in which the force is changing. Figure 10-5a shows the work done by a constant force of 20.0 N that is exerted to lift an object a distance of 1.50 m. The work done by this constant force is represented by W Fd (20.0 N)(1.50 m) 30.0 J. The shaded area under the graph is equal to (20.0 N)(1.50 m), or 30.0 J. The area under a force-displacement graph is equal to the work done by that force, even if the force changes. Figure 10-5b shows the force exerted by a spring, which varies linearly from 0.0 to 20.0 N as it is compressed 1.50 m. The work done by the force that compressed the spring is the area under the graph, which is the area of a triangle, 12 (base)(altitude), or W 12 (20.0 N)(1.50 m) 15.0 J.

0.0

1.50 Displacement (m)

■

Figure 10-6 These students are doing work at different rates while climbing the stairs.

Consider the three students in Figure 10-6. The girl hurrying up the stairs is more powerful than the boy who is walking up the stairs. Even though the same work is accomplished by both, the girl accomplishes it in less time and thus develops more power. In the case of the two students walking up the stairs, both accomplish work in the same amount of time. Power is measured in watts (W). One watt is 1 J of energy transferred in 1 s. A watt is a relatively small unit of power. For example, a glass of water weighs about 2 N. If you lift it 0.5 m to your mouth, you do 1 J of work. If you lift the glass in 1 s, you are doing work at the rate of 1 W. Because a watt is such a small unit, power often is measured in kilowatts (kW). One kilowatt is equal to 1000 W. Section 10.1 Energy and Work

263 Laura Sifferlin

Power An electric motor lifts an elevator 9.00 m in 15.0 s by exerting an upward force of 1.20104 N. What power does the motor produce in kW? 1

Analyze and Sketch the Problem • Sketch the situation showing the elevator with initial conditions. • Establish a coordinate system with up as positive. • Draw a vector diagram for the force and displacement.

2

Known:

Unknown:

d 9.00 m t 15.0 s F 1.20104 N

P?

y

d

F

F

Solve for the Unknown Solve for power. W t Fd t (1.20104 N)(9.00 m) (15.0 s)

P

Substitute W Fd

Math Handbook

Substitute F 1.20104 N, d 9.00 m, t 15.0 s

Operations with Scientific Notation pages 842—843

7.20 kW 3

Evaluate the Answer • Are the units correct? Power is measured in J/s. • Does the sign make sense? The positive sign agrees with the upward direction of the force.

9. A box that weighs 575 N is lifted a distance of 20.0 m straight up by a cable attached to a motor. The job is done in 10.0 s. What power is developed by the motor in W and kW? 10. You push a wheelbarrow a distance of 60.0 m at a constant speed for 25.0 s, by exerting a 145-N force horizontally. a. What power do you develop? b. If you move the wheelbarrow twice as fast, how much power is developed? 11. What power does a pump develop to lift 35 L of water per minute from a depth of 110 m? (1 L of water has a mass of 1.00 kg.) 12. An electric motor develops 65 kW of power as it lifts a loaded elevator 17.5 m in 35 s. How much force does the motor exert? 13. A winch designed to be mounted on a truck, as shown in Figure 10-7, is advertised as being able to exert a 6.8103-N force and to develop a power of 0.30 kW. How long would it take the truck and the winch to pull an object 15 m? 14. Your car has stalled and you need to push it. You notice as the car gets going that you need less and less force to keep it going. Suppose that for the first 15 m, your force decreased at a constant rate from 210.0 N to 40.0 N. How much work did you do on the car? Draw a force-displacement graph to represent the work done during this period.

264

Chapter 10 Energy, Work, and Simple Machines

Warn Industries Inc.

■

Figure 10-7

■

Output power

Velocity (m/s)

4

2

Velocity

200

400

1000

500

600

Output power (W)

Maximizing Power on a Multispeed Bicycle

Figure 10-8 When riding a multispeed bicycle, if the muscles in your body exert a force of 400 N and the speed is 2.6 m/s, the power output is over 1000 W.

Force (N)

You may have noticed in Example Problem 3 that when the force and displacement are in the same direction, P Fd/t. However, because the ratio d/t is the speed, power also can be calculated using P Fv. When you are riding a multispeed bicycle, how do you choose the correct gear? You want to get your body to deliver the largest amount of power. By considering the equation P Fv you can see that either zero force or zero speed results in no power delivered. The muscles cannot exert extremely large forces, nor can they move very fast. Thus, some combination of moderate force and moderate speed will produce the largest amount of power. Figure 10-8 shows that in this particular situation, the maximum power output is over 1000 W when the force is about 400 N and speed is about 2.6 m/s. All engines—not just humans—have these limitations. Simple machines often are designed to match the force and speed that the engine can deliver to the needs of the job. You will learn more about simple machines in the next section.

Tour de France A bicyclist in the Tour de France rides at about 8.94 m/s for more than 6 h a day. The power output of the racer is about 1 kW. One-fourth of that power goes into moving the bike against the resistance of the air, gears, and tires. Three-fourths of the power is used to cool the racer’s body.

10.1 Section Review 15. Work Murimi pushes a 20-kg mass 10 m across a floor with a horizontal force of 80 N. Calculate the amount of work done by Murimi. 16. Work A mover loads a 185-kg refrigerator into a moving van by pushing it up a 10.0-m, frictionfree ramp at an angle of inclination of 11.0°. How much work is done by the mover? 17. Work and Power Does the work required to lift a book to a high shelf depend on how fast you raise it? Does the power required to lift the book depend on how fast you raise it? Explain. 18. Power An elevator lifts a total mass of 1.1103 kg a distance of 40.0 m in 12.5 s. How much power does the elevator generate? 19. Work A 0.180-kg ball falls 2.5 m. How much work does the force of gravity do on the ball? physicspp.com/self_check_quiz

20. Mass A forklift raises a box 1.2 m and does 7.0 kJ of work on it. What is the mass of the box? 21. Work You and a friend each carry identical boxes from the first floor of a building to a room located on the second floor, farther down the hall. You choose to carry the box first up the stairs, and then down the hall to the room. Your friend carries it down the hall on the first floor, then up a different stairwell to the second floor. Who does more work? 22. Work and Kinetic Energy If the work done on an object doubles its kinetic energy, does it double its velocity? If not, by what ratio does it change the velocity? 23. Critical Thinking Explain how to find the change in energy of a system if three agents exert forces on the system at once. Section 10.1 Energy and Work

265

10.2 Machines

Objectives • Demonstrate a knowledge of the usefulness of simple machines. • Differentiate between ideal and real machines in terms of efficiency. • Analyze compound machines in terms of combinations of simple machines. • Calculate efficiencies for simple and compound machines.

Vocabulary machine effort force resistance force mechanical advantage ideal mechanical advantage efficiency compound machine

E

veryone uses machines every day. Some are simple tools, such as bottle openers and screwdrivers, while others are complex, such as bicycles and automobiles. Machines, whether powered by engines or people, make tasks easier. A machine eases the load by changing either the magnitude or the direction of a force to match the force to the capability of the machine or the person.

Benefits of Machines Consider the bottle opener in Figure 10-9. When you use the opener, you lift the handle, thereby doing work on the opener. The opener lifts the cap, doing work on it. The work that you do is called the input work, Wi. The work that the machine does is called the output work, Wo. Recall that work is the transfer of energy by mechanical means. You put work into a machine, such as the bottle opener. That is, you transfer energy to the opener. The opener, in turn, does work on the cap, thereby transferring energy to it. The opener is not a source of energy, and therefore, the cap cannot receive more energy than the amount of energy that you put into the opener. Thus, the output work can never be greater than the input work. The machine simply aids in the transfer of energy from you to the bottle cap. Mechanical advantage The force exerted by a person on a machine is called the effort force, Fe. The force exerted by the machine is called the resistance force, Fr. As shown in Figure 10-9a, Fe is the upward force exerted by the person using the bottle opener and Fr is the upward force exerted by the bottle opener. The ratio of resistance force to effort force, Fr /Fe , is called the mechanical advantage, MA, of the machine. Mechanical Advantage

F Fe

MA r

The mechanical advantage of a machine is equal to the resistance force divided by the effort force.

a

b Fr dr de

■

Figure 10-9 A bottle opener is an example of a simple machine. It makes opening a bottle easier, but it does not lessen the work required to do so.

266

Chapter 10 Energy, Work, and Simple Machines

Hutchings Photography

Fe

a

■ Figure 10-10 A fixed pulley has a mechanical advantage equal to 1 (a). A pulley system with a movable pulley has a mechanical advantage greater than 1 (b).

b

F F

In a fixed pulley, such as the one shown in Figure 10-10a, the forces, Fe and Fr , are equal, and consequently MA is 1. What is the advantage of this machine? The fixed pulley is useful, not because the effort force is lessened, but because the direction of the effort force is changed. Many machines, such as the bottle opener shown in Figure 10-9 and the pulley system shown in Figure 10-10b, have a mechanical advantage greater than 1. When the mechanical advantage is greater than 1, the machine increases the force applied by a person. You can write the mechanical advantage of a machine in another way using the definition of work. The input work is the product of the effort force that a person exerts, Fe , and the distance his or her hand moved, de. In the same way, the output work is the product of the resistance force, Fr, and the displacement of the load, dr. A machine can increase force, but it cannot increase energy. An ideal machine transfers all the energy, so the output work equals the input work: Wo Wi or Fr dr Fe de. This equation can be rewritten Fr /Fe de /dr. Recall that mechanical advantage is given by MA Fr /Fe. Therefore, for an ideal machine, ideal mechanical advantage, IMA, is equal to the displacement of the effort force, divided by the displacement of the load. The ideal mechanical advantage can be represented by the following equation. d dr

Ideal Mechanical Advantage IMA e The ideal mechanical advantage of an ideal machine is equal to the displacement of the effort force, divided by the displacement of the load.

Note that you measure the distances moved to calculate the ideal mechanical advantage, but you measure the forces exerted to find the actual mechanical advantage. Section 10.2 Machines

267

Efficiency In a real machine, not all of the input work is available as output work. Energy removed from the system means that there is less output work from the machine. Consequently, the machine is less efficient at accomplishing the task. The efficiency of a machine, e, is defined as the ratio of output work to input work. W Wi

o Efficiency e 100

The efficiency of a machine (in %) is equal to the output work, divided by the input work, multiplied by 100.

An ideal machine has equal output and input work, Wo /Wi 1, and its efficiency is 100 percent. All real machines have efficiencies of less than 100 percent. Efficiency can be expressed in terms of the mechanical advantage and ideal mechanical advantage. Efficiency, e Wo/Wi, can be rewritten as follows: Fd Wo rr Fede Wi

Because MA Fr/Fe and IMA de/dr, the following expression can be written for efficiency. MA IMA

Efficiency e 100 The efficiency of a machine (in %) is equal to its mechanical advantage, divided by the ideal mechanical advantage, multiplied by 100.

A machine’s design determines its ideal mechanical advantage. An efficient machine has an MA almost equal to its IMA. A less-efficient machine has a small MA relative to its IMA. To obtain the same resistance force, a greater force must be exerted in a machine of lower efficiency than in a machine of higher efficiency.

An electric pump pulls water at a rate of 0.25 m3/s from a well that is 25 m deep. The water leaves the pump at a speed of 8.5 m/s.

8.5 m/s

1. What power is needed to lift the water to the surface? 2. What power is needed to increase the pump’s kinetic energy?

3. If the pump’s efficiency is 80 percent, how much power must be delivered to the pump?

25 m

(Not to scale)

268

Chapter 10 Energy, Work, and Simple Machines

a

b

c

Fr

Fe

d

Fr

Fr

Fe

Fr

e

Fe

f Fe

Fe

Fe

_1 F 2 r

_1 F 2 r

Compound Machines Most machines, no matter how complex, are combinations of one or more of the six simple machines: the lever, pulley, wheel and axle, inclined plane, wedge, and screw. These machines are shown in Figure 10-11. The IMA of all the machines shown in Figure 10-11 is the ratio of distances moved. For machines, such as the lever and the wheel and axle, this ratio can be replaced by the ratio of the distance between the place where the force is applied and the pivot point. A common version of the wheel and axle is a steering wheel, such as the one shown in Figure 10-12. The IMA is the ratio of the radii of the wheel and axle.

Fr

■ Figure 10-11 Simple machines include the lever (a), pulley (b), wheel and axle (c), inclined plane (d), wedge (e), and screw (f).

A machine consisting of two or more simple machines linked in such a way that the resistance force of one machine becomes the effort force of the second is called a compound machine. ■

Figure 10-12 The IMA for the steering wheel is re /rr . rr Pivot point

Wheel

Fr

Axle re

Fe

Section 10.2 Machines

269

■ Figure 10-13 A series of simple machines combine to transmit the force that the rider exerts on the pedal to the road.

Fchain on gear

Fgear on chain

Frider on pedal

Frider on road

In a bicycle, the pedal and front gear act like a wheel and axle. The effort force is the force that the rider exerts on the pedal, Frider on pedal. The resistance is the force that the front gear exerts on the chain, Fgear on chain, as shown in Figure 10-13. The chain exerts an effort force on the rear gear, Fchain on gear , equal to the force exerted on the chain. This gear and the rear wheel act like another wheel and axle. The resistance force is the force that the wheel exerts on the road, Fwheel on road. According to Newton’s third law, the ground exerts an equal forward force on the wheel, which accelerates the bicycle forward. The MA of a compound machine is the product of the MAs of the simple machines from which it is made. For example, in the case of the bicycle illustrated in Figure 10-13, the following is true. MA MAmachine 1 MAmachine 2

(

Wheel and Axle The gear mechanism on your bicycle multiplies the distance that you travel. What does it do to the force? 1. Mount a wheel and axle system on a sturdy support rod. 2. Wrap a 1-m-long piece of string clockwise around the axle. 3. Wrap another piece of 1-m-long string counterclockwise around the large diameter wheel. 4. Hang a 500-g mass from the end of the string on the larger wheel. CAUTION: Avoid dropping the mass. 5. Pull the string from the axle down so that the mass is lifted by about 10 cm. Analyze and Conclude 6. What did you notice about the force on the string in your hand? 7. What did you notice about the distance that your hand needed to move to lift the mass? Explain the results in terms of the work done on both strings.

270

Fgear on chain

)(

Fwheel on road

)

Fwheel on road

MA Frider on pedal

Fchain on gear

Frider on pedal

The IMA of each wheel-and-axle machine is the ratio of the distances moved. pedal radius front gear radius

For the pedal gear, IMA rear gear radius wheel radius

For the rear wheel, IMA For the bicycle, then,

(

)( )(

pedal radius front gear radius

rear gear radius wheel radius

IMA

(

rear gear radius front gear radius

pedal radius wheel radius

)

)

Because both gears use the same chain and have teeth of the same size, you can count the number of teeth to find the IMA, as follows.

(

teeth on rear gear teeth on front gear

IMA

)(

)

pedal arm length wheel radius

Shifting gears on a bicycle is a way of adjusting the ratio of gear radii to obtain the desired IMA. You know that if the pedal of a bicycle is at the top or bottom of its circle, no matter how much downward force you exert, the pedal will not turn. The force of your foot is most effective when the force is exerted perpendicular to the arm of the pedal; that is, when the torque is largest. Whenever a force on a pedal is specified, assume that it is applied perpendicular to the arm.

Chapter 10 Energy, Work, and Simple Machines Laura Sifferlin

Mechanical Advantage You examine the rear wheel on your bicycle. It has a radius of 35.6 cm and has a gear with a radius of 4.00 cm. When the chain is pulled with a force of 155 N, the wheel rim moves 14.0 cm. The efficiency of this part of the bicycle is 95.0 percent. a. What is the IMA of the wheel and gear? b. What is the MA of the wheel and gear? c. What is the resistance force? d. How far was the chain pulled to move the rim 14.0 cm? 1

Analyze and Sketch the Problem

35.6 cm Fe

• Sketch the wheel and axle. • Sketch the force vectors. Known: re 4.00 cm rr 35.6 cm Fe 155 N 2

e 95.0% dr 14.0 cm

Unknown: IMA ? MA ?

4.00 cm

Fr ? de ? Fr

Solve for the Unknown a. Solve for IMA. r rr 4.00 cm 35.6 cm

IMA e

For a wheel-and-axle machine, IMA is equal to the ratio of radii. Substitute re 4.00 cm, rr 35.6 cm

0.112 b. Solve for MA. MA I MA

e 100

( ) ( ) 0.112

e MA IMA 100

95.0 100

Substitute e 95.0%, IMA 0.112

0.106 c. Solve for force. F Fe

MA r Fr (MA)(Fe) (0.106)(155 N) 16.4 N

Substitute MA 0.106, Fe 155 N

Math Handbook

d. Solve for distance.

Isolating a Variable page 845

d dr

IMA e de (IMA)(dr) (0.112)(14.0 cm) 1.57 cm 3

Substitute IMA 0.112, dr 14.0 cm

Evaluate the Answer • Are the units correct? Force is measured in newtons and distance in centimeters. • Is the magnitude realistic? IMA is low for a bicycle because a greater Fe is traded for a greater dr. MA is always smaller than IMA. Because MA is low, Fr also will be low. The small distance the axle moves results in a large distance covered by the wheel. Thus, de should be very small.

Section 10.2 Machines

271

24. If the gear radius in the bicycle in Example Problem 4 is doubled, while the force exerted on the chain and the distance the wheel rim moves remain the same, what quantities change, and by how much? 25. A sledgehammer is used to drive a wedge into a log to split it. When the wedge is driven 0.20 m into the log, the log is separated a distance of 5.0 cm. A force of 1.7104 N is needed to split the log, and the sledgehammer exerts a force of 1.1104 N. a. What is the IMA of the wedge? b. What is the MA of the wedge? c. Calculate the efficiency of the wedge as a machine. 26. A worker uses a pulley system to raise a 24.0-kg carton 16.5 m, as shown in Figure 10-14. A force of 129 N is exerted, and the rope is pulled 33.0 m. a. What is the MA of the pulley system?

33.0 m

24.0 kg

b. What is the efficiency of the system? 27. You exert a force of 225 N on a lever to raise a 1.25103-N rock a distance of 13 cm. If the efficiency of the lever is 88.7 percent, how far did you move your end of the lever? 28. A winch has a crank with a 45-cm radius. A rope is wrapped around a drum with a 7.5-cm radius. One revolution of the crank turns the drum one revolution.

16.5 m 129 N ■

Figure 10-14

a. What is the ideal mechanical advantage of this machine? b. If, due to friction, the machine is only 75 percent efficient, how much force would have to be exerted on the handle of the crank to exert 750 N of force on the rope?

Multi-gear bicycle On a multi-gear bicycle, the rider can change the MA of the machine by choosing the size of one or both gears. When accelerating or climbing a hill, the rider increases the ideal mechanical advantage to increase the force that the wheel exerts on the road. To increase the IMA, the rider needs to make the rear gear radius large compared to the front gear radius (refer to the IMA equation on page 270). For the same force exerted by the rider, a larger force is exerted by the wheel on the road. However, the rider must rotate the pedals through more turns for each revolution of the wheel. On the other hand, less force is needed to ride the bicycle at high speed on a level road. The rider needs to choose a gear that has a small rear gear and a large front gear that will result in a smaller IMA. Thus, for the same force exerted by the rider, a smaller force is exerted by the wheel on the road. However, in return, the rider does not have to move the pedals as far for each revolution of the wheel. An automobile transmission works in the same way. To accelerate a car from rest, large forces are needed and the transmission increases the IMA. At high speeds, however, the transmission reduces the IMA because smaller forces are needed. Even though the speedometer shows a high speed, the tachometer indicates the engine’s low angular speed. 272

Chapter 10 Energy, Work, and Simple Machines

The Human Walking Machine Movement of the human body is explained by the same principles of force and work that describe all motion. Simple machines, in the form of levers, give humans the ability to walk and run. The lever systems of the human body are complex. However each system has the following four basic parts.

1. 2. 3. 4.

a rigid bar (bone)

Biology Connection

a source of force (muscle contraction) a fulcrum or pivot (movable joints between bones) a resistance (the weight of the body or an object being lifted or moved)

Figure 10-15 shows the parts of the lever system in a human leg. Lever systems of the body are not very efficient, and mechanical advantages are low. This is why walking and jogging require energy (burn calories) and help people lose weight. When a person walks, the hip acts as a fulcrum and moves through the arc of a circle, centered on the foot. The center of mass of the body moves as a resistance around the fulcrum in the same arc. The length of the radius of the circle is the length of the lever formed by the bones of the leg. Athletes in walking races increase their velocity by swinging their hips upward to increase this radius. A tall person’s body has lever systems with less mechanical advantage than a short person’s does. Although tall people usually can walk faster than short people can, a tall person must apply a greater force to move the longer lever formed by the leg bones. How would a tall person do in a walking race? What are the factors that affect a tall person’s performance? Walking races are usually 20 or 50 km long. Because of the inefficiency of their lever systems and the length of a walking race, very tall people rarely have the stamina to win.

2

4

1 3 ■

Figure 10-15 The human walking machine.

10.2 Section Review 29. Simple Machines Classify the tools below as a lever, a wheel and axle, an inclined plane, a wedge, or a pulley. a. screwdriver c. chisel b. pliers d. nail puller 30. IMA A worker is testing a multiple pulley system to estimate the heaviest object that he could lift. The largest downward force he could exert is equal to his weight, 875 N. When the worker moves the rope 1.5 m, the object moves 0.25 m. What is the heaviest object that he could lift? 31. Compound Machines A winch has a crank on a 45-cm arm that turns a drum with a 7.5-cm radius through a set of gears. It takes three revolutions of the crank to rotate the drum through one revolution. What is the IMA of this compound machine? physicspp.com/self_check_quiz

32. Efficiency Suppose you increase the efficiency of a simple machine. Do the MA and IMA increase, decrease, or remain the same? 33. Critical Thinking The mechanical advantage of a multi-gear bicycle is changed by moving the chain to a suitable rear gear. a. To start out, you must accelerate the bicycle, so you want to have the bicycle exert the greatest possible force. Should you choose a small or large gear? b. As you reach your traveling speed, you want to rotate the pedals as few times as possible. Should you choose a small or large gear? c. Many bicycles also let you choose the size of the front gear. If you want even more force to accelerate while climbing a hill, would you move to a larger or smaller front gear? Section 10.2 Machines

273

Stair Climbing and Power Can you estimate the power you develop as you climb a flight of stairs? Climbing stairs requires energy. As the weight of the body moves through a distance, work is done. Power is a measure of the rate at which work is done. In this activity you will try to maximize the power you develop by applying a vertical force up a flight of stairs over a period of time.

QUESTION What can you do to increase the power you develop as you climb a flight of stairs?

Objectives

Safety Precautions

■ ■ ■ ■

■ Avoid wearing loose clothing.

Predict the factors that affect power. Calculate the power developed. Define power operationally. Interpret force, distance, work, time and power data. ■ Make and use graphs of work versus time, power versus force, and power versus time.

Materials meterstick (or tape measure) stopwatch bathroom scale

Procedure 1. Measure and record the mass of each person in your group using a bathroom scale. If the scale does not have kilogram units, convert the weight in pounds to kilograms. Recall that 2.2 lbs 1 kg. 2. Measure the distance from the floor to the top of the flight of stairs you will climb. Record it in the data table. 3. Have each person in your group climb the flight of stairs in a manner that he or she thinks will maximize the power developed. 4. Use your stopwatch to measure the time it takes each person to perform this task. Record your data in the data table.

274 Horizons Companies

Data Table Mass (kg)

Weight (N)

Distance (m)

Work Done (J)

Time (s)

Power Generated (W)

Analyze 1. Calculate Find each person’s weight in newtons and record it in the data table. 2. Calculate the work done by each person.

3. Why were the members of your group with more mass not necessarily the ones who developed the most power?

3. Calculate the power developed by each person in your group as he or she climbs the flight of stairs.

4. Compare and contrast your data with those of other groups in your class.

4. Make and Use Graphs Use the data you calculated to draw a graph of work versus time and draw the best-fit line.

Real-World Physics

5. Draw a graph of power versus work and draw the best-fit line. 6. Draw a graph of power versus time and draw the best-fit line.

Conclude and Apply 1. Did each person in your group have the same power rating? Why or why not? 2. Which graph(s) showed a definite relationship between the two variables? 3. Explain why this relationship exists. 4. Write an operational definition of power.

1. Research a household appliance that has a power rating equal to or less than the power you developed by climbing the stairs. 2. Suppose an electric power company in your area charges $0.06/kWh. If you charged the same amount for the power you develop climbing stairs, how much money would you earn by climbing stairs for 1 h? 3. If you were designing a stair climbing machine for the local health club, what information would you need to collect? You decide that you will design a stair climbing machine with the ability to calculate the power developed. What information would you have the machine collect in order to let the climber know how much power he or she developed?

Going Further 1. What three things can be done to increase the power you develop while climbing the flight of stairs? 2. Why were the fastest climbers not necessarily the ones who developed the most power?

To find out more about energy, work, and simple machines, visit the Web site: physicspp.com

275

Bicycle Gear Shifters In a multispeed bicycle with two or three front gears and from five to eight rear gears, front and rear derailleurs (shifters) are employed to position the chain. Changing the combination of front and rear gears varies the IMA of the system. A larger IMA reduces effort in climbing hills. A lower IMA allows for greater speed on level ground, but more effort is required. number of teeth on rear gear IMA number of teeth on front gear

Rear derailleur

Upper gear

1 The left-hand shift lever

adjusts the position of the front derailleur by means of a connecting cable.

Lower gear 2 The right-hand shift

lever adjusts the position of the rear derailleur by means of a wire cable routed along the frame from the shift lever to the derailleur.

Gear-shift cable

4 The rear derailleur,

responding to movements of the right-hand shift lever, positions the chain on the rear gears. The rear derailleur assembly includes two small gears. The lower gear, tensioned by a spring, keeps the chain tight as it moves from larger to smaller gears. The upper gear moves in or out to position the chain on the gears next to it.

276

How It Works

Front derailleur

Thinking Critically 3 The front derailleur

lifts the chain off of the gear teeth and moves it to another gear near it in response to the tightening or loosening of the cable that connects it to the left-hand shift lever.

1. Calculate What is the IMA of a multispeed bicycle in the following instances? a. when the chain is set on a front gear with 52 teeth and a rear gear with 14 teeth b. when the chain is set on a front gear with 42 teeth and a rear gear with 34 teeth 2. Apply Which setting in the previous problem, a or b, would you select to race with a friend on level ground? To climb a steep hill?

10.1 Energy and Work Vocabulary

Key Concepts

• • • •

•

work (p. 258) energy (p. 258) kinetic energy (p. 258) work-energy theorem

(p. 258) • joule (p. 259) • power (p. 263) • watt (p. 263)

Work is the transfer of energy by mechanical means. W Fd

•

A moving object has kinetic energy. 1 2

KE mv2

•

The work done on a system is equal to the change in energy of the system. W KE

•

Work is the product of the force exerted on an object and the distance the object moves in the direction of the force. W Fd cos

• •

The work done can be determined by calculating the area under a forcedisplacement graph. Power is the rate of doing work, that is the rate at which energy is transferred. W t

P

10.2 Machines Vocabulary

Key Concepts

• • • •

machine (p. 266) effort force (p. 266) resistance force (p. 266) mechanical advantage

•

(p. 266)

•

• ideal mechanical advantage (p. 267) • efficiency (p. 268) • compound machine (p. 269)

•

Machines, whether powered by engines or humans, do not change the amount of work done, but they do make the task easier. A machine eases the load, either by changing the magnitude or the direction of the force exerted to do work. The mechanical advantage, MA, is the ratio of resistance force to effort force. F Fe

MA r

•

The ideal mechanical advantage, IMA, is the ratio of the distances moved. d dr

IMA e

•

The efficiency of a machine is the ratio of output work to input work. W Wi

e o 100

• •

In all real machines, MA is less than IMA. The efficiency of a machine can by found from the real and ideal mechanical advantages. MA IMA

e 100

physicspp.com/vocabulary_puzzlemaker

277

Concept Mapping 34. Create a concept map using the following terms: force, displacement, direction of motion, work, change in kinetic energy.

Mastering Concepts

a. Which person does more work? Explain your answer. b. Which person produces more power? Explain your answer.

47. Show that power delivered can be written as P Fv cos .

35. In what units is work measured? (10.1)

48. How can you increase the ideal mechanical

36. Suppose a satellite revolves around Earth in a

advantage of a machine?

circular orbit. Does Earth’s gravity do any work on the satellite? (10.1)

49. Wedge How can you increase the mechanical

37. An object slides at constant speed on a frictionless surface. What forces act on the object? What work is done by each force? (10.1)

advantage of a wedge without changing its ideal mechanical advantage?

50. Orbits Explain why a planet orbiting the Sun does not violate the work-energy theorem.

38. Define work and power. (10.1) 39. What is a watt equivalent to in terms of kilograms, meters, and seconds? (10.1)

40. Is it possible to get more work out of a machine than you put into it? (10.2)

51. Claw Hammer A claw hammer is used to pull a nail from a piece of wood, as shown in Figure 10-16. Where should you place your hand on the handle and where should the nail be located in the claw to make the effort force as small as possible?

41. Explain how the pedals of a bicycle are a simple machine. (10.2)

Applying Concepts 42. Which requires more work, carrying a 420-N backpack up a 200-m-high hill or carrying a 210-N backpack up a 400-m-high hill? Why? ■

43. Lifting You slowly lift a box of books from the floor and put it on a table. Earth’s gravity exerts a force, magnitude mg, downward, and you exert a force, magnitude mg, upward. The two forces have equal magnitudes and opposite directions. It appears that no work is done, but you know that you did work. Explain what work was done.

44. You have an after-school job carrying cartons of

Figure 10-16

Mastering Problems 10.1 Energy and Work 52. The third floor of a house is 8 m above street level. How much work is needed to move a 150-kg refrigerator to the third floor?

new copy paper up a flight of stairs, and then carrying recycled paper back down the stairs. The mass of the paper does not change. Your physics teacher says that you do not work all day, so you should not be paid. In what sense is the physics teacher correct? What arrangement of payments might you make to ensure that you are properly compensated?

53. Haloke does 176 J of work lifting himself 0.300 m.

45. You carry the cartons of copy paper down the stairs,

2.20105 J of work in pulling team B 8.00 m. What force was team A exerting?

What is Haloke’s mass?

54. Football After scoring a touchdown, an 84.0-kg wide receiver celebrates by leaping 1.20 m off the ground. How much work was done by the wide receiver in the celebration?

55. Tug-of-War During a tug-of-war, team A does

and then along a 15-m-long hallway. Are you working now? Explain.

56. To keep a car traveling at a constant velocity, a 551-N

46. Climbing Stairs Two people of the same mass climb the same flight of stairs. The first person climbs the stairs in 25 s; the second person does so in 35 s.

278

Chapter 10 Energy, Work, and Simple Machines

Hutchings Photography

force is needed to balance frictional forces. How much work is done against friction by the car as it travels from Columbus to Cincinnati, a distance of 161 km?

For more problems, go to Additional Problems, Appendix B.

57. Cycling A cyclist exerts a force of 15.0 N as he rides a bike 251 m in 30.0 s. How much power does the cyclist develop?

58. A student librarian lifts a 2.2-kg book from the floor to a height of 1.25 m. He carries the book 8.0 m to the stacks and places the book on a shelf that is 0.35 m above the floor. How much work does he do on the book?

66. Sled Diego pulls a 4.5-kg sled across level snow with a force of 225 N on a rope that is 35.0° above the horizontal, as shown in Figure 10-18. If the sled moves a distance of 65.3 m, how much work does Diego do? 5N

22

59. A force of 300.0 N is used to push a 145-kg mass

35.0°

4.5 kg

30.0 m horizontally in 3.00 s. a. Calculate the work done on the mass. b. Calculate the power developed.

60. Wagon A wagon is pulled by a force of 38.0 N exerted on the handle at an angle of 42.0° with the horizontal. If the wagon is pulled in a circle of radius 25.0 m, how much work is done?

61. Lawn Mower Shani is pushing a lawn mower with a force of 88.0 N along a handle that makes an angle of 41.0° with the horizontal. How much work is done by Shani in moving the lawn mower 1.2 km to mow the yard?

62. A 17.0-kg crate is to be pulled a distance of 20.0 m, requiring 1210 J of work to be done. If the job is done by attaching a rope and pulling with a force of 75.0 N, at what angle is the rope held?

63. Lawn Tractor A 120-kg lawn tractor, shown in Figure 10-17, goes up a 21° incline that is 12.0 m long in 2.5 s. Calculate the power that is developed by the tractor. 0 kg

Figure 10-18

67. Escalator Sau-Lan has a mass of 52 kg. She rides up the escalator at Ocean Park in Hong Kong. This is the world’s longest escalator, with a length of 227 m and an average inclination of 31°. How much work does the escalator do on Sau-Lan?

68. Lawn Roller A lawn roller is pushed across a lawn by a force of 115 N along the direction of the handle, which is 22.5° above the horizontal. If 64.6 W of power is developed for 90.0 s, what distance is the roller pushed?

69. John pushes a crate across the floor of a factory with a horizontal force. The roughness of the floor changes, and John must exert a force of 20 N for 5 m, then 35 N for 12 m, and then 10 N for 8 m. a. Draw a graph of force as a function of distance. b. Find the work John does pushing the crate.

70. Maricruz slides a 60.0-kg crate up an inclined ramp

120.

0m

12.

21° ■

■

Figure 10-17

that is 2.0-m long and attached to a platform 1.0 m above floor level, as shown in Figure 10-19. A 400.0-N force, parallel to the ramp, is needed to slide the crate up the ramp at a constant speed. a. How much work does Maricruz do in sliding the crate up the ramp? b. How much work would be done if Maricruz simply lifted the crate straight up from the floor to the platform?

64. You slide a crate up a ramp at an angle of 30.0° by exerting a 225-N force parallel to the ramp. The crate moves at a constant speed. The coefficient of friction is 0.28. How much work did you do on the crate as it was raised a vertical distance of 1.15 m?

65. Piano A

4.2103-N

piano is to be slid up a 3.5-m frictionless plank at a constant speed. The plank makes an angle of 30.0° with the horizontal. Calculate the work done by the person sliding the piano up the plank. physicspp.com/chapter_test

.0

60 0.0

N

kg 2.0

m

1.0 m

40

■

Figure 10-19

Chapter 10 Assessment

279

71. Boat Engine An engine moves a boat through the water at a constant speed of 15 m/s. The engine must exert a force of 6.0 kN to balance the force that the water exerts against the hull. What power does the engine develop?

72. In Figure 10-20, the magnitude of the force

75. Oil Pump In 35.0 s, a pump delivers 0.550 m3 of oil into barrels on a platform 25.0 m above the intake pipe. The oil’s density is 0.820 g/cm3. a. Calculate the work done by the pump. b. Calculate the power produced by the pump.

76. Conveyor Belt A 12.0-m-long conveyor belt,

necessary to stretch a spring is plotted against the distance the spring is stretched. a. Calculate the slope of the graph, k, and show that F kd, where k 25 N/m. b. Find the amount of work done in stretching the spring from 0.00 m to 0.20 m by calculating the area under the graph from 0.00 m to 0.20 m. c. Show that the answer to part b can be calculated using the formula W 12kd2, where W is the work, k 25 N/m (the slope of the graph), and d is the distance the spring is stretched (0.20 m).

inclined at 30.0°, is used to transport bundles of newspapers from the mail room up to the cargo bay to be loaded onto delivery trucks. Each newspaper has a mass of 1.0 kg, and there are 25 newspapers per bundle. Determine the power that the conveyor develops if it delivers 15 bundles per minute.

77. A car is driven at a constant speed of 76 km/h down a road. The car’s engine delivers 48 kW of power. Calculate the average force that is resisting the motion of the car.

78. The graph in Figure 10-22 shows the force and displacement of an object being pulled. a. Calculate the work done to pull the object 7.0 m. b. Calculate the power that would be developed if the work was done in 2.0 s.

Force (N)

8.00 6.00 4.00

0.00 ■

0.10

0.20

Force (N)

2.00

0.30

Elongation (m)

Figure 10-20

40.0

20.0

73. Use the graph in Figure 10-20 to find the work needed to stretch the spring from 0.12 m to 0.28 m.

74. A worker pushes a crate weighing 93 N up an inclined plane. The worker pushes the crate horizontally, parallel to the ground, as illustrated in Figure 10-21. a. The worker exerts a force of 85 N. How much work does he do? b. How much work is done by gravity? (Be careful with the signs you use.) c. The coefficient of friction is 0.20. How much work is done by friction? (Be careful with the signs you use.)

85 N 0

5.

m

3.0 m

4.0 m

■

280

Figure 10-21

93 N

Chapter 10 Energy, Work, and Simple Machines

0.0

2.0

4.0

6.0

Displacement (m) ■

Figure 10-22

10.2 Machines 79. Piano Takeshi raises a 1200-N piano a distance of 5.00 m using a set of pulleys. He pulls in 20.0 m of rope. a. How much effort force would Takeshi apply if this were an ideal machine? b. What force is used to balance the friction force if the actual effort is 340 N? c. What is the output work? d. What is the input work? e. What is the mechanical advantage?

80. Lever Because there is very little friction, the lever is an extremely efficient simple machine. Using a 90.0percent-efficient lever, what input work is required to lift an 18.0-kg mass through a distance of 0.50 m? For more problems, go to Additional Problems, Appendix B.

81. A pulley system lifts a 1345-N weight a distance of 0.975 m. Paul pulls the rope a distance of 3.90 m, exerting a force of 375 N. a. What is the ideal mechanical advantage of the system? b. What is the mechanical advantage? c. How efficient is the system?

82. A force of 1.4 N is exerted through a distance of 40.0 cm on a rope in a pulley system to lift a 0.50-kg mass 10.0 cm. Calculate the following. a. the MA b. the IMA c. the efficiency

83. A student exerts a force of 250 N on a lever, through a distance of 1.6 m, as he lifts a 150-kg crate. If the efficiency of the lever is 90.0 percent, how far is the crate lifted?

84. What work is required to lift a 215-kg mass a distance of 5.65 m, using a machine that is 72.5 percent efficient?

85. The ramp in Figure 10-23 is 18 m long and 4.5 m high. a. What force, parallel to the ramp (FA), is required to slide a 25-kg box at constant speed to the top of the ramp if friction is disregarded? b. What is the IMA of the ramp? c. What are the real MA and the efficiency of the ramp if a parallel force of 75 N is actually required?

18 m

FA

4.5 m

■

Figure 10-23

Fg

86. Bicycle Luisa pedals a bicycle with a gear radius of 5.00 cm and a wheel radius of 38.6 cm, as shown in Figure 10-24. If the wheel revolves once, what is the length of the chain that was used?

38.6 cm

5.00 cm

■

Figure 10-24 physicspp.com/chapter_test

87. Crane A motor with an efficiency of 88 percent operates a crane with an efficiency of 42 percent. If the power supplied to the motor is 5.5 kW, with what constant speed does the crane lift a 410-kg crate of machine parts?

88. A compound machine is constructed by attaching a lever to a pulley system. Consider an ideal compound machine consisting of a lever with an IMA of 3.0 and a pulley system with an IMA of 2.0. a. Show that the IMA of this compound machine is 6.0. b. If the compound machine is 60.0 percent efficient, how much effort must be applied to the lever to lift a 540-N box? c. If you move the effort side of the lever 12.0 cm, how far is the box lifted?

Mixed Review 89. Ramps Isra has to get a piano onto a 2.0-m-high platform. She can use a 3.0-m-long frictionless ramp or a 4.0-m-long frictionless ramp. Which ramp should Isra use if she wants to do the least amount of work?

90. Brutus, a champion weightlifter, raises 240 kg of weights a distance of 2.35 m. a. How much work is done by Brutus lifting the weights? b. How much work is done by Brutus holding the weights above his head? c. How much work is done by Brutus lowering them back to the ground? d. Does Brutus do work if he lets go of the weights and they fall back to the ground? e. If Brutus completes the lift in 2.5 s, how much power is developed?

91. A horizontal force of 805 N is needed to drag a crate across a horizontal floor with a constant speed. You drag the crate using a rope held at an angle of 32°. a. What force do you exert on the rope? b. How much work do you do on the crate if you move it 22 m? c. If you complete the job in 8.0 s, what power is developed?

92. Dolly and Ramp A mover’s dolly is used to transport a refrigerator up a ramp into a house. The refrigerator has a mass of 115 kg. The ramp is 2.10 m long and rises 0.850 m. The mover pulls the dolly with a force of 496 N up the ramp. The dolly and ramp constitute a machine. a. What work does the mover do? b. What is the work done on the refrigerator by the machine? c. What is the efficiency of the machine? Chapter 10 Assessment

281

93. Sally does 11.4 kJ of work dragging a wooden crate 25.0 m across a floor at a constant speed. The rope makes an angle of 48.0° with the horizontal. a. How much force does the rope exert on the crate? b. What is the force of friction acting on the crate? c. What work is done by the floor through the force of friction between the floor and the crate?

94. Sledding An 845-N sled is pulled a distance of 185 m. The task requires 1.20104 J of work and is done by pulling on a rope with a force of 125 N. At what angle is the rope held?

95. An electric winch pulls a 875-N crate up a 15° incline at 0.25 m/s. The coefficient of friction between the crate and incline is 0.45. a. What power does the winch develop? b. If the winch is 85 percent efficient, what is the electrical power that must be delivered to the winch?

Thinking Critically carrying boxes to a storage loft that is 12 m above the ground. You have 30 boxes with a total mass of 150 kg that must be moved as quickly as possible, so you consider carrying more than one up at a time. If you try to move too many at once, you know that you will go very slowly, resting often. If you carry only one box at a time, most of the energy will go into raising your own body. The power (in watts) that your body can develop over a long time depends on the mass that you carry, as shown in Figure 10-25. This is an example of a power curve that applies to machines as well as to people. Find the number of boxes to carry on each trip that would minimize the time required. What time would you spend doing the job? Ignore the time needed to go back down the stairs and to lift and lower each box. Power v. Mass

Power (W)

25 20

problem runs the 50.0-m dash in the same time, 8.50 s. However, this time the sprinter accelerates in the first second and runs the rest of the race at a constant velocity. a. Calculate the average power produced for that first second. b. What is the maximum power that the sprinter now generates?

Writing in Physics an automobile. Find the efficiencies of the component parts of the power train (engine, transmission, wheels, and tires). Explore possible improvements in each of these efficiencies.

100. The terms force, work, power, and energy often mean the same thing in everyday use. Obtain examples from advertisements, print media, radio, and television that illustrate meanings for these terms that differ from those used in physics.

Cumulative Review 101. You are helping your grandmother with some gardening and have filled a garbage can with weeds and soil. Now you have to move the garbage can across the yard and realize it is so heavy that you will need to push it, rather than lift it. If the can has a mass of 24 kg, the coefficient of kinetic friction between the can’s bottom and the muddy grass is 0.27, and the static coefficient of friction between those same surfaces is 0.35, how hard do you have to push horizontally to get the can to just start moving? (Chapter 5) fastball horizontally at a speed of 40.3 m/s (90 mph) and it travels 18.4 m (60 ft, 6 in), how far has it dropped by the time it crosses home plate? (Chapter 6)

10

5

10

15

20

25

30

Mass (kg)

282

98. Apply Concepts The sprinter in the previous

102. Baseball If a major league pitcher throws a

15

5

■

50.0-m dash in 8.50 s. Assume that the sprinter’s acceleration is constant throughout the race. a. What is the average power of the sprinter over the 50.0 m? b. What is the maximum power generated by the sprinter? c. Make a quantitative graph of power versus time for the entire race.

99. Just as a bicycle is a compound machine, so is

96. Analyze and Conclude You work at a store,

97. Apply Concepts A sprinter of mass 75 kg runs the

Figure 10-25

Chapter 10 Energy, Work, and Simple Machines

103. People sometimes say that the Moon stays in its orbit because the “centrifugal force just balances the centripetal force, giving no net force.” Explain why this idea is wrong. (Chapter 8) For more problems, go to Additional Problems, Appendix B.

Multiple Choice 1. A pulley system consists of two fixed pulleys and two movable pulleys that lift a load that has a weight of 300 N. If the effort force used to lift the load is 100 N, what is the mechanical advantage of the system? 1 3 3 4

3

2.5 4.0

6

2. The box in the diagram is being pushed up the ramp with a force of 100.0 N. If the height of the ramp is 3.0 m, what is the work done on the box? (sin 30° 0.50, cos 30° 0.87, tan 30° 0.58) 150 J

450 J

260 J

600 J

.0

F

100

30°

6. Two people carry identical 40.0-N boxes up a ramp. The ramp is 2.00 m long and rests on a platform that is 1.00 m high. One person walks up the ramp in 2.00 s, and the other person walks up the ramp in 4.00 s. What is the difference in power the two people use to carry the boxes up the ramp?

3. A compound machine used to raise heavy boxes consists of a ramp and a pulley. The efficiency of pulling a 100-kg box up the ramp is 50%. If the efficiency of the pulley is 90%, what is the overall efficiency of the compound machine? 50%

45%

70%

4. A skater with a mass of 50.0 kg slides across an icy pond with negligible friction. As he approaches a friend, both he and his friend hold out their hands, and the friend exerts a force in the direction opposite to the skater’s movement, which slows the skater’s speed from 2.0 m/s2 to 1.0 m/s2. What is the change in the skater’s kinetic energy? 25 J

100 J

75 J

150 J physicspp.com/standardized_test

20.0 W 40.0 W

7. A 4-N soccer ball sits motionless on a field. A player’s foot exerts a force of 5 N on the ball for a distance of 0.1 m, and the ball rolls a distance of 10 m. How much kinetic energy does the ball gain from the player? 0.5 J 0.9 J

40%

5.0 10.0

5.00 W 10.0 W

3.0 m

N

5. A 20.0-N block is attached to the end of a rope, and the rope is looped around a pulley system. If you pull the opposite end of the rope a distance of 2.00 m, the pulley system raises the block a distance of 0.40 m. What is the pulley system’s ideal mechanical advantage?

9J 50 J

Extended Answer 8. The diagram shows a box being pulled by a rope with a force of 200.0 N along a horizontal surface. The angle the rope makes with the horizontal is 45°. Calculate the work done on the box and the power required to pull it a distance of 5.0 m in 10.0 s. (sin 45° cos 45° 0.7)

45°

Beat the Clock and then Go Back As you take a practice test, pace yourself to finish each section just a few minutes early so you can go back and check over your work.

Chapter 10 Standardized Test Practice

283

What You’ll Learn • You will learn that energy is a property of an object that can change the object’s position, motion, or its environment. • You will learn that energy changes from one form to another, and that the total amount of energy in a closed system remains constant.

Why It’s Important Energy turns the wheels of our world. People buy and sell energy to operate electric appliances, automobiles, and factories. Skiing The height of the ski jump determines the energy the skier has at the bottom of the ramp before jumping into the air and flying many meters down the slope. The distance that the ski jumper travels depends on his or her use of physical principles such as air resistance, balance, and energy.

Think About This How does the height of the ski ramp affect the distance that the skier can jump?

physicspp.com 284 David Madison Sports Images

How can you analyze the energy of a bouncing basketball? Question What is the relationship between the height a basketball is dropped from and the height it reaches when it bounces back? Procedure 1. Place a meterstick against a wall. Choose an initial height from which to drop a basketball. Record the height in the data table. 2. Drop the ball and record how high the ball bounced. 3. Repeat steps 1 and 2 by dropping the basketball from three other heights. 4. Make and Use Graphs Construct a graph of bounce height ( y) versus drop height (x). Find the best-fit line. Analysis Use the graph to find how high a basketball would bounce if it were dropped from a height of 10.0 m. When the ball is lifted and ready to drop, it possesses energy. What are the factors that influence this energy? Critical Thinking Why doesn’t the ball bounce back to the height from which it was dropped?

11.1 The Many Forms of Energy

T

he word energy is used in many different ways in everyday speech. Some fruit-and-cereal bars are advertised as energy sources. Athletes use energy in sports. Companies that supply your home with electricity, natural gas, or heating fuel are called energy companies. Scientists and engineers use the term energy much more precisely. As you learned in the last chapter, work causes a change in the energy of a system. That is, work transfers energy between a system and the external world. In this chapter, you will explore how objects can have energy in a variety of ways. Energy is like ice cream—it comes in different varieties. You can have vanilla, chocolate, or peach ice cream. They are different varieties, but they are all ice cream and serve the same purpose. However, unlike ice cream, energy can be changed from one variety to another. In this chapter, you will learn how energy is transformed from one variety (or form) to another and how to keep track of the changes.

Objectives • Use a model to relate work and energy. • Calculate kinetic energy. • Determine the gravitational potential energy of a system. • Identify how elastic potential energy is stored.

Vocabulary rotational kinetic energy gravitational potential energy reference level elastic potential energy

Section 11.1 The Many Forms of Energy

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Horizons Companies

A Model of the Work-Energy Theorem

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Figure 11-1 When you earn money, the amount of cash that you have increases (a). When you spend money, the amount of cash that you have decreases (b).

In the last chapter, you were introduced to the work-energy theorem. You learned that when work is done on a system, the energy of that system increases. On the other hand, if the system does work, then the energy of the system decreases. These are abstract ideas, but keeping track of energy is much like keeping track of your spending money. If you have a job, the amount of money that you have increases every time you are paid. This process can be represented with a bar graph, as shown in Figure 11-1a. The orange bar represents how much money you had to start with, and the blue bar represents the amount that you were paid. The green bar is the total amount that you possess after the payment. An accountant would say that your cash flow was positive. What happens when you spend the money that you have? The total amount of money that you have decreases. As shown in Figure 11-1b, the bar that represents the amount of money that you had before you bought that new CD is higher than the bar that represents the amount of money remaining after your shopping trip. The difference is the cost of the CD. Cash flow is shown as a bar below the axis because it represents money going out, which can be shown as a negative number. Energy is similar to your spending money. The amount of money that you have changes only when you earn more or spend it. Similarly, energy can be stored, and when energy is spent, it affects the motion of a system.

a $before Cash flow $after

b $before Cash flow $after

Throwing a ball Gaining and losing energy also can be illustrated by throwing and catching a ball. In Chapter 10, you learned that when you exert a constant force, F, on an object through a distance, d, in the direction of the force, you do an amount of work, represented by W Fd. The work is positive because the force and motion are in the same direction, and the energy of the object increases by an amount equal to W. Suppose the object is a ball, and you exert a force to throw the ball. As a result of the force you apply, the ball gains kinetic energy. This process is shown in Figure 11-2a. You can again use a bar graph to represent the process. This time, the height of the bar represents the amount of work, or energy, measured in joules. The kinetic energy after the work is done is equal to the sum of the initial kinetic energy plus the work done on the ball.

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Figure 11-2 The kinetic energy after throwing or catching a ball is equal to the kinetic energy before plus the input work.

a

Throwing a ball Begin

End vball

Stopped

b

Catching a ball Begin

End vball

d W0

W0 F

KEbefore

286

Stopped

d

W

F KEafter

Chapter 11 Energy and Its Conservation

KEbefore W

KEafter

Catching a ball What happens when you catch a ball? Before hitting your hands or glove, the ball is moving, so it has kineticc energy. In catching it, you exert a force on the ball in the direction opposite to its motion. Therefore, you do negative work on it, causing it to stop. Now that the ball is not moving, it has no kinetic energy. This process and the bar graph that represents it are shown in Figure 11-2b. Kinetic energy is always positive, so the initial kinetic energy of the ball is positive. The work done on the ball is negative and the final kinetic energy is zero. Again, the kinetic energy after the ball has stopped is equal to the sum of the initial kinetic energy plus the work done on the ball.

a

Kinetic Energy Recall that kinetic energy, KE 1 mv2, where m is the mass of the object 2 and v is the magnitude of its velocity. The kinetic energy is proportional to the object’s mass. A 7.26-kg shot put thrown through the air has much more kinetic energy than a 0.148-kg baseball with the same velocity, because the shot put has a greater mass. The kinetic energy of an object is also proportional to the square of the object’s velocity. A car speeding at 20 m/s has four times the kinetic energy of the same car moving at 10 m/s. Kinetic energy also can be due to rotational motion. If you spin a toy top in one spot, does it have kinetic energy? You might say that it does not because the top is not moving anywhere. However, to make the top rotate, someone had to do work on it. Therefore, the top has rotational kinetic energy. This is one of the several varieties of energy. Rotational kinetic energy can be calculated using KErot 1 I2, where I is the object’s 2 moment of inertia and is the object’s angular velocity. The diver, shown in Figure 11-3a, does work as she pushes off of the diving board. This work produces both linear and rotational kinetic energies. When the diver’s center of mass moves as she leaps, linear kinetic energy is produced. When she rotates about her center of mass, as shown in Figure 11-3b, rotational kinetic energy is produced. Because she is moving toward the water and rotating at the same time while in the tuck position, she has both linear and rotational kinetic energy. When she slices into the water, as shown in Figure 11-3c, she has linear kinetic energy.

1. A skater with a mass of 52.0 kg moving at 2.5 m/s glides to a stop over a distance of 24.0 m. How much work did the friction of the ice do to bring the skater to a stop? How much work would the skater have to do to speed up to 2.5 m/s again? 2. An 875.0-kg compact car speeds up from 22.0 m/s to 44.0 m/s while passing another car. What are its initial and final energies, and how much work is done on the car to increase its speed? 3. A comet with a mass of 7.851011 kg strikes Earth at a speed of 25.0 km/s. Find the kinetic energy of the comet in joules, and compare the work that is done by Earth in stopping the comet to the 4.21015 J of energy that was released by the largest nuclear weapon ever built.

b

c

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Figure 11-3 The diver does work as she pushes off of the diving board (a). This work produces rotational kinetic energy as she rotates about her center of mass (b) and she has linear kinetic energy when she slices into the water (c).

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Ken Redmond/Ken Redmond Photography

Stored Energy ■

Figure 11-4 Money in the form of bills, quarters, and pennies are different forms of the same thing. 5 $1 Bills

20 500 Quarters Pennies

$5

$5

$5

$5 $1.00 5 $5 $0.25 20 $5 $0.01 500

Imagine a group of boulders high on a hill. These boulders have been lifted up by geological processes against the force of gravity; thus, they have stored energy. In a rock slide, the boulders are shaken loose. They fall and pick up speed as their stored energy is converted to kinetic energy. In the same way, a small, spring-loaded toy, such as a jack-in-the-box, has stored energy, but the energy is stored in a compressed spring. While both of these examples represent energy stored by mechanical means, there are many other means of storing energy. Automobiles, for example, carry their energy stored in the form of chemical energy in the gasoline tank. Energy is made useful or causes motion when it changes from one form to another. How does the money model that was discussed earlier illustrate the transformation of energy from one form to another? Money, too, can come in different forms. You can have one five-dollar bill, 20 quarters, or 500 pennies. In all of these cases, you still have five dollars. The height of the bar graph in Figure 11-4 represents the amount of money in each form. In the same way, you can use a bar graph to represent the amount of energy in various forms that a system has.

Gravitational Potential Energy Look at the oranges being juggled in Figure 11-5. If you consider the system to be only one orange, then it has several external forces acting on it. The force of the juggler’s hand does work, giving the orange its original kinetic energy. After the orange leaves the juggler’s hand, only the force of gravity acts on it. How much work does gravity do on the orange as its height changes?

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Figure 11-5 Kinetic and potential energy are constantly being exchanged when juggling.

288

Work done by gravity Let h represent the orange’s height measured from the juggler’s hand. On the way up, its displacement is upward, but the force on the orange, Fg, is downward, so the work done by gravity is negative: Wg mgh. On the way back down, the force and displacement are in the same direction, so the work done by gravity is positive: Wg mgh. Thus, while the orange is moving upward, gravity does negative work, slowing the orange to a stop. On the way back down, gravity does positive work, increasing the orange’s speed and thereby increasing its kinetic energy. The orange recovers all of the kinetic energy it originally had when it returns to the height at which it left the juggler’s hand. It is as if the orange’s kinetic energy is stored in another form as the ball rises and is transformed back to kinetic energy as the ball falls. Consider a system that consists of an object plus Earth. The gravitational attraction between the object and Earth is a force that always does work on the object as it moves. If the object moves away from Earth, energy is stored in the system as a result of the gravitational force between the object and Earth. This stored energy is called gravitational potential energy and is represented by the symbol PE. The height to which the object has risen is determined by using a reference level, the position where PE is defined to be zero. For an object with mass, m, that has risen to a height, h, above the reference level, gravitational potential energy is represented by the following equation.

Chapter 11 Energy and Its Conservation

Hutchings Photography

Gravitational Potential Energy PE mgh The gravitational potential energy of an object is equal to the product of its mass, the acceleration due to gravity, and the distance from the reference level.

In the equation for gravitational potential energy, g is the acceleration due to gravity. Gravitational potential energy, like kinetic energy, is measured in joules. Kinetic energy and potential energy of a system Consider the energy of a system consisting of an orange used by the juggler plus Earth. The energy in the system exists in two forms: kinetic energy and gravitational potential energy. At the beginning of the orange’s flight, all the energy is in the form of kinetic energy, as shown in Figure 11-6a. On the way up, as the orange slows down, energy changes from kinetic energy to potential energy. At the highest point of the orange’s flight, the velocity is zero. Thus, all the energy is in the form of gravitational potential energy. On the way back down, potential energy changes back into kinetic energy. The sum of kinetic energy and potential energy is constant at all times because no work is done on the system by any external forces. Reference levels In Figure 11-6a, the reference level is the juggler’s hand. That is, the height of the orange is measured from the juggler’s hand. Thus, at the juggler’s hand, h 0 m and PE 0 J. You can set the reference level at any height that is convenient for solving a given problem. Suppose the reference level is set at the highest point of the orange’s flight. Then, h 0 m and the system’s PE 0 J at that point, as illustrated in Figure 11-6b. The potential energy of the system is negative at the beginning of the orange’s flight, zero at the highest point, and negative at the end of the orange’s flight. If you were to calculate the total energy of the system represented in Figure 11-6a, it would be different from the total energy of the system represented in Figure 11-6b. This is because the reference levels are different in each case. However, the total energy of the system in each situation would be constant at all times during the flight of the orange. Only changes in energy determine the motion of a system.

a

Begin

Middle

b

End

h

Begin

Potential Energy of an Atom It is interesting to consider the relative sizes of potential energy per atom. For instance, a carbon atom has a mass of about 21026 kg. If you lift it 1 m above the ground, its gravitational potential energy is about 21025 J. The electrostatic energy that holds the electrons on the atom has a value of about 1019 J, and the nuclear potential energy that holds the nucleus together is greater than 1012 J. The nuclear potential energy is at least a million million times greater than the gravitational potential energy.

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Figure 11-6 The energy of an orange is converted from one form to another in various stages of its flight (a). Note that the choice of a reference level is arbitrary, but that the total energy remains constant (b).

Middle

End

0 Reference level

Reference level

0 KE

PE PE

KE

h KE

KE PE

KE PE

KE

PE

PE

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Gravitational Potential Energy You lift a 7.30-kg bowling ball from the storage rack and hold it up to your shoulder. The storage rack is 0.610 m above the floor and your shoulder is 1.12 m above the floor. a. When the bowling ball is at your shoulder, what is the bowling ball’s gravitational potential energy relative to the floor? b. When the bowling ball is at your shoulder, what is its gravitational potential energy relative to the storage rack? c. How much work was done by gravity as you lifted the ball from the rack to shoulder level? 1.12 m 1

Analyze and Sketch the Problem • Sketch the situation. • Choose a reference level. • Draw a bar graph showing the gravitational potential energy with the floor as the reference level.

2

Known:

Unknown:

m 7.30 kg hr 0.610 m (relative to the floor) hs 1.12 m (relative to the floor) g 9.80 m/s2

PEs rel f ? PEs rel r ?

0.61 m

PEs rel f PEr rel f

Wby you

Solve for the Unknown a. Set the reference level to be at the floor. Solve for the potential energy of the ball at shoulder level. PE s rel f mghs Substitute m 7.30 kg, g 9.80 m/s2, hshoulder 1.12 m (7.30 kg)(9.80 m/s2)(1.12 m) 80.1 J b. Set the reference level to be at the rack height. Math Handbook Solve for the height of your shoulder relative to the rack. Order of Operations page 843 h hs hr Solve for the potential energy of the ball. PEs rel r mgh mg(hs hr) Substitute h hs hr 2 (7.30 kg)(9.80 m/s )(1.12 m 0.610 m) Substitute m 7.3 kg, g 9.80 m/s2, 36.5 J

hs 1.12 m, hr 0.610 m This also is equal to the work done by you.

c. The work done by gravity is the weight of the ball times the distance the ball was lifted. W Fd Because the weight opposes the motion of lifting, the work is negative. (mg)h (mg)(hs hr) (7.30 kg)(9.80 m/s2)(1.12 m 0.610 m) Substitute m 7.30 kg, g 9.80 m/s2, 36.5 J 3

hs 1.12 m, hr 0.610 m

Evaluate the Answer • Are the units correct? The potential energy and work are both measured in joules. • Is the magnitude realistic? The ball should have a greater potential energy relative to the floor than relative to the rack, because the ball’s distance above the reference level is greater.

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4. In Example Problem 1, what is the potential energy of the bowling ball relative to the rack when it is on the floor? 5. If you slowly lower a 20.0-kg bag of sand 1.20 m from the trunk of a car to the driveway, how much work do you do? 6. A boy lifts a 2.2-kg book from his desk, which is 0.80 m high, to a bookshelf that is 2.10 m high. What is the potential energy of the book relative to the desk? 7. If a 1.8-kg brick falls to the ground from a chimney that is 6.7 m high, what is the change in its potential energy? 8. A warehouse worker picks up a 10.1-kg box from the floor and sets it on a long, 1.1-m-high table. He slides the box 5.0 m along the table and then lowers it back to the floor. What were the changes in the energy of the box, and how did the total energy of the box change? (Ignore friction.)

Elastic Potential Energy When the string on the bow shown in Figure 11-7 is pulled, work is done on the bow, storing energy in it. Thus, the energy of the system increases. Identify the system as the bow, the arrow, and Earth. When the string and arrow are released, energy is changed into kinetic energy. The stored energy in the pulled string is called elastic potential energy, which is often stored in rubber balls, rubber bands, slingshots, and trampolines. Energy also can be stored in the bending of an object. When stiff metal or bamboo poles were used in pole-vaulting, the poles did not bend easily. Little work was done on the poles, and consequently, the poles did not store much potential energy. Since flexible fiberglass poles were introduced, however, record pole-vaulting heights have soared. ■

Figure 11-7 Elastic potential energy is stored in the string of this bow. Before the string is released, the energy is all potential (a). As the string is released, the energy is transferred to the arrow as kinetic energy (b).

a

b

Section 11.1 The Many Forms of Energy

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(r)Getty Images, (l)Luis Romero/AP Wide World Photos

A pole-vaulter runs with a flexible pole and plants its end into the socket in the ground. When the pole-vaulter bends the pole, as shown in Figure 11-8, some of the pole-vaulter’s kinetic energy is converted to elastic potential energy. When the pole straightens, the elastic potential energy is converted to gravitational potential energy and kinetic energy as the pole-vaulter is lifted as high as 6 m above the ground. Unlike stiff metal poles or bamboo poles, fiberglass poles have an increased capacity for storing elastic potential energy. Thus, pole-vaulters are able to clear bars that are set very high. Mass Albert Einstein recognized yet another form of potential energy: mass itself. He said that mass, by its very nature, is energy. This energy, E0 , is called rest energy and is represented by the following famous formula. Rest Energy E0 mc2 The rest energy of an object is equal to the object’s mass times the speed of light squared.

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Figure 11-8 When a pole-vaulter jumps, elastic potential energy is changed into kinetic energy and gravitational potential energy.

According to this formula, stretching a spring or bending a vaulting pole causes the spring or pole to gain mass. In these cases, the change in mass is too small to be detected. When forces within the nucleus of an atom are involved, however, the energy released into other forms, such as kinetic energy, by changes in mass can be quite large.

11.1 Section Review 9. Elastic Potential Energy You get a spring-loaded toy pistol ready to fire by compressing the spring. The elastic potential energy of the spring pushes the rubber dart out of the pistol. You use the toy pistol to shoot the dart straight up. Draw bar graphs that describe the forms of energy present in the following instances. a. The dart is pushed into the gun barrel, thereby compressing the spring. b. The spring expands and the dart leaves the gun barrel after the trigger is pulled. c. The dart reaches the top of its flight. 10. Potential Energy A 25.0-kg shell is shot from a cannon at Earth’s surface. The reference level is Earth’s surface. What is the gravitational potential energy of the system when the shell is at 425 m? What is the change in potential energy when the shell falls to a height of 225 m? 11. Rotational Kinetic Energy Suppose some children push a merry-go-round so that it turns twice as fast as it did before they pushed it. What are the relative changes in angular momentum and rotational kinetic energy? 292

Chapter 11 Energy and Its Conservation

Bob Daemmrich/The Image Works

12. Work-Energy Theorem How can you apply the work-energy theorem to lifting a bowling ball from a storage rack to your shoulder? 13. Potential Energy A 90.0-kg rock climber first climbs 45.0 m up to the top of a quarry, then descends 85.0 m from the top to the bottom of the quarry. If the initial height is the reference level, find the potential energy of the system (the climber and Earth) at the top and at the bottom. Draw bar graphs for both situations. 14. Critical Thinking Karl uses an air hose to exert a constant horizontal force on a puck, which is on a frictionless air table. He keeps the hose aimed at the puck, thereby creating a constant force as the puck moves a fixed distance. a. Explain what happens in terms of work and energy. Draw bar graphs. b. Suppose Karl uses a different puck with half the mass of the first one. All other conditions remain the same. How will the kinetic energy and work differ from those in the first situation? c. Explain what happened in parts a and b in terms of impulse and momentum. physicspp.com/self_check_quiz

11.2 Conservation of Energy

C

onsider a ball near the surface of Earth. The sum of gravitational potential energy and kinetic energy in that system is constant. As the height of the ball changes, energy is converted from kinetic energy to potential energy, but the total amount of energy stays the same.

Objectives • Solve problems using the law of conservation of energy. • Analyze collisions to find the change in kinetic energy.

Conservation of Energy In our everyday world, it may not seem as if energy is conserved. A hockey puck eventually loses its kinetic energy and stops moving, even on smooth ice. A pendulum stops swinging after some time. The money model can again be used to illustrate what is happening in these cases. Suppose you have a total of $50 in cash. One day, you count your money and discover that you are $3 short. Would you assume that the money just disappeared? You probably would try to remember whether you spent it, and you might even search for it. In other words, rather than giving up on the conservation of money, you would try to think of different places where it might have gone.

Vocabulary law of conservation of energy mechanical energy thermal energy elastic collision inelastic collision

Law of conservation of energy Scientists do the same thing as you would if you could not account for a sum of money. Whenever they observe energy leaving a system, they look for new forms into which the energy could have been transferred. This is because the total amount of energy in a system remains constant as long as the system is closed and isolated from external forces. The law of conservation of energy states that in a closed, isolated system, energy can neither be created nor destroyed; rather, energy is conserved. Under these conditions, energy changes from one form to another while the total energy of the system remains constant. Conservation of mechanical energy The sum of the kinetic energy and gravitational potential energy of a system is called mechanical energy. In any given system, if no other forms of energy are present, mechanical energy is represented by the following equation. Mechanical Energy of a System E KE PE

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Figure 11-9 A decrease in potential energy is equal to the increase in kinetic energy. 10.0 N

The mechanical energy of a system is equal to the sum of the kinetic energy and potential energy if no other forms of energy are present.

Imagine a system consisting of a 10.0-N ball and Earth, as shown in Figure 11-9. Suppose the ball is released from 2.00 m above the ground, which you set to be the reference level. Because the ball is not yet moving, it has no kinetic energy. Its potential energy is represented by the following equation:

0.0 J 20.0 J

2.00 m KE

PE

10.0 J 10.0 J 1.00 m KE

PE mgh (10.0 N)(2.00 m) 20.0 J

PE

20.0 J 0.0 J

The ball’s total mechanical energy, therefore, is 20.0 J. As the ball falls, it loses potential energy and gains kinetic energy. When the ball is 1.00 m above Earth’s surface: PE mgh (10.0 N)(1.00 m) 10.0 J.

0.00 m Ground

KE

Section 11.2 Conservation of Energy

PE

293

Weight 10.0 N

What is the ball’s kinetic energy when it is at a height of 1.00 m? The system consisting of the ball and Earth is closed and isolated because no external forces are acting upon it. Hence, the total energy of the system, E, remains constant at 20.0 J.

PE 20.0 J

4.0

m

2.0 m

KE 20.0 J

E KE PE, so KE E PE

KE 20.0 J

KE 20.0 J 10.0 J = 10.0 J When the ball reaches ground level, its potential energy is zero, and its kinetic energy is 20.0 J. The equation that describes conservation of mechanical energy can be written as follows.

■ Figure 11-10 The path that an object follows in reaching the ground does not affect the final kinetic energy of the object.

Conservation of Mechanical Energy KEbefore PEbefore KEafter PEafter When mechanical energy is conserved, the sum of the kinetic energy and potential energy present in the system before the event is equal to the sum of the kinetic energy and potential energy in the system after the event.

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Figure 11-11 For the simple harmonic motion of a pendulum bob (a), the mechanical energy— the sum of the potential and kinetic energies—is a constant (b).

a

What happens if the ball does not fall down, but rolls down a ramp, as shown in Figure 11-10? If there is no friction, there are no external forces acting on the system. Thus, the system remains closed and isolated. The ball still moves down a vertical distance of 2.00 m, so its loss of potential energy is 20.0 J. Therefore, it gains 20.0 J of kinetic energy. As long as there is no friction, the path that the ball takes does not matter. Roller coasters In the case of a roller coaster that is nearly at rest at the top of the first hill, the total mechanical energy in the system is the coaster’s gravitational potential energy at that point. Suppose some other hill along the track were higher than the first one. The roller coaster would not be able to climb the higher hill because the energy required to do so would be greater than the total mechanical energy of the system.

A

C B

b

Energy v. Position PE + KE

Energy

PE

KE A

B Horizontal position

294

C

Skiing Suppose you ski down a steep slope. When you begin from rest at the top of the slope, your total mechanical energy is simply your gravitational potential energy. Once you start skiing downhill, your gravitational potential energy is converted to kinetic energy. As you ski down the slope, your speed increases as more of your potential energy is converted to kinetic energy. In ski jumping, the height of the ramp determines the amount of energy that the jumper has to convert into kinetic energy at the beginning of his or her flight. Pendulums The simple oscillation of a pendulum also demonstrates conservation of energy. The system is the pendulum bob and Earth. Usually, the reference level is chosen to be the height of the bob at the lowest point, when it is at rest. If an external force pulls the bob to one side, the force does work that gives the system mechanical energy. At the instant the bob is released, all the energy is in the form of potential energy, but as the bob swings downward, the energy is converted to kinetic energy. Figure 11-11 shows a graph of the changing potential and kinetic energies of a pendulum. When the bob is at the lowest point, its gravitational potential energy is zero, and its kinetic energy is equal to the total mechanical

Chapter 11 Energy and Its Conservation

energy in the system. Note that the total mechanical energy of the system is constant if we assume that there is no friction. You will learn more about pendulums in Chapter 14. Loss of mechanical energy The oscillations of a pendulum eventually come to a stop, a bouncing ball comes to rest, and the heights of rollercoaster hills get lower and lower. Where does the mechanical energy in such systems go? Any object moving through the air experiences the forces of air resistance. In a roller coaster, there are frictional forces between the wheels and the tracks. When a ball bounces off of a surface, all of the elastic potential energy that is stored in the deformed ball is not converted back into kinetic energy after the bounce. Some of the energy is converted into thermal energy and sound energy. As in the cases of the pendulum and the roller coaster, some of the original mechanical energy in the system is converted into another form of energy within members of the system or transmitted to energy outside the system, as in air resistance. Usually, this new energy causes the temperature of objects to rise slightly. You will learn more about this form of energy, called thermal energy, in Chapter 12. The following strategies will be helpful to you when solving problems related to conservation of energy.

Conservation of Energy When solving problems related to the conservation of energy, use the following strategies.

1. Carefully identify the system. Make sure it is closed. In a closed system, no objects enter or leave the system. 2. Identify the forms of energy in the system. 3. Identify the initial and final states of the system. 4. Is the system isolated? a. If there are no external forces acting on the system, then the system is isolated and the total energy of the system is constant. Ebefore Eafter

Energy Bar Graphs Initial

b. If there are external forces, then the following is true. Ebefore W Eafter 5. If mechanical energy is conserved, decide on the reference level for potential energy. Draw bar graphs showing initial and final energy like the bar graphs shown to the right.

Final

KEi

Total energy

PEf KEf

PEi

Section 11.2 Conservation of Energy

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Conservation of Mechanical Energy During a hurricane, a large tree limb, with a mass of 22.0 kg and a height of 13.3 m above the ground, falls on a roof that is 6.0 m above the ground. a. Ignoring air resistance, find the kinetic energy of the limb when it reaches the roof. b. What is the speed of the limb when it reaches the roof? 1

Before (initial) vi 0.0 m/s hi 13.3 m

Ground reference

Analyze and Sketch the Problem • Sketch the initial and final conditions. • Choose a reference level. • Draw a bar graph.

After (final) vf

Known: m 22.0 kg hlimb 13.3 m hroof 6.0 m

g 9.80 m/s2 vi 0.0 m/s KEi 0.0 J

Unknown: PEi ? PEf ? 2

hf 6.0 m

PEi KEi

KEf ? vf ?

PEf

KEf

Solve for the Unknown a. Set the reference level as the height of the roof. Solve for the initial height of the limb relative to the roof. h hlimb hroof 13.3 m 6.0 m Substitute hlimb 13.3 m, hroof 6.0 m 7.3 m

Bar Graph

Solve for the initial potential energy of the limb. PEi mgh (22.0 kg)(9.80 m/s2)(7.3 m) Substitute m 22.0 kg, g 9.80 m/s2, h 7.3 m 1.6103 J Identify the initial kinetic energy of the limb. KEi 0.0 J The tree limb is initially at rest. The kinetic energy of the limb when it reaches the roof is equal to its initial potential energy because energy is conserved. KEf PEi PEf 0.0 J because h 0.0 m at the reference level. 1.6103 J b. Solve for the speed of the limb. 1 2

KEf mvf2 vf

Math Handbook

2KE f m 2(1.61 03 J) 22.0 kg

Square and Cube Roots pages 839–840 Substitute KEf 1.6103 J, m 22.0 kg

12 m/s 3

Evaluate the Answer • Are the units correct? Velocity is measured in m/s and energy is measured in kgm2/s2 J. • Do the signs make sense? KE and the magnitude of velocity are always positive.

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Chapter 11 Energy and Its Conservation

15. A bike rider approaches a hill at a speed of 8.5 m/s. The combined mass of the bike and the rider is 85.0 kg. Choose a suitable system. Find the initial kinetic energy of the system. The rider coasts up the hill. Assuming there is no friction, at what height will the bike come to rest? 16. Suppose that the bike rider in problem 15 pedaled up the hill and never came to a stop. In what system is energy conserved? From what form of energy did the bike gain mechanical energy? 17. A skier starts from rest at the top of a 45.0-m-high hill, skis down a 30° incline into a valley, and continues up a 40.0-m-high hill. The heights of both hills are measured from the valley floor. Assume that you can neglect friction and the effect of the ski poles. How fast is the skier moving at the bottom of the valley? What is the skier’s speed at the top of the next hill? Do the angles of the hills affect your answers? 18. In a belly-flop diving contest, the winner is the diver who makes the biggest splash upon hitting the water. The size of the splash depends not only on the diver’s style, but also on the amount of kinetic energy that the diver has. Consider a contest in which each diver jumps from a 3.00-m platform. One diver has a mass of 136 kg and simply steps off the platform. Another diver has a mass of 102 kg and leaps upward from the platform. How high would the second diver have to leap to make a competitive splash?

Analyzing Collisions A collision between two objects, whether the objects are automobiles, hockey players, or subatomic particles, is one of the most common situations analyzed in physics. Because the details of a collision can be very complex during the collision itself, the strategy is to find the motion of the objects just before and just after the collision. What conservation laws can be used to analyze such a system? If the system is isolated, then momentum and energy are conserved. However, the potential energy or thermal energy in the system may decrease, remain the same, or increase. Therefore, you cannot predict whether or not kinetic energy is conserved. Figure 11-12 and Figure 11-13 on the next page show three different kinds of collisions. In case 1, the momentum of the system before and after the collision is represented by the following: pi pCi pDi (1.00 kg)(1.00 m/s) (1.00 kg)(0.00 m/s) 1.00 kgm/s pf pCf pDf (1.00 kg)(0.20 m/s) (1.00 kg)(1.20 m/s) 1.00 kgm/s Thus, in case 1, the momentum is conserved. Look again at Figure 11-13 and verify for yourself that momentum is conserved in cases 2 and 3. Before (initial) mD 1.00 kg mC 1.00 kg C

D

vCi 1.00 m/s

vDi 0.00 m/s

Case 1

After (final)

C

vCf 0.20 m/s

■ Figure 11-12 Two moving objects can have different types of collisions. Case 1: the two objects move apart in opposite directions.

D

vDf 1.20 m/s

Section 11.2 Conservation of Energy

297

■ Figure 11-13 Case 2: the moving object comes to rest and the stationary object begins to move. Case 3: the two objects are stuck together and move as one.

Case 2

Before (initial) mC 1.00 kg

After (final)

mD 1.00 kg

C

D

vCi 1.00 m/s

C

vDi 0.00 m/s

D

vCf 0.00 m/s

vDf 1.00 m/s

Case 3

C

D

vCi 1.00 m/s

C

vDi 0.00 m/s

D

vCf vDf 0.50 m/s

Next, consider the kinetic energy of the system in each of these cases. For case 1 the kinetic energy of the system before and after the collision is represented by the following equations: 1 2

1 2

KECi KEDi (1.00 kg)(1.00 m/s)2 (1.00 kg)(0.00 m/s)2 0.50 J 1 2

1 2

KECf KEDf (1.00 kg)(0.20 m/s)2 (1.00 kg)(1.20 m/s)2 0.74 J

■

Figure 11-14 Bar graphs can be drawn to represent the three kinds of collisions. Before KE Other energy

After KE Other energy

Case 1: KE increases

Case 2: KE is constant

Case 3: KE decreases

298

In case 1, the kinetic energy of the system increased. If energy in the system is conserved, then one or more of the other forms of energy must have decreased. Perhaps when the two carts collided, a compressed spring was released, adding kinetic energy to the system. This kind of collision is sometimes called a superelastic or explosive collision. After the collision in case 2, the kinetic energy is equal to: 1 2

KECf KEDf (1.0 kg)(0.00 m/s)2 (1.0 kg)(1.0 m/s)2 0.50 J Kinetic energy remained the same after the collision. This type of collision, in which the kinetic energy does not change, is called an elastic collision. Collisions between hard, elastic objects, such as those made of steel, glass, or hard plastic, often are called nearly elastic collisions. After the collision in case 3, the kinetic energy is equal to: 1 2

1 2

KECf KEDf (1.00 kg)(0.50 m/s)2 (1.00 kg)(0.50 m/s)2 0.25 J Kinetic energy decreased and some of it was converted to thermal energy. This kind of collision, in which kinetic energy decreases, is called an inelastic collision. Objects made of soft, sticky material, such as clay, act in this way. The three kinds of collisions can be represented using bar graphs, such as those shown in Figure 11-14. Although the kinetic energy before and after the collisions can be calculated, only the change in other forms of energy can be found. In automobile collisions, kinetic energy is transferred into other forms of energy, such as heat and sound.

Chapter 11 Energy and Its Conservation

Kinetic Energy In an accident on a slippery road, a compact car with a mass of 575 kg moving at 15.0 m/s smashes into the rear end of a car with mass 1575 kg moving at 5.00 m/s in the same direction. a. What is the final velocity if the wrecked cars lock together? b. How much kinetic energy was lost in the collision? c. What fraction of the original kinetic energy was lost? 1

Analyze and Sketch the Problem

Before (initial) mAvAi

vBi

vAi

• Sketch the initial and final conditions. • Sketch the momentum diagram.

After (final) (mA mB)vf

Known: mA 575 kg vAi 15.0 m/s

mBvBi

mB 1575 kg vBi 5.00 m/s vAf vBf vf

Unknown:

vf ? KE KEf KEi ? Fraction of KEi lost, KE/KEi ? 2

vf

Solve for the Unknown a. Use the conservation of momentum equation to find the final velocity. pAi pBi pAf pBf mAvAi mBvBi (mA mB)vf (m v m v ) (mA mB) (575 kg)(15.0 m/s) (1575 kg)(5.00 m/s) (575 kg 1575 kg)

B Bi A Ai vf

Math Handbook Isolating a Variable page 845

Substitute mA 575 kg, vAi 15.0 m/s, mB 1575 kg, vBi 5.00 m/s

7.67 m/s, in the direction of the motion before the collision b. To determine the change in kinetic energy of the system, KEf and KEi are needed. 1 KEf mv 2 2 1 (mA mB)vf2 2 1 (575 kg + 1575 kg)(7.67 m/s)2 2

Substitute m mA mB Substitute mA 575 kg, mB 1575 kg, vf 7.67 m/s

6.32104 J KEi KEAi KEBi

1 1 1 1 Substitute KEAi mAvAi2, KEBi mBvBi2 2 2 2 2 1 1 2 2 Substitute mA 575 kg, mB 1575 kg, (575 kg)(15.0 m/s) (1575 kg)(5.00 m/s) 2 2

mAvAi2 mBvBi2

vAi 15.0 m/s, vBi 5.00 m/s 8.44104 J Solve for the change in kinetic energy of the system. KE KEf KEi 6.32104 J 8.44104 J Substitute KEf 6.32104 J, KEi 8.44104 J 4 2.1210 J

c. Calculate the fraction of the original kinetic energy that is lost. 2.12104 J KE 8.44104 J K Ei

Substitute KE 2.11104 J, KEi 8.44104 J

0.251 25.1% of the original kinetic energy in the system was lost.

3

Evaluate the Answer • Are the units correct? Velocity is measured in m/s; energy is measured in J. • Does the sign make sense? Velocity is positive, consistent with the original velocities.

Section 11.2 Conservation of Energy

299

19. An 8.00-g bullet is fired horizontally into a 9.00-kg block of wood on an air table and is embedded in it. After the collision, the block and bullet slide along the frictionless surface together with a speed of 10.0 cm/s. What was the initial speed of the bullet? 20. A 0.73-kg magnetic target is suspended on a string. A 0.025-kg magnetic dart, shot horizontally, strikes the target head-on. The dart and the target together, acting like a pendulum, swing 12.0 cm above the initial level before instantaneously coming to rest. a. Sketch the situation and choose a system. b. Decide what is conserved in each part and explain your decision. c. What was the initial velocity of the dart? 21. A 91.0-kg hockey player is skating on ice at 5.50 m/s. Another hockey player of equal mass, moving at 8.1 m/s in the same direction, hits him from behind. They slide off together. a. What are the total energy and momentum in the system before the collision? b. What is the velocity of the two hockey players after the collision? c. How much energy was lost in the collision?

In collisions, you can see how momentum and energy are really very different. Momentum is almost always conserved in a collision. Energy is conserved only in elastic collisions. Momentum is what makes objects stop. A 10.0-kg object moving at 5.00 m/s will stop a 20.0-kg object moving at 2.50 m/s if they have a head-on collision. However, in this case, the smaller object has much more kinetic energy. The kinetic energy of the smaller object is KE 12 (10.0 kg)(5.0 m/s)2 125 J. The kinetic energy of the larger object is KE 12 (20.0 kg)(2.50 m/s)2 62.5 J. Based on the work-energy theorem, you can conclude that it takes more work to make the 10.0-kg object move at 5.00 m/s than it does to move the 20.0-kg object at 2.50 m/s. It sometimes is said that in automobile collisions, the momentum stops the cars but it is the energy in the collision that causes the damage. It also is possible to have a collision in which nothing collides. If two lab carts sit motionless on a table, connected by a compressed spring, their total momentum is zero. If the spring is released, the carts will be forced to move away from each other. The potential energy of the spring will be transformed into the kinetic energy of the carts. The carts will still move away from each other so that their total momentum is zero.

A bullet of mass m, moving at speed v1, goes through a motionless wooden block and exits with speed v2. After the collision, the block, which v 1 has mass mB, is moving. 1. What is the final speed, vB, of the block? 2. How much energy was lost to the bullet? 3. How much energy was lost to friction inside the block?

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Chapter 11 Energy and Its Conservation

Initial

Final v2 vB

It is useful to remember two simple examples of collisions. One is the elastic collision between two objects of equal mass, such as when a cue ball with velocity, v, hits a motionless billiard ball head-on. In this case, after the collision, the cue ball is motionless and the other ball rolls off at velocity, v. It is easy to prove that both momentum and energy are conserved in this collision. The other simple example is to consider a skater of mass m, with velocity v, running into another skater of equal mass who happens to be standing motionless on the ice. If they hold on to each other after the collision, they will slide off at a velocity of 12v because of the conservation of momentum. The final kinetic energy of the pair would be equal to KE 21(2m)(21v)2 41mv2, which is half the initial kinetic energy. This is because the collision was inelastic. You have investigated examples in which the conservation of energy, and sometimes the conservation of momentum, can be used to calculate the motions of a system of objects. These systems would be too complicated to comprehend using only Newton’s second law of motion. The understanding of the forms of energy and how energy flows from one form to another is one of the most useful concepts in science. The term energy conservation appears in everything from scientific papers to electric appliance commercials. Scientists use the concept of energy to explore topics much more complicated than colliding billiard balls.

Energy Exchange 1. Select different-sized steel balls and determine their masses. 2. Stand a spring-loaded laboratory cart on end with the spring mechanism pointing upward. 3. Place a ball on top of the spring mechanism and press down until the ball is touching the cart. 4. Quickly release the ball so that the spring shoots it upward. CAUTION: Stay clear of the ball when launching. 5. Repeat the process several times, and measure the average height. 6. Estimate how high the other sizes of steel balls will rise. Analyze and Conclude 7. Classify the balls by height attained. What can you conclude?

11.2 Section Review 22. Closed Systems Is Earth a closed, isolated system? Support your answer. 23. Energy A child jumps on a trampoline. Draw bar graphs to show the forms of energy present in the following situations. a. The child is at the highest point. b. The child is at the lowest point.

27. Energy As shown in Figure 11-15, a 36.0-kg child slides down a playground slide that is 2.5 m high. At the bottom of the slide, she is moving at 3.0 m/s. How much energy was lost as she slid down the slide?

36.0 kg

24. Kinetic Energy Suppose a glob of chewing gum and a small, rubber ball collide head-on in midair and then rebound apart. Would you expect kinetic energy to be conserved? If not, what happens to the energy? 2.5 m

25. Kinetic Energy In table tennis, a very light but hard ball is hit with a hard rubber or wooden paddle. In tennis, a much softer ball is hit with a racket. Why are the two sets of equipment designed in this way? Can you think of other ball-paddle pairs in sports? How are they designed? 26. Potential Energy A rubber ball is dropped from a height of 8.0 m onto a hard concrete floor. It hits the floor and bounces repeatedly. Each time it hits the floor, it loses 15 of its total energy. How many times will it bounce before it bounces back up to a height of only about 4 m? physicspp.com/self_check_quiz

■

Figure 11-15

28. Critical Thinking A ball drops 20 m. When it has fallen half the distance, or 10 m, half of its energy is potential and half is kinetic. When the ball has fallen for half the amount of time it takes to fall, will more, less, or exactly half of its energy be potential energy? Section 11.2 Conservation of Energy

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Horizons Companies

Conservation of Energy Alternate CBL instructions can be found on the Web site. physicspp.com

There are many examples of situations where energy is conserved. One such example is a rock falling from a given height. If the rock starts at rest, at the moment the rock is dropped, it only has potential energy. As it falls, its potential energy decreases as its height decreases, but its kinetic energy increases. The sum of potential energy and kinetic energy remains constant if friction is neglected. When the rock is about to hit the ground, all of its potential energy has been converted to kinetic energy. In this experiment, you will model a falling object and calculate its speed as it hits the ground.

QUESTION How does the transfer of an object’s potential energy to kinetic energy demonstrate conservation of energy?

Objectives

Materials

■ Calculate the speed of a falling object as it hits

grooved track (two sections) marble or steel ball stopwatch block of wood

the ground by using a model. ■ Interpret data to find the relationship between potential energy and kinetic energy of a falling object.

Procedure

Safety Precautions

1. Place the two sections of grooved track together, as shown in Figure 1. Raise one end of the track and place the block under it, about 5 cm from the raised end. Make sure the ball can roll smoothly across the junction of the two tracks.

Figure 1

Figure 2

Figure 3 302

electronic balance metric ruler graphing calculator

2. Record the length of the level portion of the track in the data table. Place a ball on the track directly above the point supported by the block. Release the ball. Start the stopwatch when the ball reaches the level section of track. Stop timing when the ball reaches the end of the level portion of the track. Record the time required for the ball to travel that distance in the data table. 3. Move the support block so that it is under the midsection of the inclined track, as shown in Figure 2. Place the ball on the track just above the point supported by the block. Release the ball and measure the time needed for the ball to roll the length of the level portion of the track and record it in the data table. Notice that even though the incline is steeper, the ball is released from the same height as in step 2. 4. Calculate the speed of the ball on the level portion of the track in steps 2 and 3. Move the support block to a point about three-quarters down the length of the inclined track, as shown in Figure 3.

Data Table Release Height (m)

Distance (m)

Time (s)

Speed (m/s)

0.05 0.05 0.05 0.01 0.02 0.03

5. Predict the amount of time the ball will take to travel the length of the level portion of the track. Record your prediction. Test your prediction.

Conclude and Apply

6. Place the support block at the midpoint of the inclined track (Figure 2). Measure a point on the inclined portion of the track that is 1.0 cm above the level portion of the track. Be sure to measure 1.0 cm above the level portion, and not 1.0 cm above the table.

2. How does the equation found in the previous question relate to the power law regression calculated earlier?

7. Release the ball from this point and measure the time required for the ball to travel on the level portion of the track and record it in the data table. 8. Use a ruler to measure a point that is 2.0 cm above the level track. Release the ball from this point and measure the time required for the ball to travel on the level portion of the track. Record the time in the data table.

1. Solve for speed, y, in terms of height, x. Begin by setting PEi KEf.

3. Suppose you want the ball to roll twice as fast on the level part of the track as it did when you released it from the 2-cm mark. Using the power law regression performed earlier, calculate the height from which you should release the ball. 4. Explain how this experiment only models dropping a ball and finding its kinetic energy just as it hits the ground.

9. Repeat step 8 for 3.0 cm, 4.0 cm, 5.0 cm, 6.0 cm, 7.0 cm, and 8.0 cm. Record the times.

5. Compare and Contrast Compare the potential energy of the ball before it is released (step 8) to the kinetic energy of the ball on the level track (step 9). Explain why they are the same or why they are different.

Analyze

6. Draw Conclusions Does this experiment demonstrate conservation of energy? Explain.

1. Infer What effect did changing the slope of the inclined plane in steps 2–6 have on the speed of the ball on the level portion of the track? 2. Analyze Perform a power law regression for this graph using your graphing calculator. Record the equation of this function. Graph this by inputting the equation into Y=. Draw a sketch of the graph. 3. Using the data from step 9 for the release point of 8.0 cm, find the potential energy of the ball before it was released. Use an electronic balance to find the mass of the ball. Note that height must be in m, and mass in kg. 4. Using the speed data from step 9 for the release point of 8.0 cm calculate the kinetic energy of the ball on the level portion of the track. Remember, speed must be in m/s and mass in kg.

Going Further What are potential sources of error in this experiment, and how can they be reduced?

Real-World Physics How does your favorite roller coaster demonstrate the conservation of energy by the transfer of potential energy to kinetic energy?

To find out more about energy, visit the Web site: physicspp.com

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Running Smarter The Physics of Running Shoes Today’s running shoes are high-tech marvels. They enhance performance and protect your body by acting as shock absorbers. How do running shoes help you win a race? They Upper reduce your energy consumption, as well as allow you to use energy more efficiently. Good running shoes must be flexible enough to bend with your feet as you run, support your feet, and hold them in place. They must be lightInsert weight and provide traction to prevent slipping.

Running Shoes as Shock Absorbers Today, much of the

Midsole

focus of running shoe technology centers on the cushioned midsole Outsole that plays a key role as a shockabsorber and performance enhancer. Each time a runner’s foot hits the ground, the ground exerts an equal and opposite force on the runner’s foot. This force can be nearly four times a runner’s weight, causing aches and pains, shin splints, and damage to knees and ankles over long distances. Cushioning is used in running shoes to decrease the force absorbed by the runner. As a runner’s foot hits the ground and comes to a stop, its momentum changes. The change in momentum is p Ft, where F is the force on that object and t is the time during which the force acts. The cushioning causes the change of momentum to occur over an extended time and reduces the force of the foot on the ground. The decreased force reduces the damage to the runner’s body.

Running Shoes Boost Performance A shoe’s cushioning system also affects energy consumption. The bones, muscles, ligaments, 304

Technology and Society

Horizons Companies

and tendons of the foot and leg are a natural cushioning system. But operating this system requires the body to use stored energy to contract muscles. So if a shoe can be worn that assists a runner’s natural cushioning system, the runner does not expend as much of his or her own stored energy. The energy the runner saved can be spent to run farther or faster. The cushioned midsole uses the law of conservation of energy to return as much of the energy to the runner as possible. The runner’s kinetic energy transforms to elastic potential energy, plus heat, when the runner’s foot hits the running surface. If the runner can reduce the amount of energy that is lost as heat, the runner’s elastic potential energy can be converted back to useful kinetic energy. Bouncy, springy, elastic materials that resist crushing over time commonly are used to create the cushioned midsole. Options now range from silicon gel pads to complex fluid-filled systems and even springs to give a runner extra energy efficiency.

Going Further 1. Use Scientific Explanations Use physics to explain why manufacturers put cushioned midsoles in running shoes. 2. Analyze Which surface would provide more cushioning when running: a grassy field or a concrete sidewalk? Explain why that surface provides better cushioning. 3. Research Some people prefer to run barefoot, even in marathon races. Why might this be so?

11.1 The Many Forms of Energy Vocabulary

Key Concepts

• rotational kinetic energy

•

(p. 287)

• gravitational potential energy (p. 288) • reference level (p. 288) • elastic potential energy (p. 291)

• • •

The kinetic energy of an object is proportional to its mass and the square of its velocity. The rotational kinetic energy of an object is proportional to the object’s moment of inertia and the square of its angular velocity. When Earth is included in a system, the work done by gravity is replaced by gravitational potential energy. The gravitational potential energy of an object depends on the object’s weight and its distance from Earth’s surface. PE mgh

• • •

The reference level is the position where the gravitational potential energy is defined to be zero. Elastic potential energy may be stored in an object as a result of its change in shape. Albert Einstein recognized that mass itself has potential energy. This energy is called rest energy. E0 mc2

11.2 Conservation of Energy Vocabulary

Key Concepts

• law of conservation of energy (p. 293) • mechanical energy

•

(p. 293)

• thermal energy (p. 295) • elastic collision (p. 298) • inelastic collision (p. 298)

The sum of kinetic and potential energy is called mechanical energy. E KE PE

• • •

If no objects enter or leave a system, the system is considered to be a closed system. If there are no external forces acting on a system, the system is considered to be an isolated system. The total energy of a closed, isolated system is constant. Within the system, energy can change form, but the total amount of energy does not change. Thus, energy is conserved. KEbefore PEbefore KEafter PEafter

• • •

The type of collision in which the kinetic energy after the collision is less than the kinetic energy before the collision is called an inelastic collision. The type of collision in which the kinetic energy before and after the collision is the same is called an elastic collision. Momentum is conserved in collisions if the external force is zero. The mechanical energy may be unchanged or decreased by the collision, depending on whether the collision is elastic or inelastic.

physicspp.com/vocabulary_puzzlemaker

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Concept Mapping 29. Complete the concept map using the following terms: gravitational potential energy, elastic potential energy, kinetic energy.

Energy

39. You throw a clay ball at a hockey puck on ice. The smashed clay ball and the hockey puck stick together and move slowly. (11.2) a. Is momentum conserved in the collision? Explain. b. Is kinetic energy conserved? Explain.

40. Draw energy bar graphs for the following processes. (11.2)

potential

linear

rotational

a. An ice cube, initially at rest, slides down a frictionless slope. b. An ice cube, initially moving, slides up a frictionless slope and instantaneously comes to rest.

41. Describe the transformations from kinetic energy to

Mastering Concepts Unless otherwise directed, assume that air resistance is negligible.

30. Explain how work and a change in energy are related. (11.1)

31. What form of energy does a wound-up watch spring have? What form of energy does a functioning mechanical watch have? When a watch runs down, what has happened to the energy? (11.1)

32. Explain how energy change and force are related. (11.1) 33. A ball is dropped from the top of a building. You choose the top of the building to be the reference level, while your friend chooses the bottom. Explain whether the energy calculated using these two reference levels is the same or different for the following situations. (11.1) a. the ball’s potential energy at any point b. the change in the ball’s potential energy as a result of the fall c. the kinetic energy of the ball at any point

34. Can the kinetic energy of a baseball ever be negative? (11.1)

35. Can the gravitational potential energy of a baseball ever be negative? Explain without using a formula. (11.1)

36. If a sprinter’s velocity increases to three times the original velocity, by what factor does the kinetic energy increase? (11.1)

37. What energy transformations take place when an athlete is pole-vaulting? (11.2)

38. The sport of pole-vaulting was drastically changed when the stiff, wooden poles were replaced by flexible, fiberglass poles. Explain why. (11.2)

306

potential energy and vice versa for a roller-coaster ride. (11.2)

42. Describe how the kinetic energy and elastic potential energy are lost in a bouncing rubber ball. Describe what happens to the motion of the ball. (11.2)

Applying Concepts 43. The driver of a speeding car applies the brakes and the car comes to a stop. The system includes the car but not the road. Apply the work-energy theorem to the following situations. a. The car’s wheels do not skid. b. The brakes lock and the car’s wheels skid.

44. A compact car and a trailer truck are both traveling at the same velocity. Did the car engine or the truck engine do more work in accelerating its vehicle?

45. Catapults Medieval warriors used catapults to assault castles. Some catapults worked by using a tightly wound rope to turn the catapult arm. What forms of energy are involved in catapulting a rock to the castle wall?

46. Two cars collide and come to a complete stop. Where did all of their energy go?

47. During a process, positive work is done on a system, and the potential energy decreases. Can you determine anything about the change in kinetic energy of the system? Explain.

48. During a process, positive work is done on a system, and the potential energy increases. Can you tell whether the kinetic energy increased, decreased, or remained the same? Explain.

49. Skating Two skaters of unequal mass have the same speed and are moving in the same direction. If the ice exerts the same frictional force on each skater, how will the stopping distances of their bodies compare?

Chapter 11 Energy and Its Conservation For more problems, go to Additional Problems, Appendix B.

50. You swing a 55-g mass on the end of a 0.75-m string around your head in a nearly horizontal circle at constant speed, as shown in Figure 11-16. a. How much work is done on the mass by the tension of the string in one revolution? b. Is your answer to part a in agreement with the work-energy theorem? Explain.

57. Tony has a mass of 45 kg and is moving with a speed of 10.0 m/s. a. Find Tony’s kinetic energy. b. Tony’s speed changes to 5.0 m/s. Now what is his kinetic energy? c. What is the ratio of the kinetic energies in parts a and b? Explain.

58. Katia and Angela each have a mass of 45 kg, and

0.75 m 55 g

they are moving together with a speed of 10.0 m/s. a. What is their combined kinetic energy? b. What is the ratio of their combined mass to Katia’s mass? c. What is the ratio of their combined kinetic energy to Katia’s kinetic energy? Explain.

59. Train In the 1950s, an experimental train, which

■

Figure 11-16

51. Give specific examples that illustrate the following processes. a. Work is done on a system, thereby increasing kinetic energy with no change in potential energy. b. Potential energy is changed to kinetic energy with no work done on the system. c. Work is done on a system, increasing potential energy with no change in kinetic energy. d. Kinetic energy is reduced, but potential energy is unchanged. Work is done by the system.

52. Roller Coaster You have been hired to make a roller coaster more exciting. The owners want the speed at the bottom of the first hill doubled. How much higher must the first hill be built?

had a mass of 2.50104 kg, was powered across a level track by a jet engine that produced a thrust of 5.00105 N for a distance of 509 m. a. Find the work done on the train. b. Find the change in kinetic energy. c. Find the final kinetic energy of the train if it started from rest. d. Find the final speed of the train if there had been no friction.

60. Car Brakes A 14,700-N car is traveling at 25 m/s. The brakes are applied suddenly, and the car slides to a stop, as shown in Figure 11-17. The average braking force between the tires and the road is 7100 N. How far will the car slide once the brakes are applied? Before (initial) v 25 m/s

After (final) v 0.0 m/s

53. Two identical balls are thrown from the top of a cliff, each with the same speed. One is thrown straight up, the other straight down. How do the kinetic energies and speeds of the balls compare as they strike the ground?

Mastering Problems Unless otherwise directed, assume that air resistance is negligible.

11.1 The Many Forms of Energy 54. A 1600-kg car travels at a speed of 12.5 m/s. What is its kinetic energy?

55. A racing car has a mass of 1525 kg. What is its kinetic energy if it has a speed of 108 km/h?

56. Shawn and his bike have a combined mass of 45.0 kg. Shawn rides his bike 1.80 km in 10.0 min at a constant velocity. What is Shawn’s kinetic energy? physicspp.com/chapter_test

m 14,700 N ■

Figure 11-17

61. A 15.0-kg cart is moving with a velocity of 7.50 m/s down a level hallway. A constant force of 10.0 N acts on the cart, and its velocity becomes 3.20 m/s. a. What is the change in kinetic energy of the cart? b. How much work was done on the cart? c. How far did the cart move while the force acted?

62. How much potential energy does DeAnna with a mass of 60.0 kg, gain when she climbs a gymnasium rope a distance of 3.5 m?

63. Bowling A 6.4-kg bowling ball is lifted 2.1 m into a storage rack. Calculate the increase in the ball’s potential energy. Chapter 11 Assessment

307

64. Mary weighs 505 N. She walks down a flight of stairs to a level 5.50 m below her starting point. What is the change in Mary’s potential energy?

65. Weightlifting A weightlifter raises a 180-kg barbell

Before (initial) vi 12.0 m/s

After (final) vf 0.0 m/s

to a height of 1.95 m. What is the increase in the potential energy of the barbell?

66. A 10.0-kg test rocket is fired vertically from Cape

m 2.00103 kg

Canaveral. Its fuel gives it a kinetic energy of 1960 J by the time the rocket engine burns all of the fuel. What additional height will the rocket rise?

■

Figure 11-19

67. Antwan raised a 12.0-N physics book from a table

72. A constant net force of 410 N is applied upward to a

75 cm above the floor to a shelf 2.15 m above the floor. What was the change in the potential energy of the system?

stone that weighs 32 N. The upward force is applied through a distance of 2.0 m, and the stone is then released. To what height, from the point of release, will the stone rise?

68. A hallway display of energy is constructed in which several people pull on a rope that lifts a block 1.00 m. The display indicates that 1.00 J of work is done. What is the mass of the block?

69. Tennis It is not uncommon during the serve of a professional tennis player for the racket to exert an average force of 150.0 N on the ball. If the ball has a mass of 0.060 kg and is in contact with the strings of the racket, as shown in Figure 11-18, for 0.030 s, what is the kinetic energy of the ball as it leaves the racket? Assume that the ball starts from rest.

11.2 Conservation of Energy 73. A 98.0-N sack of grain is hoisted to a storage room 50.0 m above the ground floor of a grain elevator. a. How much work was done? b. What is the increase in potential energy of the sack of grain at this height? c. The rope being used to lift the sack of grain breaks just as the sack reaches the storage room. What kinetic energy does the sack have just before it strikes the ground floor?

74. A 20-kg rock is on the edge of a 100-m cliff, as

150.0 N

shown in Figure 11-20. a. What potential energy does the rock possess relative to the base of the cliff? b. The rock falls from the cliff. What is its kinetic energy just before it strikes the ground? c. What speed does the rock have as it strikes the ground? 20 kg

■

Figure 11-18

70. Pam, wearing a rocket pack, stands on frictionless ice. She has a mass of 45 kg. The rocket supplies a constant force for 22.0 m, and Pam acquires a speed of 62.0 m/s. a. What is the magnitude of the force? b. What is Pam’s final kinetic energy?

100 m

■

Figure 11-20

71. Collision A 2.00103-kg car has a speed of 12.0 m/s. The car then hits a tree. The tree doesn’t move, and the car comes to rest, as shown in Figure 11-19. a. Find the change in kinetic energy of the car. b. Find the amount of work done as the front of the car crashes into the tree. c. Find the size of the force that pushed in the front of the car by 50.0 cm.

308

75. Archery An archer puts a 0.30-kg arrow to the bowstring. An average force of 201 N is exerted to draw the string back 1.3 m. a. Assuming that all the energy goes into the arrow, with what speed does the arrow leave the bow? b. If the arrow is shot straight up, how high does it rise?

Chapter 11 Energy and Its Conservation For more problems, go to Additional Problems, Appendix B.

76. A 2.0-kg rock that is initially at rest loses 407 J of potential energy while falling to the ground. Calculate the kinetic energy that the rock gains while falling. What is the rock’s speed just before it strikes the ground?

77. A physics book of unknown mass is dropped 4.50 m. What speed does the book have just before it hits the ground?

83. A person weighing 635 N climbs up a ladder to a height of 5.0 m. Use the person and Earth as the system. a. Draw energy bar graphs of the system before the person starts to climb the ladder and after the person stops at the top. Has the mechanical energy changed? If so, by how much? b. Where did this energy come from?

78. Railroad Car A railroad car with a mass of 5.0105 kg collides with a stationary railroad car of equal mass. After the collision, the two cars lock together and move off at 4.0 m/s, as shown in Figure 11-21. a. Before the collision, the first railroad car was moving at 8.0 m/s. What was its momentum? b. What was the total momentum of the two cars after the collision? c. What were the kinetic energies of the two cars before and after the collision? d. Account for the loss of kinetic energy. m 5.0105 kg v 4.0 m/s

Mixed Review 84. Suppose a chimpanzee swings through the jungle on vines. If it swings from a tree on a 13-m-long vine that starts at an angle of 45°, what is the chimp’s velocity when it reaches the ground?

85. An 0.80-kg cart rolls down a frictionless hill of height 0.32 m. At the bottom of the hill, the cart rolls on a flat surface, which exerts a frictional force of 2.0 N on the cart. How far does the cart roll on the flat surface before it comes to a stop?

86. High Jump The world record for the men’s high jump is about 2.45 m. To reach that height, what is the minimum amount of work that a 73.0-kg jumper must exert in pushing off the ground?

87. A stuntwoman finds that she can safely break her

■

Figure 11-21

79. From what height would a compact car have to be dropped to have the same kinetic energy that it has when being driven at 1.00102 km/h?

80. Kelli weighs 420 N, and she is sitting on a playground swing that hangs 0.40 m above the ground. Her mom pulls the swing back and releases it when the seat is 1.00 m above the ground. a. How fast is Kelli moving when the swing passes through its lowest position? b. If Kelli moves through the lowest point at 2.0 m/s, how much work was done on the swing by friction?

81. Hakeem throws a 10.0-g ball straight down from a height of 2.0 m. The ball strikes the floor at a speed of 7.5 m/s. What was the initial speed of the ball?

82. Slide Lorena’s mass is 28 kg. She climbs the 4.8-m ladder of a slide and reaches a velocity of 3.2 m/s at the bottom of the slide. How much work was done by friction on Lorena? physicspp.com/chapter_test

fall from a one-story building by landing in a box filled to a 1-m depth with foam peanuts. In her next movie, the script calls for her to jump from a fivestory building. How deep a box of foam peanuts should she prepare?

88. Football A 110-kg football linebacker has a head-on collision with a 150-kg defensive end. After they collide, they come to a complete stop. Before the collision, which player had the greater momentum and which player had the greater kinetic energy?

89. A 2.0-kg lab cart and a 1.0-kg lab cart are held together by a compressed spring. The lab carts move at 2.1 m/s in one direction. The spring suddenly becomes uncompressed and pushes the two lab carts apart. The 2-kg lab cart comes to a stop, and the 1.0-kg lab cart moves ahead. How much energy did the spring add to the lab carts?

90. A 55.0-kg scientist roping through the top of a tree in the jungle sees a lion about to attack a tiny antelope. She quickly swings down from her 12.0-m-high perch and grabs the antelope (21.0 kg) as she swings. They barely swing back up to a tree limb out of reach of the lion. How high is this tree limb? Chapter 11 Assessment

309

91. An 0.80-kg cart rolls down a 30.0° hill from a vertical height of 0.50 m as shown in Figure 11-22. The distance that the cart must roll to the bottom of the hill is 0.50 m/sin 30.0° 1.0 m. The surface of the hill exerts a frictional force of 5.0 N on the cart. Does the cart roll to the bottom of the hill? m 0.80 kg F 5.0 N

0.50 m 30.0° ■

Figure 11-22

92. Object A, sliding on a frictionless surface at 3.2 m/s, hits a 2.0-kg object, B, which is motionless. The collision of A and B is completely elastic. After the collision, A and B move away from each other at equal and opposite speeds. What is the mass of object A?

93. Hockey A 90.0-kg hockey player moving at 5.0 m/s collides head-on with a 110-kg hockey player moving at 3.0 m/s in the opposite direction. After the collision, they move off together at 1.0 m/s. How much energy was lost in the collision?

Thinking Critically 94. Apply Concepts A golf ball with a mass of 0.046 kg rests on a tee. It is struck by a golf club with an effective mass of 0.220 kg and a speed of 44 m/s. Assuming that the collision is elastic, find the speed of the ball when it leaves the tee.

95. Apply Concepts A fly hitting the windshield of a moving pickup truck is an example of a collision in which the mass of one of the objects is many times larger than the other. On the other hand, the collision of two billiard balls is one in which the masses of both objects are the same. How is energy transferred in these collisions? Consider an elastic collision in which billiard ball m1 has velocity v1 and ball m2 is motionless. a. If m1 m2, what fraction of the initial energy is transferred to m2? b. If m1 m2, what fraction of the initial energy is transferred to m2? c. In a nuclear reactor, neutrons must be slowed down by causing them to collide with atoms. (A neutron is about as massive as a proton.) Would hydrogen, carbon, or iron atoms be more desirable to use for this purpose?

310

96. Analyze and Conclude In a perfectly elastic collision, both momentum and mechanical energy are conserved. Two balls, with masses mA and mB, are moving toward each other with speeds vA and vB, respectively. Solve the appropriate equations to find the speeds of the two balls after the collision.

97. Analyze and Conclude A 25-g ball is fired with an initial speed of v1 toward a 125-g ball that is hanging motionless from a 1.25-m string. The balls have a perfectly elastic collision. As a result, the 125-g ball swings out until the string makes an angle of 37.0° with the vertical. What is v1?

Writing in Physics 98. All energy comes from the Sun. In what forms has this solar energy come to us to allow us to live and to operate our society? Research the ways that the Sun’s energy is turned into a form that we can use. After we use the Sun’s energy, where does it go? Explain.

99. All forms of energy can be classified as either kinetic or potential energy. How would you describe nuclear, electric, chemical, biological, solar, and light energy, and why? For each of these types of energy, research what objects are moving and how energy is stored in those objects.

Cumulative Reveiw 100. A satellite is placed in a circular orbit with a radius of 1.0107 m and a period of 9.9103 s. Calculate the mass of Earth. Hint: Gravity is the net force on such a satellite. Scientists have actually measured the mass of Earth this way. (Chapter 7)

101. A 5.00-g bullet is fired with a velocity of 100.0 m/s toward a 10.00-kg stationary solid block resting on a frictionless surface. (Chapter 9) a. What is the change in momentum of the bullet if it is embedded in the block? b. What is the change in momentum of the bullet if it ricochets in the opposite direction with a speed of 99 m/s? c. In which case does the block end up with a greater speed?

102. An automobile jack must exert a lifting force of at least 15 kN. (Chapter 10) a. If you want to limit the effort force to 0.10 kN, what mechanical advantage is needed? b. If the jack is 75% efficient, over what distance must the effort force be exerted in order to raise the auto 33 cm?

Chapter 11 Energy and Its Conservation For more problems, go to Additional Problems, Appendix B.

Multiple Choice 1. A bicyclist increases her speed from 4.0 m/s to 6.0 m/s. The combined mass of the bicyclist and the bicycle is 55 kg. How much work did the bicyclist do in increasing her speed? 11 J 28 J

55 J 550 J

2. The illustration below shows a ball swinging freely in a plane. The mass of the ball is 4.0 kg. Ignoring friction, what is the maximum kinetic energy of the ball as it swings back and forth? 0.14 m/s 21 m/s

7.0 m/s 49 m/s

6. A ball of mass m rolls along a flat surface with a speed of v1. It strikes a padded wall and bounces back in the opposite direction. The energy of the ball after striking the wall is half its initial energy. Ignoring friction, which of the following expressions gives the ball’s new speed as a function of its initial speed? 1 v1 2 2 (v1) 2

2(v1) 2v1

7. The illustration below shows a ball on a curved track. The ball starts with zero velocity at the top of the track. It then rolls from the top of the track to the horizontal part at the ground. Ignoring friction, its velocity just at the moment it reaches the ground is 14 m/s. What is the height, h, from the ground to the top of the track? 7m 14 m

10 m 20 m

h 2.5 m

3. You lift a 4.5-kg box from the floor and place it on a shelf that is 1.5 m above the ground. How much energy did you use in lifting the box? 9.0 J 49 J

11 J 66 J

4. You drop a 6.0102-kg ball from a height of 1.0 m above a hard, flat surface. The ball strikes the surface and loses 0.14 J of its energy. It then bounces back upward. How much kinetic energy does the ball have just after it bounces off the flat surface? 0.20 J 0.59 J

0.45 J 0.73 J

5. You move a 2.5-kg book from a shelf that is 1.2 m above the ground to a shelf that is 2.6 m above the ground. What is the change in the book’s potential energy? 1.4 J 25 J

h

3.5 J 34 J physicspp.com/standardized_test

Extended Answer 8. A box sits on a platform supported by a compressed spring. The box has a mass of 1.0 kg. When the spring is released, it gives 4.9 J of energy to the box, and the box flies upward. What will be the maximum height above the platform reached by the box before it begins to fall?

Use the Process of Elimination On any multiple-choice test, there are two ways to find the correct answer to each question. Either you can choose the right answer immediately or you can eliminate the answers that you know are wrong.

Chapter 11 Standardized Test Practice

311

What You’ll Learn • You will learn how temperature relates to the potential and kinetic energies of atoms and molecules. • You will distinguish heat from work. • You will calculate heat transfer and the absorption of thermal energy.

Why It’s Important Thermal energy is vital for living creatures, chemical reactions, and the working of engines. Solar Energy A strategy used to produce electric power from sunlight concentrates the light with many mirrors onto one collector that becomes very hot. The energy collected at a high temperature is then used to drive an engine, which turns an electric generator.

Think About This What forms of energy does light from the Sun take in the process of converting solar energy into useful work through an engine?

physicspp.com 312 CORBIS

What happens when you provide thermal energy by holding a glass of water? Question What happens to the temperature of water when you hold a glass of water in your hand? Procedure

Analysis

1. You will need to use a 250-mL beaker and 150 mL of water. 2. Fill the beaker with the 150 mL of water. 3. Record the initial temperature of the water by holding a thermometer in the water in the beaker. Note that the bulb end of the thermometer must not touch the bottom or sides of the beaker, nor should it touch a table or your hands. 4. Remove the thermometer and hold the beaker of water for 2 min by cupping it with both hands, as shown in the figure. 5. Have your lab partner record the final temperature of the water by placing the thermometer in the beaker. Be sure that the bulb end of the thermometer is not touching the bottom or sides of the beaker.

Calculate the change in temperature of the water. If you had more water in the beaker, would it affect the change in temperature? Critical Thinking Explain what caused the water temperature to change.

12.1 Temperature and Thermal Energy

T

he study of heat transformations into other forms of energy, called thermodynamics, began with the eighteenth-century engineers who built the first steam engines. These steam engines were used to power trains, factories, and water pumps for coal mines, and thus they contributed greatly to the Industrial Revolution in Europe and in the United States. In learning to design more efficient engines, the engineers developed new concepts about how heat is related to useful work. Although the study of thermodynamics began in the eighteenth century, it was not until around 1900 that the concepts of thermodynamics were linked to the motions of atoms and molecules in solids, liquids, and gases. Today, the concepts of thermodynamics are widely used in various applications that involve heat and temperature. Engineers use the laws of thermodynamics to continually develop higher performance refrigerators, automobile engines, aircraft engines, and numerous other machines.

Objectives • Describe thermal energy and compare it to potential and kinetic energies. • Distinguish temperature from thermal energy. • Define specific heat and calculate heat transfer.

Vocabulary conduction thermal equilibrium heat convection radiation specific heat

Section 12.1 Temperature and Thermal Energy

313

Horizons Companies

Thermal Energy You already have studied how objects collide and trade kinetic energies. For example, the many molecules present in a gas have linear and rotational kinetic energies. The molecules also may have potential energy in their vibrations and bending. The gas molecules collide with each other and with the walls of their container, transferring energy among each other in the process. There are numerous molecules moving freely in a gas, resulting in many collisions. Therefore, it is convenient to discuss the total energy of the molecules and the average energy per molecule. The total energy of the molecules is called thermal energy, and the average energy per molecule is related to the temperature of the gas. Helium balloon ■ Figure 12-1 Helium atoms in a balloon collide with the rubber wall and cause the balloon to expand.

Hot object

KEhot KEcold

Cold object ■ Figure 12-2 Particles in a hot object have greater kinetic and potential energies than particles in a cold object do.

314

Chapter 12 Thermal Energy

Hot objects What makes an object hot? When you fill up a balloon with helium, the rubber in the balloon is stretched by the repeated pounding from helium atoms. Each of the billions of helium atoms in the balloon collides with the rubber wall, bounces back, and hits the other side of the balloon, as shown in Figure 12-1. If you put a balloon in sunlight, you might notice that the balloon gets slightly larger. The energy from the Sun makes each of the gas atoms move faster and bounce off the rubber walls of the balloon more often. Each atomic collision with the balloon wall puts a greater force on the balloon and stretches the rubber. Thus, the balloon expands. On the other hand, if you refrigerate a balloon, you will find that it shrinks slightly. Lowering the temperature slows the movement of the helium atoms. Hence, their collisions do not transfer enough momentum to stretch the balloon quite as much. Even though the balloon contains the same number of atoms, the balloon shrinks. Solids The atoms in solids also have kinetic energy, but they are unable to move freely as gas atoms do. One way to illustrate the molecular structure of a solid is to picture a number of atoms that are connected to each other by springs. Because of the springs, the atoms bounce back and forth, with some bouncing more than others. Each atom has some kinetic energy and some potential energy from the springs that are attached to it. If a solid has N number of atoms, then the total thermal energy in the solid is equal to the average kinetic and potential energy per atom times N.

Thermal Energy and Temperature According to the previous discussion of gases and solids, a hot object has more thermal energy than a similar cold object, as shown in Figure 12-2. This means that, as a whole, the particles in a hot object have greater thermal energy than the particles in a cold object. This does not mean that all the particles in an object have exactly the same amount of energy; they have a wide range of energies. However, the average energy of the particles in a hot object is higher than the average energy of the particles in a cold object. To understand this, consider the heights of students in a twelfth-grade class. Although the students’ heights vary, you can calculate the average height of the students in the class. This average is likely to be greater than the average height of students in a ninth-grade class, even though some ninth-grade students may be taller than some twelfth-grade students.

Temperature Temperature depends only on the average kinetic energy of the particles in the object. Because temperature depends on average kinetic energy, it does not depend on the number of atoms in an object. To understand this, consider two blocks of steel. The first block has a mass of 1 kg, and the second block has a mass of 2 kg. If the 1-kg block is at the same temperature as the 2-kg block, the average kinetic energy of the particles in each block is the same. However, the 2-kg block has twice the mass of the 1-kg block. Hence, the 2-kg block has twice the amount of particles as the 1kg block. Thus, the total amount of kinetic energy of the particles in the 2kg block is twice that of the 1-kg mass. Total kinetic energy is divided by the total number of particles in an object to calculate its average kinetic energy. Therefore, the thermal energy in an object is proportional to the number of particles in it. Temperature, however, is not dependent on the number of particles in an object.

Before Thermal Equilibrium Hot object (A)

Cold object (B)

KEA KEB

After Thermal Equilibrium

Equilibrium and Thermometry How do you measure your body temperature? For example, if you suspect that you have a fever, you might place a thermometer in your mouth and wait for a few minutes before checking the thermometer for your temperature reading. The microscopic process involved in measuring temperature involves collisions and energy transfers between the thermometer and your body. Your body is hot compared to the thermometer, which means that the particles in your body have greater thermal energy and are moving faster than the particles in the thermometer. When the cold glass tube of the thermometer touches your skin, which is warmer than the glass, the faster-moving particles in your skin collide with the slower-moving particles in the glass. Energy is then transferred from your skin to the glass particles by the process of conduction, which is the transfer of kinetic energy when particles collide. The thermal energy of the particles that make up the thermometer increases, while at the same time, the thermal energy of the particles in your skin decreases. Thermal equilibrium As the particles in the glass gain more energy, they begin to give some of their energy back to the particles in your body. At some point, the rate of transfer of energy between the glass and your body becomes equal, and your body and the thermometer are then at the same temperature. At this point, your body and the thermometer are said to have reached thermal equilibrium, the state in which the rate of energy flow between two objects is equal and the objects are at the same temperature, as shown in Figure 12-3. The operation of a thermometer depends on some property, such as volume, which changes with temperature. Many household thermometers contain colored alcohol that expands when heated and rises in a narrow tube. The hotter the thermometer, the more the alcohol expands and the higher it rises in the tube. In liquid-crystal thermometers, such as the one shown in Figure 12-4, a set of different kinds of liquid crystals is used. Each crystal’s molecules rearrange at a specific temperature, which causes the color of the crystal to change and indicates the temperature by color. Medical thermometers and the thermometers that monitor automobile engines use very small, temperature-sensitive electronic circuits to take rapid measurements.

KEA KEB ■ Figure 12-3 Thermal energy is transferred from a hot object to a cold object. When thermal equilibrium is reached, the transfer of energy between objects is equal.

■ Figure 12-4 Thermometers use a change in physical properties to measure temperature. A liquidcrystal thermometer changes color with a temperature change.

Section 12.1 Temperature and Thermal Energy

315 Tom Pantages

Interstellar space

Human body

Surface of the Sun Flames

108 106 104 102 1

Lowest temperature in laboratory

10

Helium liquefies

■

Figure 12-5 There is an extremely wide range of temperatures throughout the universe. Note that the scale has been expanded in areas of particular interest.

K

°C

380

373.15100

370

110

90

360

80

350

70

340

60

330

50

320

40

310

30

300

20

290 280

10

273.15

270

10

260

250

20

260

10 0

°F 100.00 210

270

0.00

0.00

200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10

420 430 440 450 460

273.15

212.00

32.00

459.67

■ Figure 12-6 The three mostcommon temperature scales are Kelvin, Celsius, and Fahrenheit.

316

Chapter 12 Thermal Energy

(cl)Getty Images, (r)FPG/Getty Images, (others)CORBIS

100

103

Superconductivity Life below this exists temperature

104

Nuclear bomb

Center of the Sun 105

106

107

108

Uncharged atoms exist below this temperature

Supernova explosions 109 1010

Nuclei exist below this temperature

Temperature (K)

Temperature Scales: Celsius and Kelvin Over the years, scientists developed temperature scales so that they could compare their measurements with those of other scientists. A scale based on the properties of water was devised in 1741 by Swedish astronomer and physicist Anders Celsius. On this scale, now called the Celsius scale, the freezing point of pure water is defined to be 0°C. The boiling point of pure water at sea level is defined to be 100°C. Temperature limits The wide range of temperatures present in the universe is shown in Figure 12-5. Temperatures do not appear to have an upper limit. The interior of the Sun is at least 1.5107°C. Temperatures do, however, have a lower limit. Generally, materials contract as they cool. If an ideal gas, such as the helium in a balloon is cooled, it contracts in such a way that it occupies a volume that is only the size of the helium atoms at 273.15°C. At this temperature, all the thermal energy that can be removed has been removed from the gas. It is impossible to reduce the temperature any further. Therefore, there can be no temperature lower than 273.15°C, which is called absolute zero. The Celsius scale is useful for day-to-day measurements of temperature. It is not conducive for working on science and engineering problems, however, because it has negative temperatures. Negative temperatures suggest a molecule could have negative kinetic energy, which is not possible because kinetic energy is always positive. The solution to this issue is to use a temperature scale based on absolute zero. The zero point of the Kelvin scale is defined to be absolute zero. On the Kelvin scale, the freezing point of water (0°C) is about 273 K and the boiling point of water is about 373 K. Each interval on this scale, called a kelvin, is equal to 1°C. Thus, TC 273 TK. Figure 12-6 shows representative temperatures on the three most-common scales: Fahrenheit, Celsius, and Kelvin. Very cold temperatures are reached by liquefying gases. Helium liquefies at 4.2 K, or 269°C. Even colder temperatures can be reached by making use of special properties of solids, helium isotopes, and atoms and lasers.

1. Convert the following Kelvin temperatures to Celsius temperatures. a. 115 K

c. 125 K

e. 425 K

b. 172 K

d. 402 K

f. 212 K

2. Find the Celsius and Kelvin temperatures for the following. a. room temperature

c. a hot summer day in North Carolina

b. a typical refrigerator

d. a winter night in Minnesota

Heat and the Flow of Thermal Energy When two objects come in contact with each other, they transfer energy. This energy that is transferred between the objects is called heat. Heat is described as the energy that always flows from the hotter object to the cooler object. Left to itself heat never flows from a colder object to a hotter object. The symbol Q is used to represent an amount of heat, which, like other forms of energy, is measured in joules. If Q has a negative value, heat has left the object; if Q has a positive value, heat has been absorbed by the object. Conduction If you place one end of a metal rod in a flame, the hot gas particles in the flame conduct heat to the rod. The other end of the rod also becomes warm within a short period of time. Heat is conducted because the particles in the rod are in direct contact with each other. Convection Thermal energy transfer can occur even if the particles in an object are not in direct contact with each other. Have you ever looked into a pot of water just about to boil? The water at the bottom of the pot is heated by conduction and rises to the top, while the colder water at the top sinks to the bottom. Heat flows between the rising hot water and the descending cold water. This motion of fluid in a liquid or gas caused by temperature differences is called convection. Atmospheric turbulence is caused by convection of gases in the atmosphere. Thunderstorms are excellent examples of large-scale atmospheric convection. Ocean currents that cause changes in weather patterns also result from convection. Radiation The third method of thermal transfer, unlike the first two, does not depend on the presence of matter. The Sun warms Earth from over 150 million km away via radiation, which is the transfer of energy by electromagnetic waves. These waves carry the energy from the hot Sun through the vacuum of space to the much cooler Earth.

Steam Heating In a steam heating system of a building, water is turned into steam in a boiler located in a maintenance area or the basement. The steam then flows through insulated pipes to each room in the building. In the radiator, the steam is condensed as liquid water and then flows back through pipes to the boiler to be revaporized. The hot steam physically carries the heat from the boiler, and that energy is released when the steam condenses in the radiator. Some disadvantages of steam heating are that it requires expensive boilers and pipes must contain steam under pressure.

Meteorology Connection

Specific Heat Some objects are easier to heat than others. On a bright summer day, the Sun warms the sand on a beach and the ocean water. However, the sand on the beach gets quite hot, while the ocean water stays relatively cool. When heat flows into an object, its thermal energy and temperature increase. The amount of the increase in temperature depends on the size of the object and on the material from which the object is made. Section 12.1 Temperature and Thermal Energy

317

The specific heat of a material is the amount of energy that must Specific Heat of Common Substances be added to the material to raise Specific Heat Specific Heat the temperature of a unit mass by Material (J/kgK) (J/kgK) one temperature unit. In SI units, 897 Lead 130 specific heat, represented by C, is measured in J/kgK. Table 12-1 376 Methanol 2450 provides values of specific heat 710 Silver 235 for some common substances. For 385 Steam 2020 example, 897 J must be added to 840 Water 4180 1 kg of aluminum to raise its 2060 Zinc 388 temperature by 1 K. The specific 450 heat of aluminum is therefore 897 J/kgK. The heat gained or lost by an object as its temperature changes depends on the mass, the change in temperature, and the specific heat of the substance. By using the following equation, you can calculate the amount of heat, Q, that must be transferred to change the temperature of an object.

Table 12-1

Material Aluminum Brass Carbon Copper Glass Ice Iron

Heat Transfer

Q mCT mC(Tf Ti)

Heat transfer is equal to the mass of an object times the specific heat of the object times the difference between the final and initial temperatures.

Liquid water has a high specific heat compared to the other substance in Table 12-1. When the temperature of 10.0 kg of water is increased by 5.0 K, the heat absorbed is Q (10.0 kg)(4180 J/kgK)(5.0 K) = 2.1105 J. Remember that the temperature interval for kelvins is the same as that for Celsius degrees. For this reason, you can calculate T in kelvins or in degrees Celsius.

Heat Transfer A 5.10-kg cast-iron skillet is heated on the stove from 295 K to 450 K. How much heat had to be transferred to the iron? 1

m 5.10 kg

Analyze and Sketch the Problem • Sketch the flow of heat into the skillet from the stove top. Known: m = 5.10 kg Ti = 295 K

2

C = 450 J/kgK Tf = 450 K

Unknown:

Ti 295 K Tf 450 K

Solve for the Unknown Q mC(Tf Ti ) (5.10 kg)(450 J/kgK)(450 K 295 K) 3.6105 J

3

Q

Q=?

Substitute m 5.10 kg, C 450 J/kgK, Tf 450 K, Ti 295 K

Evaluate the Answer • Are the units correct? Heat is measured in J. • Does the sign make sense? Temperature increased, so Q is positive.

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Chapter 12 Thermal Energy

Math Handbook Order of Operations page 843

3. When you turn on the hot water to wash dishes, the water pipes have to heat up. How much heat is absorbed by a copper water pipe with a mass of 2.3 kg when its temperature is raised from 20.0°C to 80.0°C? 4. The cooling system of a car engine contains 20.0 L of water (1 L of water has a mass of 1 kg). a. What is the change in the temperature of the water if the engine operates until 836.0 kJ of heat is added? b. Suppose that it is winter, and the car’s cooling system is filled with methanol. The density of methanol is 0.80 g/cm3. What would be the increase in temperature of the methanol if it absorbed 836.0 kJ of heat? c. Which is the better coolant, water or methanol? Explain. 5. Electric power companies sell electricity by the kWh, where 1 kWh 3.6106 J. Suppose that it costs $0.15 per kWh to run an electric water heater in your neighborhood. How much does it cost to heat 75 kg of water from 15°C to 43°C to fill a bathtub?

Calorimetry: Measuring Specific Heat A simple calorimeter, such as the one shown in Figure 12-7, is a device used to measure changes in Insulated outer thermal energy. A calorimeter is carefully insulated container so that heat transfer to the external world is kept to a minimum. A measured mass of a substance that has been heated to a high temperature is placed in the calorimeter. The calorimeter also contains a known mass of cold water at a measured temperature. The heat released by the substance is transferred to the cooler water. The change in thermal energy of the substance is calculated using the resulting increase in the water temperature. More sophisticated types of calorimeters are used to Inner measure chemical reactions and the energy content container of various foods. The operation of a calorimeter depends on the conservation of energy in an isolated, closed system. Energy can neither enter nor leave this system. As a result, if the energy of one part of the system increases, the energy of another part of the system must decrease by the same amount. Consider a system composed of two blocks of metal, block A and block B, shown in Figure 12-8a on the next page. The total energy of the system is constant, as represented by the following equation.

Stirrer

Thermometer Lid

Water

Test substance ■ Figure 12-7 A calorimeter provides an isolated, closed system in which to measure energy transfer.

Conservation of Energy EA EB constant In an isolated, closed system, the thermal energy of object A plus the thermal energy of object B is constant.

Section 12.1 Temperature and Thermal Energy

319

■ Figure 12-8 A system is composed of two model blocks at different temperatures that initially are separated (a). When the blocks are brought together, heat flows from the hotter block to the colder block (b). Total energy remains constant.

a A

EA

B

Insulation

EB

b AB

EAB EA EB

Suppose that the two blocks initially are separated but can be placed in contact with each other. If the thermal energy of block A changes by an amount EA, then the change in thermal energy of block B, EB, must be related by the equation, EA EB 0. Thus, EA EB. The change in energy of one block is positive, while the change in energy of the other block is negative. For the block whose thermal energy change is positive, the temperature of the block rises. For the block whose thermal energy change is negative, the temperature falls. Assume that the initial temperatures of the two blocks are different. When the blocks are brought together, heat flows from the hotter block to the colder block, as shown in Figure 12-8b. The heat flow continues until the blocks are in thermal equilibrium, which is when the blocks have the same temperature. In an isolated, closed system, the change in thermal energy is equal to the heat transferred because no work is done. Therefore, the change in energy for each block can be expressed by the following equation: E Q mCT The increase in thermal energy of block A is equal to the decrease in thermal energy of block B. Thus, the following relationship is true: mACATA mBCBTB 0 The change in temperature is the difference between the final and initial temperatures; that is, T Tf Ti. If the temperature of a block increases, Tf Ti, and T is positive. If the temperature of the block decreases, Tf Ti, and T is negative. The final temperatures of the two blocks are equal. The following is the equation for the transfer of energy: mACA(Tf TA) mBCB(Tf TB) 0 To solve for Tf, expand the equation. mACATf mACATA mBCBTf mBCBTB 0 Tf(mACA mBCB) mACATA mBCBTB m C T m C T

A A A B B B Tf m C m C A A

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Chapter 12 Thermal Energy

B B

Transferring Heat in a Calorimeter A calorimeter contains 0.50 kg of water at 15°C. A 0.040-kg block of zinc at 115°C is placed in the water. What is the final temperature of the system? 1

Before block of zinc is placed

Analyze and Sketch the Problem Water

• Let zinc be sample A and water be sample B. • Sketch the transfer of heat from the hotter zinc to the cooler water.

2

After block of zinc is placed

Known:

Unknown:

mA 0.040 kg CA 388 J/kg°C TA 115°C mB 0.50 kg CB 4180 J/kg°C TB 15.0°C

Tf ?

Zinc mB 0.50 kg TB 15°C

mA 0.040 kg TA 115°C Tf ?

Math Handbook Operations with Significant Digits pages 835—836

Solve for the Unknown Determine the final temperature using the following equation. m C T m C T mACA mBCB

A A A B B B Tf

(0.040 kg)(388 J/kg°C)(115°C) (0.50 kg)(4180 J/kg°C)(15.0°C) (0.040 kg)(388 J/kg°C) (0.50 kg)(4180 J/kg°C)

16°C 3

Substitute mA 0.040 kg, CA 388 J/kg°C, TA 115°C, mB 0.50 kg, CB 4180 J/kg°C, TB 15°C

Evaluate the Answer • Are the units correct? Temperature is measured in Celsius. • Is the magnitude realistic? The answer is between the initial temperatures of the two samples, as is expected when using a calorimeter.

6. A 2.00102-g sample of water at 80.0°C is mixed with 2.00102 g of water at 10.0°C. Assume that there is no heat loss to the surroundings. What is the final temperature of the mixture? 7. A 4.00102-g sample of methanol at 16.0°C is mixed with 4.00102 g of water at 85.0°C. Assume that there is no heat loss to the surroundings. What is the final temperature of the mixture? 8. Three lead fishing weights, each with a mass of 1.00102 g and at a temperature of 100.0°C, are placed in 1.00102 g of water at 35.0°C. The final temperature of the mixture is 45.0°C. What is the specific heat of the lead in the weights? 9. A 1.00102-g aluminum block at 100.0°C is placed in 1.00102 g of water at 10.0°C. The final temperature of the mixture is 25.0°C. What is the specific heat of the aluminum?

Section 12.1 Temperature and Thermal Energy

321

■ Figure 12-9 A lizard regulates its body temperature by hiding under a rock when the atmosphere is hot (a) and sunbathing when the atmosphere gets cold (b).

Biology Connection

a

b

Animals can be divided into two groups based on their body temperatures. Most are cold-blooded animals whose body temperatures depend on the environment. The others are warm-blooded animals whose body temperatures are controlled internally. That is, a warm-blooded animal’s body temperature remains stable regardless of the temperature of the environment. In contrast, when the temperature of the environment is high, the body temperature of a cold-blooded animal also becomes high. A cold-blooded animal, such as the lizard shown in Figure 12-9, regulates this heat flow by hiding under a rock or crevice, thereby reducing its body temperature. Humans are warm-blooded and have a body temperature of about 37°C. To regulate its body temperature, a warm-blooded animal increases or decreases the level of its metabolic processes. Thus, a warm-blooded animal may hibernate in winter and reduce its body temperature to approach the freezing point of water.

12.1 Section Review 10. Temperature Make the following conversions. a. 5°C to kelvins b. 34 K to degrees Celsius c. 212°C to kelvins d. 316 K to degrees Celsius 11. Conversions Convert the following Celsius temperatures to Kelvin temperatures. a. 28°C b. 154°C c. 568°C d. 55°C e. 184°C 12. Thermal Energy Could the thermal energy of a bowl of hot water equal that of a bowl of cold water? Explain your answer. 13. Heat Flow On a dinner plate, a baked potato always stays hot longer than any other food. Why? 322

Chapter 12 Thermal Energy

(l)John Cancalosi/Peter Arnold, Inc., (r)Jenny Hager/The Image Works

14. Heat The hard tile floor of a bathroom always feels cold to bare feet even though the rest of the room is warm. Is the floor colder than the rest of the room? 15. Specific Heat If you take a plastic spoon out of a cup of hot cocoa and put it in your mouth, you are not likely to burn your tongue. However, you could very easily burn your tongue if you put the hot cocoa in your mouth. Why? 16. Heat Chefs often use cooking pans made of thick aluminum. Why is thick aluminum better than thin aluminum for cooking? 17. Heat and Food It takes much longer to bake a whole potato than to cook french fries. Why? 18. Critical Thinking As water heats in a pot on a stove, the water might produce some mist above its surface right before the water begins to roll. What is happening, and where is the coolest part of the water in the pot? physicspp.com/self_check_quiz

12.2 Changes of State and the Laws of Thermodynamics

E

ighteenth-century steam-engine builders used heat to turn liquid water into steam. The steam pushed a piston to turn the engine, and then the steam was cooled and condensed into a liquid again. Adding heat to the liquid water changed not only its temperature, but also its structure. You will learn that changing state means changing form as well as changing the way in which atoms store thermal energy.

• Define heats of fusion and vaporization. • State the first and second laws of thermodynamics. • Distinguish between heat and work. • Define entropy.

Changes of State

The three most common states of matter are solids, liquids, and gases. As the temperature of a solid is raised, it usually changes to a liquid. At even higher temperatures, it becomes a gas. How can these changes be explained? Consider a material in a solid state. When the thermal energy of the solid is increased, the motion of the particles also increases, as does the temperature. Figure 12-10 diagrams the changes of state as thermal energy is added to 1.0 g of water starting at 243 K (ice) and continuing until it reaches 473 K (steam). Between points A and B, the ice is warmed to 273 K. At some point, the added thermal energy causes the particles to move rapidly enough that their motion overcomes the forces holding the particles together in a fixed location. The particles are still touching each other, but they have more freedom of movement. Eventually, the particles become free enough to slide past each other.

Vocabulary heat of fusion heat of vaporization first law of thermodynamics heat engine entropy second law of thermodynamics

Melting point At this point, the substance has changed from a solid to a liquid. The temperature at which this change occurs is the melting point of the substance. When a substance is melting, all of the added thermal energy goes to overcome the forces holding the particles together in the solid state. None of the added thermal energy increases the kinetic energy of the particles. This can be observed between points B and C in Figure 12-10, where the added thermal energy melts the ice at a constant 273 K. Because the kinetic energy of the particles does not increase, the temperature does not increase between points B and C.

■ Figure 12-10 A plot of temperature versus heat added when 1.0 g of ice is converted to steam. Note that the scale is broken between points D and E.

D

373 Temperature (K)

Boiling point Once a solid is completely melted, there are no more forces holding the particles in the solid state. Adding more thermal energy again increases the motion of the particles, and the temperature of the liquid rises. In the diagram, this process occurs between points C and D. As the temperature increases further, some particles in the liquid acquire enough energy to break free from the other particles. At a specific temperature, known as the boiling point, further addition of energy causes the substance to undergo another change of state. All the added thermal energy converts the substance from the liquid state to the gaseous state.

Objectives

E Water Steam Steam

323

273

B

C

Water

Ice Water 243 A

61.7

395.7

813.7

3073

Heat (J)

Section 12.2 Changes of State and the Laws of Thermodynamics

323

Table 12-2 Heats of Fusion and Vaporization of Common Substances Material Copper Mercury Gold Methanol Iron Silver Lead Water (ice)

Heat of Fusion Hf (J/kg)

Heat of Vaporization Hv (J/kg)

2.05105 1.15104 6.30104 1.09105 2.66105 1.04105 2.04104 3.34105

5.07106 2.72105 1.64106 8.78105 6.29106 2.36106 8.64105 2.26106

As in melting, the temperature does not rise while a liquid boils. In Figure 12-10, this transition is represented between points D and E. After the material is entirely converted to gas, any added thermal energy again increases the motion of the particles, and the temperature rises. Above point E, steam is heated to temperatures greater than 373 K.

Heat of fusion The amount of energy needed to melt 1 kg of a substance is called the heat of fusion of that substance. For example, the heat of fusion of ice is 3.34105 J/kg. If 1 kg of ice at its melting point, 273 K, absorbs 3.34105 J, the ice becomes 1 kg of water at the same temperature, 273 K. The added energy causes a change in state but not in temperature. The horizontal distance in Figure 12-10 from point B to point C represents the heat of fusion. Heat of vaporization At normal atmospheric pressure, water boils at 373 K. The thermal energy needed to vaporize 1 kg of a liquid is called the heat of vaporization. For water, the heat of vaporization is 2.26106 J/kg. The distance from point D to point E in Figure 12-10 represents the heat of vaporization. Every material has a characteristic heat of vaporization. Between points A and B, there is a definite slope to the line as the temperature is raised. This slope represents the specific heat of the ice. The slope between points C and D represents the specific heat of water, and the slope above point E represents the specific heat of steam. Note that the slope for water is less than those of both ice and steam. This is because water has a greater specific heat than does ice or steam. The heat, Q, required to melt a solid of mass m is given by the following equation. Heat Required to Melt a Solid

Melting 1. Label two foam cups A and B. 2. Measure and pour 75 mL of room-temperature water into each cup. Wipe up any spilled liquid. 3. Add an ice cube to cup A, and add ice water to cup B until the water levels are equal. 4. Measure the temperature of the water in each cup at 1-min intervals until the ice has melted. 5. Record the temperatures in a data table and plot a graph. Analyze and Conclude 6. Do the samples reach the same final temperature? Why?

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Chapter 12 Thermal Energy

Q mHf

The heat required to melt a solid is equal to the mass of the solid times the heat of fusion of the solid.

Similarly, the heat, Q, required to vaporize a mass, m, of liquid is given by the following equation. Heat Required to Vaporize a Liquid Q mHv The heat required to vaporize a liquid is equal to the mass of the liquid times the heat of vaporization of the liquid.

When a liquid freezes, an amount of heat, Q mHf , must be removed from the liquid to turn it into a solid. The negative sign indicates that the heat is transferred from the sample to the external world. In the same way, when a vapor condenses to a liquid, an amount of heat, Q mHv , must be removed from the vapor. The values of some heats of fusion, Hf , and heats of vaporization, Hv, are shown in Table 12-2.

Heat Suppose that you are camping in the mountains. You need to melt 1.50 kg of snow at 0.0°C and heat it to 70.0°C to make hot cocoa. How much heat will be needed? 1

Analyze and Sketch the Problem • Sketch the relationship between heat and water in its solid and liquid states. • Sketch the transfer of heat as the temperature of the water increases. Known: m 1.50 kg Ti 0.0°C C 4180 J/kg°C

2

Q

Unknown:

Hf 3.34105 J/kg Tf 70.0°C

Qmelt ice ? Qheat liquid ? Qtotal ?

Solve for the Unknown Calculate the heat needed to melt ice. Qmelt ice mHf (1.50 kg)(3.34105 J/kg) 5.01105 J 5.01102 kJ Calculate the temperature change. T Tf Ti 70.0°C 0.0°C 70.0°C

1.5 kg Snow

Hf

Ti 0.0°C

Tf 70.0°C

Math Handbook Operations with Scientific Notation pages 842—843

Substitute m 1.50 kg, Hf 3.34105 J/kg

Substitute Tf 70.0°C, Ti 0.0°C

Calculate the heat needed to raise the water temperature. Qheat liquid mCT (1.50 kg)(4180 J/kg°C)(70.0°C) Substitute m 1.50 kg, C = 4180 J/kg°C, T 70.0°C 4.39105 J 4.39102 kJ Calculate the total amount of heat needed. Qtotal Qmelt ice + Qheat liquid 5.01102 kJ + 4.39102 kJ Substitute Qmelt ice 5.01102 kJ, Qheat liquid 4.39102 kJ 2 9.4010 kJ 3

Evaluate the Answer • Are the units correct? Energy units are in joules. • Does the sign make sense? Q is positive when heat is absorbed. • Is the magnitude realistic? The amount of heat needed to melt the ice is greater than the amount of heat needed to increase the water temperature by 70.0°C. It takes more energy to overcome the forces holding the particles in the solid state than to raise the temperature of water.

19. How much heat is absorbed by 1.00102 g of ice at 20.0°C to become water at 0.0°C? 20. A 2.00102-g sample of water at 60.0°C is heated to steam at 140.0°C. How much heat is absorbed? 21. How much heat is needed to change 3.00102 g of ice at 30.0°C to steam at 130.0°C?

Section 12.2 Changes of State and the Laws of Thermodynamics

325

The First Law of Thermodynamics Before thermal energy was linked to the motion of atoms, the study of heat and temperature was considered to be a separate science. The first law developed for this science was a statement about what thermal energy is and where it can go. As you know, you can heat a nail by holding it over a flame or by pounding it with a hammer. That is, you can increase the nail’s thermal energy by adding heat or by doing work on it. We do not normally think that the nail does work on the hammer. However, the work done by the nail on the hammer is equal to the negative of the work done by the hammer on the nail. The first law of thermodynamics states that the change in thermal energy, U, of an object is equal to the heat, Q, that is added to the object minus the work, W, done by the object. Note that U, Q, and W are all measured in joules, the unit of energy. The First Law of Thermodynamics U Q W The change in thermal energy of an object is equal to the heat added to the object minus the work done by the object.

Thermodynamics also involves the study of the changes in thermal properties of matter. The first law of thermodynamics is merely a restatement of the law of conservation of energy, which states that energy is neither created nor destroyed, but can be changed into other forms. Another example of changing the amount of thermal energy in a system is a hand pump used to inflate a bicycle tire. As a person pumps, the air and the hand pump become warm. The mechanical energy in the moving piston is converted into thermal energy of the gas. Similarly, other forms of energy, such as light, sound, and electric energy, can be changed into thermal energy. For example, a toaster converts electric energy into heat when it toasts bread, and the Sun warms Earth with light from a distance of over 150 million km away. Hot reservoir at TH QH W

Heat engine QL

QH W QL

Cold reservoir at TL

■ Figure 12-11 A heat engine transforms heat at high temperature into mechanical energy and low-temperature waste heat.

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Chapter 12 Thermal Energy

Heat engines The warmth that you experience when you rub your hands together is a result of the conversion of mechanical energy into thermal energy. The conversion of mechanical energy into thermal energy occurs easily. However, the reverse process, the conversion of thermal energy into mechanical energy, is more difficult. A device that is able to continuously convert thermal energy to mechanical energy is called a heat engine. A heat engine requires a high-temperature source from which thermal energy can be removed; a low-temperature receptacle, called a sink, into which thermal energy can be delivered; and a way to convert the thermal energy into work. A diagram of a heat engine is shown in Figure 12-11. An automobile internal-combustion engine, such as the one shown in Figure 12-12, is one example of a heat engine. In the engine, a mixture of air and gasoline vapor is ignited and produces a high-temperature flame. Input heat, QH, flows from the flame to the air in the cylinder. The hot air expands and pushes on a piston, thereby changing thermal energy into mechanical energy. To obtain continuous mechanical energy, the engine must be returned to its starting condition. The heated air is expelled and replaced by new air, and the piston is returned to the top of the cylinder.

Spark plug Air and gasoline vapor

Exhaust

Piston ■

Intake

Compression

Spark

Power

Exhaust

Figure 12-12 The heat produced by burning gasoline causes the gases that are produced to expand and to exert force and do work on the piston.

The entire cycle is repeated many times each minute. The thermal energy from the burning of gasoline is converted into mechanical energy, which eventually results in the movement of the car. Not all of the thermal energy from the high-temperature flame in an automobile engine is converted into mechanical energy. When the automobile engine is functioning, the exhaust gases and the engine parts become hot. As the exhaust comes in contact with outside air and transfers heat to it, the temperature of the outside air is raised. In addition, heat from the engine is transferred to a radiator. Outside air passes through the radiator and the air temperature is raised. All of this energy, QL, transferred out of the automobile engine is called waste heat, that is, heat that has not been converted into work. When the engine is working continuously, the internal energy of the engine does not change, or U 0 Q W. The net heat going into the engine is Q QH QL. Thus, the work done by the engine is W QH QL. In an automobile engine, the thermal energy in the flame produces the mechanical energy and the waste heat that is expelled. All heat engines generate waste heat, and therefore no engine can ever convert all of the energy into useful motion or work. Efficiency Engineers and car salespeople often talk about the fuel efficiency of automobile engines. They are referring to the amount of the input heat, QH, that is turned into useful work, W. The actual efficiency of an engine is given by the ratio W/QH. The efficiency could equal 100 percent only if all of the input heat were turned into work by the engine. Because there is always waste heat, even the most efficient engines fall short of 100-percent efficiency. In solar collectors, heat is collected at high temperatures and used to drive engines. The Sun’s energy is transmitted as electromagnetic waves and increases the internal energy of the solar collectors. This energy is then transmitted as heat to the engine, where it is turned into useful work and waste heat. Refrigerators Heat flows spontaneously from a warm object to a cold object. However, it is possible to remove thermal energy from a colder object and add it to a warmer object if work is done. A refrigerator is a common example of a device that accomplishes this transfer with the use of mechanical work. Electric energy runs a motor that does work on a gas and compresses it. Section 12.2 Changes of State and the Laws of Thermodynamics

327

Hot reservoir at TH QH

Refrigerator W

QL

QH W QL

Cold reservoir at TL

The gas draws heat from the interior of the refrigerator, passes from the compressor through the condenser coils on the outside of the refrigerator, and cools into a liquid. Thermal energy is transferred into the air in the room. The liquid reenters the interior, vaporizes, and absorbs thermal energy from its surroundings. The gas returns to the compressor and the process is repeated. The overall change in the thermal energy of the gas is zero. Thus, according to the first law of thermodynamics, the sum of the heat removed from the refrigerator’s contents and the work done by the motor is equal to the heat expelled, as shown in Figure 12-13. Heat pumps A heat pump is a refrigerator that can be run in two directions. In the summer, the pump removes heat from a house and thus cools the house. In the winter, heat is removed from the cold outside air and transferred into the warmer house. In both cases, mechanical energy is required to transfer heat from a cold object to a warmer one.

■

Figure 12-13 A refrigerator absorbs heat, QL, from the cold reservoir and gives off heat, QH, to the hot reservoir. Work, W, is done on the refrigerator.

22. A gas balloon absorbs 75 J of heat. The balloon expands but stays at the same temperature. How much work did the balloon do in expanding? 23. A drill bores a small hole in a 0.40-kg block of aluminum and heats the aluminum by 5.0°C. How much work did the drill do in boring the hole? 24. How many times would you have to drop a 0.50-kg bag of lead shot from a height of 1.5 m to heat the shot by 1.0°C? 25. When you stir a cup of tea, you do about 0.050 J of work each time you circle the spoon in the cup. How many times would you have to stir the spoon to heat a 0.15-kg cup of tea by 2.0°C? 26. How can the first law of thermodynamics be used to explain how to reduce the temperature of an object?

The Second Law of Thermodynamics Many processes that are consistent with the first law of thermodynamics have never been observed to occur spontaneously. Three such processes are presented in Figure 12-14. For example, the first law of thermodynamics does not prohibit heat flowing from a cold object to a hot object. However, when hot objects have been placed in contact with cold objects, the hot objects have never been observed to become hotter. Similarly, the cold objects have never been observed to become colder. Entropy If heat engines completely converted thermal energy into mechanical energy with no waste heat, then the first law of thermodynamics would be obeyed. However, waste heat is always generated, and randomly distributed particles of a gas are not observed to spontaneously arrange themselves in specific ordered patterns. In the nineteenth century, French engineer Sadi Carnot studied the ability of engines to convert thermal energy into mechanical energy. He developed a logical proof that even an ideal engine would generate some waste heat. Carnot’s result is best described in terms of a quantity called entropy, which is a measure of the disorder in a system. 328

Chapter 12 Thermal Energy

Consistent with the First Law of Thermodynamics but do not occur spontaneously

Hot

Cold

Occur spontaneously

Hot

Cold Q

Q

Heat engine

■ Figure 12-14 Many processes that do not violate the first law of thermodynamics do not occur spontaneously. The spontaneous processes obey both the first and second law of thermodynamics.

QH W

Heat engine

Time

QH W QL

Time

When a baseball is dropped and falls due to gravity, it possesses potential and kinetic energies that can be recovered to do work. However, when the baseball falls through the air, it collides with many air molecules that absorb some of its energy. This causes air molecules to move in random directions and at random speeds. The energy absorbed from the baseball causes more disorder among the molecules. The greater the range of speeds exhibited by the molecules, the greater the disorder, which in turn increases the entropy. It is highly unlikely that the molecules that have been dispersed in all directions will come back together, give their energies back to the baseball, and cause it to rise. Entropy, like thermal energy, is contained in an object. If heat is added to an object, entropy is increased. If heat is removed from an object, entropy is decreased. If an object does work with no change in temperature, the entropy does not change, as long as friction is ignored. The change in entropy, S, is expressed by the following equation, in which entropy has units of J/K and the temperature is measured in kelvins. Q T The change in entropy of an object is equal to the heat added to the object divided by the temperature of the object in kelvins.

Change in Entropy S

Entropy has some interesting properties. Compare the following situations. Explain how and why these changes in entropy are different. 1. Heating 1.0 kg of water from 273 K to 274 K. 2. Heating 1.0 kg of water from 353 K to 354 K. 3. Completely melting 1.0 kg of ice at 273 K. 4. Heating 1.0 kg of lead from 273 K to 274 K.

Q

1 kg

1 kg

Ti

Tf

Section 12.2 Changes of State and the Laws of Thermodynamics

329

■

Figure 12-15 The spontaneous mixing of the food coloring and water is an example of the second law of thermodynamics.

The second law of thermodynamics states that natural processes go in a direction that maintains or increases the total entropy of the universe. That is, all things will become more and more disordered unless some action is taken to keep them ordered. The increase in entropy and the second law of thermodynamics can be thought of as statements of the probability of events happening. Figure 12-15 illustrates an increase in entropy as food-coloring molecules, originally separate from the clear water, are thoroughly mixed with the water molecules over time. Figure 12-16 shows an example of the second law of thermodynamics that might be familiar to many teenagers. The second law of thermodynamics predicts that heat flows spontaneously only from a hot object to a cold object. Consider a hot iron bar and a cold cup of water. On the average, the particles in the iron will be moving very fast, whereas the particles in the water will be moving slowly. When the bar is plunged into the water and thermal equilibrium is eventually reached, the average kinetic energy of the particles in the iron and the water will be the same. More particles now have an increased random motion than was true for the initial state. This final state is less ordered than the initial state. The fast particles are no longer confined solely to the iron, and the slower particles are no longer confined only to the water; all speeds are evenly distributed. The entropy of the final state is greater than that of the initial state. Violations of the second law We take for granted many daily events that occur spontaneously, or naturally, in one direction. We would be shocked, however, if the reverse of the same events occurred spontaneously. You are not surprised when a metal spoon, heated at one end, soon becomes uniformly hot. Consider your reaction, however, if a spoon lying on a table suddenly, on its own, became red hot at one end and icy cold at the other. If you dive into a swimming pool, you take for granted that you push the water molecules away as you enter the water. However, you would be amazed if you were swimming in the pool and all the water molecules spontaneously threw you up onto the diving board. Neither of these imagined reverse processes would violate the first law of thermodynamics. They are simply examples of the countless events that do not occur because their processes would violate the second law of thermodynamics.

■ Figure 12-16 If no work is done on a system, entropy spontaneously reaches a maximum.

330

Chapter 12 Thermal Energy

(t)Doug Martin, (others)Richard Hutchings/CORBIS

The second law of thermodynamics and the increase in entropy also give new meaning to what has been commonly called the energy crisis. The energy crisis refers to the continued use of limited resources of fossil fuels, such as natural gas and petroleum. When you use a resource, such as natural gas to heat your home, you do not use up the energy in the gas. As the gas ignites, the internal chemical energy contained in the molecules of the gas is converted into thermal energy of the flame. The thermal energy of the flame is then transferred to thermal energy in the air of your home. Even if this warm air leaks to the outside, the energy is not lost. Energy has not been used up. The entropy, however, has increased. The chemical structure of natural gas is very ordered. As you have learned, when a substance becomes warmer, the average kinetic energy of the particles in the substance increases. In contrast, the random motion of warmed air is very disordered. While it is mathematically possible for the original chemical order to be reestablished, the probability of this occurring is essentially zero. For this reason, entropy often is used as a measure of the unavailability of useful energy. The energy in the warmed air in a home is not as available to do mechanical work or to transfer heat to other objects as the original gas molecules were. The lack of usable energy is actually a surplus of entropy.

12.2 Section Review 27. Heat of Vaporization Old-fashioned heating systems sent steam into radiators in each room of a house. In the radiators, the steam condensed back to water. Analyze this process and explain how it heated a room. 28. Heat of Vaporization How much heat is needed to change 50.0 g of water at 80.0°C to steam at 110.0°C?

32. Mechanical Energy and Thermal Energy Water flows over a fall that is 125.0 m high, as shown in Figure 12-17. If the potential energy of the water is all converted to thermal energy, calculate the temperature difference between the water at the top and the bottom of the fall.

29. Heat of Vaporization The specific heat of mercury is 140 J/kg°C. Its heat of vaporization is 3.06105 J/kg. How much energy is needed to heat 1.0 kg of mercury metal from 10.0°C to its boiling point and vaporize it completely? The boiling point of mercury is 357°C.

125.0 m

30. Mechanical Energy and Thermal Energy James Joule carefully measured the difference in temperature of water at the top and bottom of a waterfall. Why did he expect a difference?

33. Entropy Evaluate why heating a home with natural gas results in an increased amount of disorder.

31. Mechanical Energy and Thermal Energy A man uses a 320-kg hammer moving at 5.0 m/s to smash a 3.0-kg block of lead against a 450-kg rock. When he measured the temperature he found that it had increased by 5.0°C. Explain how this happened.

34. Critical Thinking A new deck of cards has all the suits (clubs, diamonds, hearts, and spades) in order, and the cards are ordered by number within the suits. If you shuffle the cards many times, are you likely to return the cards to their original order? Explain. Of what physical law is this an example?

physicspp.com/self_check_quiz

■

Figure 12-17

Section 12.2 Changes of State and the Laws of Thermodynamics

331

Heating and Cooling Alternate CBL instructions can be found on the Web site. physicspp.com

When a beaker of water is set on a hot plate and the hot plate is turned on, heat is transferred. It first is transferred to the beaker and then to the water at the bottom of the beaker by conduction. The water then transfers heat from the bottom to the top by moving hot water to the top through convection. Once the heat source is removed or shut off, the water radiates thermal energy until it reaches room temperature. How quickly the water heats up is a function of the amount of heat added, the mass of the water, and the specific heat of water.

QUESTION How does the constant supply of thermal energy affect the temperature of water?

Objectives

Procedure

■ Measure, in SI, temperature and mass. ■ Make and use graphs to help describe the

1. Set the hot plate to the highest setting, or as recommended by your teacher. Allow a few minutes for the plate to heat up.

change in temperature of water as it heats up and cools down. ■ Explain any similarities and differences in these two changes.

Safety Precautions

2. Measure the mass of the empty beaker. 3. Pour 150 mL of water into the beaker and measure the combined mass of the water and the beaker. 4. Calculate and record the mass of the water in the beaker. 5. Create a data and observations table.

■ Be careful when using a hot plate. It can

burn the skin.

Materials hot plate (or Bunsen burner) 250-mL ovenproof glass beaker 50–200 g of water two thermometers (non-mercury) stopwatch (or timer)

6. Record the initial temperature of the water and the air in the classroom. Note that the bulb end of the thermometers must not touch the bottom or sides of the beaker, nor should it touch a table or your hands. 7. Place the beaker on the hot plate and record the temperature every minute for 5 min. 8. Carefully remove the beaker from the hot plate and record the temperature every minute for the next 10 min. 9. At the end of 10 min, record the temperature of the air. 10. Turn off the hot plate. 11. When finished, allow the equipment to cool and dispose of the water as instructed by your teacher.

332 Horizons Companies

Data Table Mass of water Initial air temperature Final air temperature Change in air temperature Time (min)

Temperature (°C)

Heating or Cooling

Analyze

Going Further

1. Calculate the change in air temperature to determine if air temperature may be an extraneous variable.

1. Does placing your thermometer at the top of the water in your beaker result in different readings than if it is placed at the bottom of the beaker? Explain.

2. Make a scatter-plot graph of temperature (vertical axis) versus time (horizontal axis). Use a computer or a calculator to construct the graph, if possible. 3. Calculate What was the change in water temperature as the water heated up? 4. Calculate What was the drop in water temperature when the heat source was removed? 5. Calculate the average slope for the temperature increase by dividing change in temperature by the amount of time the water was heating up. 6. Calculate the average slope for the temperature decrease by dividing change in temperature by the amount of time the heat source was removed.

Conclude and Apply

2. Hypothesize what the temperature changes might look like if you had the following amounts of water in the beaker: 50 mL, 250 mL. 3. Suppose you insulated the beaker you were using. How would the beaker’s ability to heat up and cool down be affected?

Real-World Physics 1. Suppose you were to use vegetable oil in the beaker instead of water. Hypothesize what the temperature changes might look like if you were to follow the same steps and perform the experiment. 2. If you were to take soup at room temperature and cook it in a microwave oven for 3 min, would the soup return to room temperature in 3 min? Explain your answer.

1. Summarize What was the change in water temperature when a heat source was applied? 2. Summarize What was the change in water temperature once the heat source was removed? 3. What would happen to the water temperature after the next 10 min? Would it continue cooling down forever? 4. Did the water appear to heat up or cool down quicker? Why do you think this is so? Hint: Examine the slopes you calculated.

To find out more about thermal energy, visit the Web site: physicspp.com

5. Hypothesize Where did the thermal energy in the water go once the water began to cool down? Support your hypothesis. 333

The Heat Pump Heat pumps, also called reversible air conditioners, were invented in the 1940s. They are used to heat and cool homes and hotel rooms. Heat pumps change from heaters to air conditioners by reversing the flow of refrigerant through the system.

5 The fan cools the coil

during cooling and warms the coil during heating. 1 Cooling The thin

capillary tube sprays liquid refrigerant into a larger coil inside.

Inside

Outside Receiver: tank stores refrigerant Air grating Fan motor

Airflow to rooms Fan motor Airflow from rooms

1

4

3

2

Compressor pump refrigerant

2 Cooling Valves 1

and 2 are opened and valves 3 and 4 are closed for cooling. The refrigerant flows downward. The inside coil functions as an evaporator and the outside coil functions as a condenser.

334

How It Works

Unit cabinet Air grating 4 Heating Valves 3 and 4

are opened and valves 1 and 2 are closed for heating. The refrigerant flows upward. The inside coil functions as a condenser and the outside coil functions as an evaporator.

3 Heating The thin

capillary tube sprays liquid refrigerant into a larger diameter pipe in an outer coil for heating.

Thinking Critically 1. Observe Trace the flow of refrigerant through the entire system for both heating and cooling. Start at the compressor. 2. Analyze Would a heat pump be able to heat an entire house when the outside temperature drops to extremely cold levels?

12.1 Temperature and Thermal Energy Vocabulary

Key Concepts

• • • • • •

•

conduction (p. 315) thermal equilibrium (p. 315) heat (p. 317) convection (p. 317) radiation (p. 317) specific heat (p. 318)

• • • • •

The temperature of a gas is proportional to the average kinetic energy of its particles. Thermal energy is a measure of the internal motion of an object’s particles. A thermometer reaches thermal equilibrium with the object that it comes in contact with, and then a temperature-dependent property of the thermometer indicates the temperature. The Celsius and Kelvin temperature scales are used in scientific work. The magnitude of 1 K is equal to the magnitude of 1°C. At absolute zero, no more thermal energy can be removed from a substance. Heat is energy transferred because of a difference in temperature. Q mCT mC(Tf Ti)

• •

Specific heat is the quantity of heat required to raise the temperature of 1 kg of a substance by 1 K. In a closed, isolated system, heat may flow and change the thermal energy of parts of the system, but the total energy of the system is constant. EA EB constant

12.2 Changes of State and the Laws of Thermodynamics Vocabulary

Key Concepts

• heat of fusion (p. 324) • heat of vaporization

•

The heat of fusion is the quantity of heat needed to change 1 kg of a substance from its solid to liquid state at its melting point.

(p. 324)

• first law of thermodynamics (p. 326) • heat engine (p. 326) • entropy (p. 328) • second law of thermodynamics (p. 330)

Q mHf

•

The heat of vaporization is the quantity of heat needed to change 1 kg of a substance from its liquid to gaseous state at its boiling point. Q mHv

• •

Heat transferred during a change of state does not change the temperature of a substance. The change in energy of an object is the sum of the heat added to it minus the work done by the object. U Q W

• • • •

A heat engine continuously converts thermal energy to mechanical energy. A heat pump and a refrigerator use mechanical energy to transfer heat from a region of lower temperature to one of higher temperature. Entropy is a measure of the disorder of a system. The change in entropy of an object is defined to be the heat added to the object divided by the temperature of the object. Q T

S

physicspp.com/vocabulary_puzzlemaker

335

Concept Mapping 35. Complete the following concept map using the following terms: heat, work, internal energy. First law of thermodynamics

47. Equal masses of aluminum and lead are heated to the same temperature. The pieces of metal are placed on a block of ice. Which metal melts more ice? Explain.

48. Why do easily vaporized liquids, such as acetone and methanol, feel cool to the skin?

49. Explain why fruit growers spray their trees with water when frost is expected to protect the fruit from freezing.

50. Two blocks of lead have the same temperature. entropy

temperature

external forces

Block A has twice the mass of block B. They are dropped into identical cups of water of equal temperatures. Will the two cups of water have equal temperatures after equilibrium is achieved? Explain.

51. Windows Often, architects design most of the

Mastering Concepts 36. Explain the differences among the mechanical energy of a ball, its thermal energy, and its temperature. (12.1)

37. Can temperature be assigned to a vacuum? Explain. (12.1)

38. Do all of the molecules or atoms in a liquid have the same speed? (12.1)

windows of a house on the north side. How does putting windows on the south side affect the heating and cooling of the house?

Mastering Problems 12.1 Temperature and Thermal Energy 52. How much heat is needed to raise the temperature

39. Is your body a good judge of temperature? On a cold winter day, a metal doorknob feels much colder to your hand than a wooden door does. Explain why this is true. (12.1)

40. When heat flows from a warmer object in contact with a colder object, do the two have the same temperature changes? (12.1)

41. Can you add thermal energy to an object without increasing its temperature? Explain. (12.2)

42. When wax freezes, does it absorb or release energy? (12.2)

of 50.0 g of water from 4.5°C to 83.0°C?

53. A 5.00102-g block of metal absorbs 5016 J of heat when its temperature changes from 20.0°C to 30.0°C. Calculate the specific heat of the metal.

54. Coffee Cup A 4.00102-g glass coffee cup is 20.0°C at room temperature. It is then plunged into hot dishwater at a temperature of 80.0°C, as shown in Figure 12-18. If the temperature of the cup reaches that of the dishwater, how much heat does the cup absorb? Assume that the mass of the dishwater is large enough so that its temperature does not change appreciably.

43. Explain why water in a canteen that is surrounded by dry air stays cooler if it has a canvas cover that is kept wet. (12.2)

20.0°C

80.0°C

44. Which process occurs at the coils of a running air conditioner inside a house, vaporization or condensation? Explain. (12.2) 4.00102 g

Applying Concepts

■

Figure 12-18

45. Cooking Sally is cooking pasta in a pot of boiling water. Will the pasta cook faster if the water is boiling vigorously or if it is boiling gently?

46. Which liquid would an ice cube cool faster, water or methanol? Explain.

336

Chapter 12 Thermal Energy

55. A 1.00102-g mass of tungsten at 100.0°C is placed in 2.00102 g of water at 20.0°C. The mixture reaches equilibrium at 21.6°C. Calculate the specific heat of tungsten.

For more problems, go to Additional Problems, Appendix B.

56. A 6.0102-g sample of water at 90.0°C is mixed with 4.00102 g of water at 22.0°C. Assume that there is no heat loss to the surroundings. What is the final temperature of the mixture?

57. A 10.0-kg piece of zinc at 71.0°C is placed in a container of water, as shown in Figure 12-19. The water has a mass of 20.0 kg and a temperature of 10.0°C before the zinc is added. What is the final temperature of the water and the zinc?

60. Car Engine A 2.50102-kg cast-iron car engine contains water as a coolant. Suppose that the engine’s temperature is 35.0°C when it is shut off, and the air temperature is 10.0°C. The heat given off by the engine and water in it as they cool to air temperature is 4.40106 J. What mass of water is used to cool the engine?

12.2 Changes of State and the Laws of Thermodynamics 61. Years ago, a block of ice with a mass of about 20.0 kg was used daily in a home icebox. The temperature of the ice was 0.0°C when it was delivered. As it melted, how much heat did the block of ice absorb?

62. A 40.0-g sample of chloroform is condensed from a 10.0 kg

20.0 kg

10.0°C ■

vapor at 61.6°C to a liquid at 61.6°C. It liberates 9870 J of heat. What is the heat of vaporization of chloroform?

63. A 750-kg car moving at 23 m/s brakes to a stop. The brakes contain about 15 kg of iron, which absorbs the energy. What is the increase in temperature of the brakes?

Figure 12-19

58. The kinetic energy of a compact car moving at 100 km/h is 2.9105 J. To get a feeling for the amount of energy needed to heat water, what volume of water (in liters) would 2.9105 J of energy warm from room temperature (20.0°C) to boiling (100.0°C)?

59. Water Heater A 3.0102-W electric immersion heater is used to heat a cup of water, as shown in Figure 12-20. The cup is made of glass, and its mass is 3.00102 g. It contains 250 g of water at 15°C. How much time is needed to bring the water to the boiling point? Assume that the temperature of the cup is the same as the temperature of the water at all times and that no heat is lost to the air.

64. How much heat is added to 10.0 g of ice at 20.0°C to convert it to steam at 120.0°C?

65. A 4.2-g lead bullet moving at 275 m/s strikes a steel plate and comes to a stop. If all its kinetic energy is converted to thermal energy and none leaves the bullet, what is its temperature change?

66. Soft Drink A soft drink from Australia is labeled “Low-Joule Cola.” The label says “100 mL yields 1.7 kJ.” The can contains 375 mL of cola. Chandra drinks the cola and then wants to offset this input of food energy by climbing stairs. How high would Chandra have to climb if she has a mass of 65.0 kg?

Mixed Review 67. What is the efficiency of an engine that produces

3.00102 W

2200 J/s while burning enough gasoline to produce 5300 J/s? How much waste heat does the engine produce per second?

68. Stamping Press A metal stamping machine in a

15°C

250 g

69. A 1500-kg automobile comes to a stop from 2

3.0010 g ■

factory does 2100 J of work each time it stamps out a piece of metal. Each stamped piece is then dipped in a 32.0-kg vat of water for cooling. By how many degrees does the vat heat up each time a piece of stamped metal is dipped into it?

Figure 12-20 physicspp.com/chapter_test

25 m/s. All of the energy of the automobile is deposited in the brakes. Assuming that the brakes are about 45 kg of aluminum, what would be the change in temperature of the brakes? Chapter 12 Assessment

337

70. Iced Tea To make iced tea, you start by brewing the tea with hot water. Then you add ice. If you start with 1.0 L of 90°C tea, what is the minimum amount of ice needed to cool it to 0°C? Would it be better to let the tea cool to room temperature before adding the ice?

71. A block of copper at 100.0°C comes in contact with a block of aluminum at 20.0°C, as shown in Figure 12-21. The final temperature of the blocks is 60.0°C. What are the relative masses of the blocks?

100.0°C

20.0°C

Copper

Aluminum

76. Analyze and Conclude Chemists use calorimeters to measure the heat produced by chemical reactions. For instance, a chemist dissolves 1.01022 molecules of a powdered substance into a calorimeter containing 0.50 kg of water. The molecules break up and release their binding energy to the water. The water temperature increases by 2.3°C. What is the binding energy per molecule for this substance?

77. Apply Concepts All of the energy on Earth comes from the Sun. The surface temperature of the Sun is approximately 104 K. What would be the effect on our world if the Sun’s surface temperature were 103 K?

Writing in Physics 78. Our understanding of the relationship between heat

■

60.0°C

60.0°C

Copper

Aluminum

Figure 12-21

72. A 0.35-kg block of copper sliding on the floor hits an identical block moving at the same speed from the opposite direction. The two blocks come to a stop together after the collision. Their temperatures increase by 0.20°C as a result of the collision. What was their velocity before the collision?

73. A 2.2-kg block of ice slides across a rough floor. Its initial velocity is 2.5 m/s and its final velocity is 0.50 m/s. How much of the ice block melted as a result of the work done by friction?

Thinking Critically 74. Analyze and Conclude A certain heat engine removes 50.0 J of thermal energy from a hot reservoir at temperature TH 545 K and expels 40.0 J of heat to a colder reservoir at temperature TL 325 K. In the process, it also transfers entropy from one reservoir to the other. a. How does the operation of the engine change the total entropy of the reservoirs? b. What would be the total entropy change in the reservoirs if TL 205 K?

75. Analyze and Conclude During a game, the metabolism of basketball players often increases by as much as 30.0 W. How much perspiration must a player vaporize per hour to dissipate this extra thermal energy?

338

Chapter 12 Thermal Energy

and energy was influenced by a soldier named Benjamin Thompson, Count Rumford; and a brewer named James Prescott Joule. Both relied on experimental results to develop their ideas. Investigate what experiments they did and evaluate whether or not it is fair that the unit of energy is called the Joule and not the Thompson.

79. Water has an unusually large specific heat and large heats of fusion and vaporization. Our weather and ecosystems depend upon water in all three states. How would our world be different if water’s thermodynamic properties were like other materials, such as methanol?

Cumulative Review 80. A rope is wound around a drum with a radius of 0.250 m and a moment of inertia of 2.25 kg m2. The rope is connected to a 4.00-kg block. (Chapter 8) a. Find the linear acceleration of the block. b. Find the angular acceleration of the drum. c. Find the tension, FT , in the rope. d. Find the angular velocity of the drum after the block has fallen 5.00 m.

81. A weight lifter raises a 180-kg barbell to a height of 1.95 m. How much work is done by the weight lifter in lifting the barbell? (Chapter 10)

82. In a Greek myth, the man Sisyphus is condemned by the gods to forever roll an enormous rock up a hill. Each time he reaches the top, the rock rolls back down to the bottom. If the rock has a mass of 215 kg, the hill is 33 m in height, and Sisyphus can produce an average power of 0.2 kW, how many times in 1 h can he roll the rock up the hill? (Chapter 11)

For more problems, go to Additional Problems, Appendix B.

Multiple Choice 1. Which of the following temperature conversions is incorrect? 273°C 0 K

298 K 571°C

273°C 546 K

88 K 185°C

2. What are the units of entropy? J/K

J

K/J

kJ

3. Which of the following statements about thermal equilibrium is false? When two objects are at equilibrium, heat radiation between the objects continues to occur. Thermal equilibrium is used to create energy in a heat engine. The principle of thermal equilibrium is used for calorimetry calculations. When two objects are not at equilibrium, heat will flow from the hotter object to the cooler object. 4. How much heat is required to heat 87 g of methanol ice at 14 K to vapor at 340 K? (melting point 97.6°C, boiling point 64.6°C) 17 kJ

1.4102 kJ

69 kJ

1.5102 kJ

5. Which statement is true about energy, entropy, and changes of state? Freezing ice increases in energy as it gains molecular order as a solid. The higher the specific heat capacity of a substance, the higher its melting point will be. States of matter with increased kinetic energy have higher entropy. Energy and entropy cannot increase at the same time. 6. How much heat is needed to warm 363 mL of water in a baby bottle from 24°C to 38°C? 21 kJ

121 kJ

36 kJ

820 kJ physicspp.com/standardized_test

7. Why is there always some waste heat in a heat engine? Heat cannot flow from a cold object to a hot object. Friction slows the engine down. The entropy increases at each stage. The heat pump uses energy. 8. How much heat is absorbed from the surroundings when 81 g of 0.0°C ice in a beaker melts and warms to 10°C? 0.34 kJ 27 kJ

30 kJ 190 kJ

Ice

m 81 g Ti 0.0°C

9. You do 0.050 J of work on the coffee in your cup each time you stir it. What would be the increase in entropy in 125 mL of coffee at 65°C when you stir it 85 times? 0.013 J/K 0.050 J

0.095 J/K 4.2 J

Extended Answer 10. What is the difference in heat required to melt 454 g of ice at 0.00°C, and to turn 454 g of water at 100.0°C into steam? Is the amount of this difference greater or less than the amount of energy required to heat the 454 g of water from 0.00°C to 100.0°C?

Your Mistakes Can Teach You The mistakes you make before the test are helpful because they show you areas in which you need more work. When calculating the heat needed to melt and warm a substance, remember to calculate the heat needed for melting as well as the heat needed for raising the temperature of the substance.

Chapter 12 Standardized Test Practice

339

What You’ll Learn • You will explain the expansion and contraction of matter caused by changes in temperature. • You will apply Pascal’s, Archimedes’, and Bernoulli’s principles in everyday situations.

Why It’s Important Fluids and the forces that they exert enable us to swim and dive, balloons to float, and planes to fly. Thermal expansion affects the designs of buildings, roads, bridges, and machines. Submarines A nuclear submarine is designed to maneuver at all levels in the ocean. It must withstand great differences in pressure and temperature as it moves deeper under water.

Think About This How is the submarine able to float at the surface of the ocean and to dive far beneath it?

physicspp.com 340 Steve Kaufman/Peter Arnold, Inc.

Does it float or sink? Question How can you measure the buoyancy of objects? Procedure

Analysis

1. Obtain a small vial (with a cap or a seal) and a 500-mL graduated cylinder. Attach a rubber band to the vial in order to suspend it from a spring scale. 2. Use the spring scale to find the weight of the vial. Then, use the graduated cylinder to find the volume of water displaced by the sealed vial when it is floating. Record both of these figures. Immediately wipe up any spilled liquid. 3. Place one nickel into the vial and close the lid. Repeat the procedures in step 2, recording the weight of the vial and the nickel, as well as the volume of water displaced. Also, record whether the vial floats or sinks. 4. Repeat steps 2 and 3, each time adding one nickel until the vial no longer floats. When the vial sinks, use the spring scale to find the apparent weight of the vial. Be sure that the vial is not touching the graduated cylinder when it is suspended under water.

Use the information you recorded to calculate the density of the vial-and-nickel system in each of your trials. Also, calculate the mass of the water displaced by the system in each trial. How does density appear to be related to floating? Critical Thinking How does the mass of the vialnickel system appear to be related to the mass of the water displaced by the system? Does this relationship hold regardless of whether the system is floating?

13.1 Properties of Fluids

W

ater and air are probably two of the most common substances in the everyday lives of people. We feel their effects when we drink, when we bathe, and literally with every breath we take. In your everyday experience, it might not seem that water and air have a great deal in common. If you think further about them, however, you will recognize that they have common properties. Both water and air flow, and unlike solids, neither one of them has a definite shape. Gases and liquids are two states of matter in which atoms and molecules have great freedom to move. In this chapter, you will explore states of matter. Beginning with gases and liquids, you will learn about the principles that explain how matter responds to changes in temperature and pressure, how hydraulic systems can multiply forces, and how huge metallic ships can float on water. You also will investigate the properties of solids, discovering how they expand and contract, why some solids are elastic, and why some solids seem to straddle the line between solid and liquid.

Objectives • Describe how fluids create pressure. • Calculate the pressure, volume, and number of moles of a gas. • Compare gases and plasma.

Vocabulary fluids pressure pascal combined gas law ideal gas law thermal expansion plasma

Section 13.1 Properties of Fluids

341

Horizons Companies

Pressure Suppose that you put an ice cube in an empty glass. The ice cube has a certain mass and shape, and neither of these quantities depends on the size or shape of the glass. What happens, however, when the ice melts? Its mass remains the same, but its shape changes. The water flows to take the shape of its container and forms a definite, flat, upper surface, as in Figure 13-1. If you boiled the water, it would change into a gas in the form of water vapor, and it also would flow and expand to fill the room. However, the water vapor would not have any definite surface. Both liquids and gases are fluids, which are materials that flow and have no definite shape of their own. For now, you can assume that you are dealing with ideal fluids, whose particles take up no space and have no intermolecular attractive forces.

■ Figure 13-1 The ice cubes, which are solids, have definite shapes. However, the liquid water, a fluid, takes the shape of its container. What fluid is filling the space above the water?

■ Figure 13-2 The astronaut and the landing module both exert pressure on the lunar surface. If the lunar module had a mass of approximately 7300 kg and rested on four pads that were each 91 cm in diameter, what pressure did it exert on the Moon’s surface? How could you estimate the pressure exerted by the astronaut?

Pressure in fluids You have applied the law of conservation of energy to solid objects. Can this law also be applied to fluids? Work and energy can be defined if we introduce the concept of pressure, which is the force on a surface, divided by the area of the surface. Since pressure is force exerted over a surface, anything that exerts pressure is capable of producing change and doing work. F A Pressure equals force divided by surface area.

Pressure P

Pressure (P) is a scalar quantity. In the SI system, the unit of pressure is the pascal (Pa), which is 1 N/m2. Because the pascal is a small unit, the kilopascal (kPa), equal to 1000 Pa, is more commonly used. The force, F, on a surface is assumed to be perpendicular to the surface area, A. Figure 13-2 illustrates the relationships between force, area, and pressure. Table 13-1 shows how pressures vary in different situations. Solids, liquids, and pressure Imagine that you are standing on the surface of a frozen lake. The forces that your feet exert on the ice are spread over the area of your shoes, resulting in pressure on the ice. Ice is a solid that is made up of vibrating water molecules, and the forces that hold the water molecules in place cause the ice to exert upward forces on your feet that equal your weight. If the ice melted, most of the bonds between the water molecules would be weakened. Although the molecules would continue to vibrate and remain close to each other, they also would slide past one another, and you would break through the surface. The moving water molecules would continue to exert forces on your body. Gas particles and pressure The pressure exerted by a gas can be understood by applying the kinetic-molecular theory of gases. The kineticmolecular theory explains the properties of an ideal gas. In reality, the particles of a gas take up space and have intermolecular attractive forces, but an ideal gas is an accurate model of a real gas under most conditions. According to the kinetic-molecular theory, the particles in a gas are in random motion at high speeds and undergoing elastic collisions with each other. When a gas particle hits a container’s surface, it rebounds, which changes its momentum. The impulses exerted by many of these collisions result in gas pressure on the surface.

342

Chapter 13 States of Matter

(t)Gerard Photography, (b)NASA’s Goddard Space Flight Center

Atmospheric pressure On every square cenTable 13-1 timeter of Earth’s surface at sea level, the Some Typical Pressures atmospheric gas exerts a force of approximately Location Pressure (Pa) 10 N, about the weight of a 1-kg object. The The center of the Sun pressure of Earth’s atmosphere on your body is 31016 so well balanced by your body’s outward forces The center of Earth 41011 that you seldom notice it. You probably The deepest ocean trench 1.1108 become aware of this pressure only when your Standard atmosphere 1.01325105 ears pop as the result of pressure changes, as Blood pressure 1.6104 when you ride an elevator in a tall building Air pressure on top of Mt. Everest 3104 or fly in an airplane. Atmospheric pressure The best vacuum 11013 is about 10 N per 1 cm2 (104 m2), which is about 1.0105 N/m2, or 100 kPa. Other planets in our solar system also have atmospheres. The pressure exerted by these atmospheres, however, varies widely. For example, the pressure at the surface of Venus is about 92 times the pressure at the surface of Earth, while the pressure at the surface of Mars is less than 1 percent of Earth’s.

Calculating Pressure A child weighs 364 N and sits on a three-legged stool, which weighs 41 N. The bottoms of the stool’s legs touch the ground over a total area of 19.3 cm2. a. What is the average pressure that the child and stool exert on the ground? b. How does the pressure change when the child leans over so that only two legs of the stool touch the floor? 1

Analyze and Sketch the Problem • Sketch the child and the stool, labeling the total force that they exert on the ground. • List the variables, including the force that the child and stool exert on the ground and the areas for parts a and b. Known:

Unknown:

Fg child 364 N Fg stool 41 N Fg total Fg child Fg stool 364 N 41 N 405 N 2

AA 19.3 cm2 2 AB 19.3 cm2 3 12.9 cm2

PA ? PB ? Fg 405 N Math Handbook

Solve for the Unknown

Dimensional Calculations pages 846—847

Find each pressure. P F/A 405 N (100 cm)2 a. PA 2 2

( 19.3 cm )(

(1 m)

)

Substitute F Fg total 405 N, A AA 19.3 cm2

)

Substitute F Fg total 405 N, A AB 12.9 cm2

2.10102 kPa 405 N (100 cm)2 b. PB 2 2

( 12.9 cm )(

(1 m)

3.14102 kPa 3

Evaluate the Answer • Are the units correct? The units for pressure should be Pa, and 1 N/m2 1 Pa.

Section 13.1 Properties of Fluids

343

1. The atmospheric pressure at sea level is about 1.0105 Pa. What is the force at sea level that air exerts on the top of a desk that is 152 cm long and 76 cm wide? 2. A car tire makes contact with the ground on a rectangular area of 12 cm by 18 cm. If the car’s mass is 925 kg, what pressure does the car exert on the ground as it rests on all four tires? 3. A lead brick, 5.0 cm 10.0 cm 20.0 cm, rests on the ground on its smallest face. Lead has a density of 11.8 g/cm3. What pressure does the brick exert on the ground? 4. In a tornado, the pressure can be 15 percent below normal atmospheric pressure. Suppose that a tornado occurred outside a door that is 195 cm high and 91 cm wide. What net force would be exerted on the door by a sudden 15 percent drop in normal atmospheric pressure? In what direction would the force be exerted? 5. In industrial buildings, large pieces of equipment must be placed on wide steel plates that spread the weight of the equipment over larger areas. If an engineer plans to install a 454-kg device on a floor that is rated to withstand additional pressure of 5.0104 Pa, how large should the steel support plate be?

The Gas Laws

■ Figure 13-3 The gas in the tank on the diver’s back is at high pressure. This pressure is reduced by the regulator so that the pressure of the gas the diver breathes is equal to the water pressure. In the photo, you can see bubbles coming from the regulator.

As scientists first studied gases and pressure, they began to notice some interesting relationships. The first relationship to emerge was named Boyle’s law, after seventeenth-century chemist and physicist Robert Boyle. Boyle’s law states that for a fixed sample of gas at constant temperature, the volume of the gas varies inversely with the pressure. Because the product of inversely related variables is a constant, Boyle’s law can be written PV constant, or P1V1 P2V2. The subscripts that you see in the gas laws will help you keep track of different variables, such as pressure and volume, as they change throughout a problem. These variables can be rearranged to solve for an unknown pressure or volume. As shown in Figure 13-3, the relationship between the pressure and the volume of a gas is critical to the sport of scuba diving. A second relationship was discovered about 100 years after Boyle’s work 1 by Jacques Charles. When Charles cooled a gas, the volume shrank by 273 of its original volume for every degree cooled, which is a linear relationship. At the time, Charles could not cool gases to the extremely low temperatures achieved in modern laboratories. In order to see what lower limits might be possible, he extended, or extrapolated, the graph of his data to these temperatures. This extrapolation suggested that if the temperature were reduced to 273°C, a gas would have zero volume. The temperature at which a gas would have zero volume is now called absolute zero, which is represented by the zero of the Kelvin temperature scale. These experiments indicated that under constant pressure, the volume of a sample of gas varies directly with its Kelvin temperature, a result that is now called Charles’s law. Charles’s law can be written V/T constant, or V1/T1 V2/T2.

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Combining Boyle’s law and Charles’s law relates the pressure, temperature, and volume of a fixed amount of ideal gas, which leads to the equation called the combined gas law.

Pressure Combined Gas Law

P V1 P V 1 2 2 = constant T1 T2

For a fixed amount of an ideal gas, the pressure times the volume, divided by the Kelvin temperature equals a constant.

As shown in Figure 13-4, the combined gas law reduces to Boyle’s law under conditions of constant temperature and to Charles’s law under conditions of constant pressure. The ideal gas law You can use the kinetic-molecular theory to discover how the constant in the combined gas law depends on the number of particles, N. Suppose that the volume and temperature of an ideal gas are held constant. If the number of particles increases, the number of collisions that the particles make with the container will increase, thereby increasing the pressure. Removing particles decreases the number of collisions, and thus, decreases the pressure. You can conclude that the constant in the combined gas law equation is proportional to N. PV kN T

How much pressure do you exert when standing on one foot? Have a partner trace your foot, and then use the outline to estimate its area. 1. Determine your weight in newtons and the area of the outline in m2. 2. Calculate the pressure. 3. Compare and contrast the pressure you exert on the ground with the pressure exerted by various objects. For example, you could weigh a brick and determine the pressure it exerts when resting on different faces. Analyze and Conclude 4. How do shoes with high heels affect the pressure that a person exerts on the ground?

The constant, k, is called Boltzmann’s constant, and its value is 1.381023 Pam3/K. Of course, N, the number of particles, is a very large number. Instead of using N, scientists often use a unit called a mole. One mole (abbreviated mol and represented in equations by n) is similar to one dozen, except that instead of representing 12 items, one mole repre■ Figure 13-4 You can use the sents 6.0221023 particles. This number is called Avogadro’s number, after combined gas law to derive both Italian scientist Amedeo Avogadro. Boyle’s and Charles’s laws. What Avogadro’s number is numerically equal to the number of particles in happens if you hold volume a sample of matter whose mass equals the molar mass of the substance. constant? You can use this relationship to convert between mass and n, the number of moles presP V1 P V ent. Using moles instead of the number of par1 2 2 T1 T2 ticles changes Boltzmann’s constant. This new constant is abbreviated R, and it has the value Combined gas law 8.31 Pam3/molK. Rearranging, you can write the ideal gas law in its most familiar form. Ideal Gas Law PV nRT

If temperature is constant

If pressure is constant

For an ideal gas, the pressure times the volume is equal to the number of moles multiplied by the constant R and the Kelvin temperature.

Note that with the given value of R, volume must be expressed in m3, temperature in K, and pressure in Pa. In practice, the ideal gas law predicts the behavior of gases remarkably well, except under conditions of high pressures or low temperatures.

P1V1 P2V2

V1 V 2 T1 T2

Boyle’s law

Charles’s law

Section 13.1 Properties of Fluids

345

Gas Laws A 20.0-L sample of argon gas at 273 K is at atmospheric pressure, 101.3 kPa. The temperature is lowered to 120 K, and the pressure is increased to 145 kPa. a. What is the new volume of the argon sample? b. Find the number of moles of argon atoms in the argon sample. c. Find the mass of the argon sample. The molar mass, M, of argon is 39.9 g/mol. 1

Analyze and Sketch the Problem • Sketch the situation. Indicate the conditions in the container of argon before and after the change in temperature and pressure. • List the known and unknown variables.

2

Known:

Unknown:

V1 20.0 L P1 101.3 kPa T1 273 K P2 145 kPa T2 120 K R 8.31 Pam3/molK Margon 39.9 g/mol

V2 ? moles of argon ? mass of argon sample ?

T1 273 K P1 101.3 kPa V1 20.0 L

T2 120 K P2 145 kPa V2 ?

Solve for the Unknown a. Use the combined gas law and solve for V2. PV PV 11 22 T1 T2

Math Handbook Isolating a Variable page 845

PVT P2T1

1 1 2 V2

(101.3 kPa)(20.0 L)(120 K) (145 kPa)(273 K)

6.1 L

Substitute P1 101.3 kPa, P2 145 kPa, V1 20.0 L, T1 273 K, T2 120 K

b. Use the ideal gas law and solve for n. PV nRT PV RT (101.3103 Pa)(0.0200 m3) (8.31 Pam3/molK)(273 K)

n

Substitute P 101.3103 Pa, V 0.0200 m3, R 8.31 m3/molK, T 273 K

0.893 mol c. Use the molar mass to convert from moles of argon in the sample to mass of the sample. m Mn Substitute M 39.9 g/mol, n 0.893 mol margon sample (39.9 g/mol)(0.893 mol) 35.6 g 3

Evaluate the Answer • Are the units correct? The volume, V2, is in liters, and the mass of the sample is in grams. • Is the magnitude realistic? The change in volume is consistent with an increase in pressure and decrease in temperature. The calculated mass of the argon sample is reasonable.

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6. A tank of helium gas used to inflate toy balloons is at a pressure of 15.5106 Pa and a temperature of 293 K. The tank’s volume is 0.020 m3. How large a balloon would it fill at 1.00 atmosphere and 323 K? 7. What is the mass of the helium gas in the previous problem? The molar mass of helium gas is 4.00 g/mol. 8. A tank containing 200.0 L of hydrogen gas at 0.0°C is kept at 156 kPa. The temperature is raised to 95°C, and the volume is decreased to 175 L. What is the new pressure of the gas? 9. The average molar mass of the components of air (mainly diatomic oxygen gas and diatomic nitrogen gas) is about 29 g/mol. What is the volume of 1.0 kg of air at atmospheric pressure and 20.0°C?

Thermal Expansion As you applied the combined gas law, you discovered how gases expand as their temperatures increase. When heated, all forms of matter—solids, liquids, and gases—generally become less dense and expand to fill more space. This property, known as thermal expansion, has many useful applications, such as circulating air in a room. When the air near the floor of a room is warmed, gravity pulls the denser, colder air near the ceiling down, which pushes the warmer air upward. This circulation of air within a room is called a convection current. Figure 13-5 shows convection currents starting as the hot air above the flames rises. You also can see convection currents in a pot of hot, but not boiling, water on a stove. When the pot is heated from the bottom, the colder and denser water sinks to the bottom where it is warmed and then pushed up by the continuous flow of cooler water from the top. This thermal expansion occurs in most liquids. A good model for all liquids does not exist, but it is useful to think of a liquid as a finely ground solid. Groups of two, three, or more particles move together as if they were tiny pieces of a solid. When a liquid is heated, particle motion causes these groups to expand in the same way that particles in a solid are pushed apart. The spaces between groups increase. As a result, the whole liquid expands. With an equal change in temperature, liquids expand considerably more than solids, but not as much as gases.

■

Figure 13-5 This image was made by a special technique that enables you to see different densities in the air. Convection currents are set up as warmer, less dense air rises and cooler, denser air sinks.

Why ice floats Because matter expands as it is heated, you might predict that ice would be more dense than water, and therefore, it should sink. However, when water is heated from 0°C to 4°C, instead of expanding, it contracts as the forces between particles increase and the ice crystals collapse. These forces between water molecules are strong, and the crystals that make up ice have a very open structure. Even when ice melts, tiny crystals remain. These remaining crystals are melting, and the volume of the water decreases until the temperature reaches 4°C. However, once the temperature of water moves above 4°C, its volume increases because of greater molecular motion. The practical result is that water is most dense at 4°C and ice floats. This fact is very important to our lives and environment. If ice sank, lakes would freeze from the bottom each winter and many would never melt completely in the summer. Section 13.1 Properties of Fluids

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■ Figure 13-6 The colorful lighting effects in neon signs are caused by luminous plasmas formed in the glass tubing.

Plasma If you heat a solid, it melts to form a liquid. Further heating results in a gas. What happens if you increase the temperature still further? Collisions between the particles become violent enough to tear the electrons off the atoms, thereby producing positively charged ions. The gaslike state of negatively charged electrons and positively charged ions is called plasma. Plasma is considered to be another fluid state of matter. The plasma state may seem to be uncommon; however, most of the matter in the universe is plasma. Stars consist mostly of plasma at extremely high temperatures. Much of the matter between stars and galaxies consists of energetic hydrogen that has no electrons. This hydrogen is in the plasma state. The primary difference between gas and plasma is that plasma can conduct electricity, whereas gas cannot. Lightning bolts are in the plasma state. Neon signs, such as the one shown in Figure 13-6, fluorescent bulbs, and sodium vapor lamps all contain glowing plasma.

13.1 Section Review 10. Pressure and Force Suppose that you have two boxes. One is 20 cm 20 cm 20 cm. The other is 20 cm 20 cm 40 cm. a. How does the pressure of the air on the outside of the two boxes compare? b. How does the magnitude of the total force of the air on the two boxes compare?

13. Density and Temperature Starting at 0°C, how will the density of water change if it is heated to 4°C? To 8°C?

11. Meteorology A weather balloon used by meteorologists is made of a flexible bag that allows the gas inside to freely expand. If a weather balloon containing 25.0 m3 of helium gas is released from sea level, what is the volume of gas when the balloon reaches a height of 2100 m, where the pressure is 0.82105 Pa? Assume that the temperature is unchanged.

15. The Air in a Refrigerator How many moles of air are in a refrigerator with a volume of 0.635 m3 at a temperature of 2.00°C? If the average molar mass of air is 29 g/mol, what is the mass of the air in the refrigerator?

12. Gas Compression In a certain internal-combustion engine, 0.0021 m3 of air at atmospheric pressure and 303 K is rapidly compressed to a pressure of 20.1105 Pa and a volume of 0.0003 m3. What is the final temperature of the compressed gas? 348

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14. The Standard Molar Volume What is the volume of 1 mol of a gas at atmospheric pressure and a temperature of 273 K?

16. Critical Thinking Compared to the particles that make up carbon dioxide gas, the particles that make up helium gas are very small. What can you conclude about the number of particles in a 2.0-L sample of carbon dioxide gas compared to the number of particles in a 2.0-L sample of helium gas if both samples are at the same temperature and pressure? physicspp.com/self_check_quiz

13.2 Forces Within Liquids

T

he liquids considered thus far have been ideal liquids, in which the particles are totally free to slide past one another. The unexpected behavior of water between 0°C and 4°C, however, illustrates that in real fluids, particles exert electromagnetic forces of attraction, called cohesive forces, on each other. These and other forces affect the behavior of fluids.

Cohesive Forces

• Explain how cohesive forces cause surface tension. • Explain how adhesive forces cause capillary action. • Discuss evaporative cooling and the role of condensation in cloud formation.

Have you ever noticed that dewdrops on spiderwebs and falling drops of oil are nearly spherical? What happens when rain falls on a freshly washed and waxed car? The water drops bead up into rounded shapes, as shown in the spiderweb in Figure 13-7. All of these phenomena are examples of surface tension, which is the tendency of the surface of a liquid to contract to the smallest possible area. Surface tension is a result of the cohesive forces among the particles of a liquid. Notice that beneath the surface of the liquid shown in Figure 13-8a on the next page, each particle of the liquid is attracted equally in all directions by neighboring particles, and even to the particles of the wall of the container. As a result, no net force acts on any of the particles beneath the surface. At the surface, however, the particles are attracted downward and to the sides, but not upward. There is a net downward force, which acts on the top layers and causes the surface layer to be slightly compressed. The surface layer acts like a tightly stretched rubber sheet or a film that is strong enough to support the weight of very light objects, such as the water strider in Figure 13-8b on the next page. The surface tension of water also can support a steel paper clip, even though the density of steel is nine times greater than that of water. Try it! Why does surface tension produce spherical drops? The force pulling the surface particles into a liquid causes the surface to become as small as possible, and the shape that has the least surface for a given volume is a sphere. The higher the surface tension of the liquid, the more resistant the liquid is to having its surface broken. For example, liquid mercury has much stronger cohesive forces than water does. Thus, liquid mercury forms spherical drops, even when it is placed on a smooth surface. On the other hand, liquids such as alcohol and ether have weaker cohesive forces. A drop of either of these liquids flattens out on a smooth surface. Viscosity In nonideal fluids, the cohesive forces and collisions between fluid molecules cause internal friction that slows the fluid flow and dissipates mechanical energy. The measure of this internal friction is called the viscosity of the liquid. Water is not very viscous, but motor oil is very viscous. As a result of its viscosity, motor oil flows slowly over the parts of an engine to coat the metal and reduce rubbing. Lava, molten rock that flows from a volcano or vent in Earth’s surface, is one of the most viscous fluids. There are several types of lava, and the viscosity of each type varies with composition and temperature.

Objectives

Vocabulary cohesive forces adhesive forces

■

Figure 13-7 Rainwater beads up on a spider’s web because water drops have surface tension.

Geology Connection

Section 13.2 Forces Within Liquids

349 Frank Cezus

■ Figure 13-8 Molecules in the interior of a liquid are attracted in all directions (a). A water strider can walk on water because molecules at the surface have a net inward attraction that results in surface tension (b).

a

b

Adhesive Forces

Plants Cohesive forces in liquids actually allow them to be stretched just like rubber bands. This stretching is difficult to achieve in the laboratory, but it is common in plants. The strength of the cohesive forces in water keeps the water from breaking and bubbling as it goes through the plant tissue to the leaves. If not for these forces, trees could not grow higher than about 10 m.

Similar to cohesive forces, adhesive forces are electromagnetic attractive forces that act between particles of different substances. If a glass tube with a small inner diameter is placed in water, the water rises inside the tube. The water rises because the adhesive forces between glass and water molecules are stronger than the cohesive forces between water molecules. This phenomenon is called capillary action. The water continues to rise until the weight of the water that is lifted balances the total adhesive force between the glass and water molecules. If the radius of the tube increases, the volume and the weight of the water will increase proportionally faster than the surface area of the tube. Thus, water is lifted higher in a narrow tube than in a wider one. Capillary action causes molten wax to rise in a candle’s wick and water to move up through the soil and into the roots of plants. When a glass tube is placed in a beaker of water, the surface of the water climbs the outside of the tube, as shown in Figure 13-9a. The adhesive forces between the glass molecules and water molecules are greater than the cohesive forces between the water molecules. In contrast, the cohesive forces between mercury molecules are greater than the adhesive forces between the mercury and glass molecules, so the liquid does not climb the tube. These forces also cause the center of the mercury’s surface to depress, as shown in Figure 13-9b.

Evaporation and Condensation Why does a puddle of water disappear on a hot, dry day? As you learned in Chapter 12, the particles in a liquid are moving at random speeds. If a fast-moving particle can break through the surface layer, it will escape from the liquid. Because there is a net downward cohesive force at the surface, however, only the most energetic particles escape. This escape of particles is called evaporation. a

■ Figure 13-9 Water climbs the outside wall of this glass tube (a), while the mercury is depressed by the rod (b). The forces of attraction between mercury atoms are stronger than any adhesive forces between the mercury and the glass.

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(t)Runk/Schoenberger/Grant Heilman Photography, (others)Matt Meadows

b

Evaporative cooling Evaporation has a cooling effect. On a hot day, your body perspires, and the evaporation of your sweat cools you down. In a puddle of water, evaporation causes the remaining liquid to cool down. Each time a particle with higher-than-average kinetic energy escapes from the water, the average kinetic energy of the remaining particles decreases. As you learned in Chapter 12, a decrease in average kinetic energy is a decrease in temperature. You can test this cooling effect by pouring a small amount of rubbing alcohol in the palm of your hand. Alcohol molecules evaporate easily because they have weak cohesive forces. As the molecules evaporate, the cooling effect is quite noticeable. A liquid that evaporates quickly is called a volatile liquid. Have you ever wondered why humid days feel warmer than dry days at the same temperature? On a day that is humid, the water vapor content of the air is high. Because there are already many water molecules in the air, the water molecules in perspiration are less likely to evaporate from the skin. Evaporation is the body’s primary cooling mechanism, so the body is not able to cool itself as effectively on a humid day. Particles of liquid that have evaporated into the air can also return to the liquid phase if the kinetic energy or temperature decreases, a process called condensation. What happens if you bring a cold glass into a hot, humid area? The outside of the glass soon becomes coated with condensed water. Water molecules moving randomly in the air surrounding the glass strike the cold surface, and if they lose enough energy, the cohesive forces become strong enough to prevent their escape. The air above any body of water, as shown in Figure 13-10, contains evaporated water vapor, which is water in the form of gas. If the temperature is reduced, the water vapor condenses around tiny dust particles in the air and produces droplets only 0.01 mm in diameter. A cloud of these droplets is called fog. Fog often forms when moist air is chilled by the cold ground. Fog also can form in your home. When a carbonated drink is opened, the sudden decrease in pressure causes the temperature of the gas in the container to drop, which condenses the water vapor dissolved in that gas.

■ Figure 13-10 Warm, moist, surface air rises until it reaches a height where the temperature is at the point at which water vapor condenses and forms clouds.

13.2 Section Review 17. Evaporation and Cooling In the past, when a baby had a high fever, the doctor might have suggested gently sponging off the baby with rubbing alcohol. Why would this help? 18. Surface Tension A paper clip, which has a density greater than that of water, can be made to stay on the surface of water. What procedures must you follow for this to happen? Explain. 19. Language and Physics The English language includes the terms adhesive tape and working as a cohesive group. In these terms, are adhesive and cohesive being used in the same context as their meanings in physics? physicspp.com/self_check_quiz

20. Adhesion and Cohesion In terms of adhesion and cohesion, explain why alcohol clings to the surface of a glass rod but mercury does not. 21. Floating How can you tell that the paper clip in problem 18 was not floating? 22. Critical Thinking On a hot, humid day, Beth sat on the patio with a glass of cold water. The outside of the glass was coated with water. Her younger sister, Jo, suggested that the water had leaked through the glass from the inside to the outside. Suggest an experiment that Beth could do to show Jo where the water came from. Section 13.2 Forces Within Liquids

351 Orville Andrews

13.3 Fluids at Rest and in Motion

Objectives • Relate Pascal’s principle to simple machines and occurrences. • Apply Archimedes’ principle to buoyancy. • Apply Bernoulli’s principle to airflow.

Vocabulary Pascal’s principle buoyant force Archimedes’ principle Bernoulli’s principle streamlines

■ Figure 13-11 The pressure exerted by the force of the small piston is transmitted throughout the fluid and results in a multiplied force on the larger piston.

F2 F1

Y

ou have learned how fluids exert pressure, the force per unit area. You also know that the pressure exerted by fluids changes; for example, atmospheric pressure drops as you climb a mountain. In this section, you will learn about the forces exerted by resting and moving fluids.

Fluids at Rest If you have ever dived deep into a swimming pool or lake, you know that your body, especially your ears, is sensitive to changes in pressure. You may have noticed that the pressure you felt on your ears did not depend on whether your head was upright or tilted, but that if you swam deeper, the pressure increased. Pascal’s principle Blaise Pascal, a French physician, noted that the pressure in a fluid depends upon the depth of the fluid and has nothing to do with the shape of the fluid’s container. He also discovered that any change in pressure applied at any point on a confined fluid is transmitted undiminished throughout the fluid, a fact that is now known as Pascal’s principle. Every time you squeeze a tube of toothpaste, you demonstrate Pascal’s principle. The pressure that your fingers exert at the bottom of the tube is transmitted through the toothpaste and forces the paste out at the top. Likewise, if you squeeze one end of a helium balloon, the other end of the balloon expands. When fluids are used in machines to multiply forces, Pascal’s principle is being applied. In a common hydraulic system, a fluid is confined to two connecting chambers, as shown in Figure 13-11. Each chamber has a piston that is free to move, and the pistons have different surface areas. If a force, F1, is exerted on the first piston with a surface area of A1, the pressure, P1, exerted on the fluid can be determined by using the following equation. F A1

P1 1 This equation is simply the definition of pressure: pressure equals the force per unit area. The pressure exerted by the fluid on the second piston, with a surface area A2, can also be determined. F A2

2 P2

A1

A2

According to Pascal’s principle, pressure is transmitted without change throughout a fluid, so pressure P2 is equal in value to P1. You can determine the force exerted by the second piston by using F1/A1 F2/A2 and solving for F2. This force is shown by the following equation. Force Exerted by a Hydraulic Lift

Piston 1

352

Piston 2

Chapter 13 States of Matter

FA A1

F2 1 2

The force exerted by the second piston is equal to the force exerted by the first piston multiplied by the ratio of the area of the second piston to the area of the first piston.

23. Dentists’ chairs are examples of hydraulic-lift systems. If a chair weighs 1600 N and rests on a piston with a cross-sectional area of 1440 cm2, what force must be applied to the smaller piston, with a cross-sectional area of 72 cm2, to lift the chair? 24. A mechanic exerts a force of 55 N on a 0.015 m2 hydraulic piston to lift a small automobile. The piston that the automobile sits on has an area of 2.4 m2. What is the weight of the automobile? 25. By multiplying a force, a hydraulic system serves the same purpose as a lever or seesaw. If a 400-N child standing on one piston is balanced by a 1100-N adult standing on another piston, what is the ratio of the areas of their pistons? 26. In a machine shop, a hydraulic lift is used to raise heavy equipment for repairs. The system has a small piston with a cross-sectional area of 7.0102 m2 and a large piston with a cross-sectional area of 2.1101 m2. An engine weighing 2.7103 N rests on the large piston. a. What force must be applied to the small piston to lift the engine? b. If the engine rises 0.20 m, how far does the smaller piston move?

Swimming Under Pressure When you are swimming, you feel the pressure of the water increase as you dive deeper. This pressure is actually a result of gravity; it is related to the weight of the water above you. The deeper you go, the more water there is above you, and the greater the pressure. The pressure of the water is equal to the weight, Fg, of the column of water above you divided by the column’s cross-sectional area, A. Even though gravity pulls only in the downward direction, the fluid transmits the pressure in all directions: up, down, and to the sides. You can find the pressure of the water by applying the following equation. Fg

P A

■

The weight of the column of water is Fg mg, and the mass is equal to the density, , of the water times its volume, m V. You also know that the volume of the water is the area of the base of the column times its height, V Ah. Therefore, Fg Ahg. Substituting Ahg for Fg in the equation for water pressure gives P Fg /A Ahg/A. Divide A from the numerator and denominator to arrive at the simplest form of the equation for the pressure exerted by a column of water on a submerged body.

Figure 13-12 In 1960, the Trieste, a crewed submersible, descended to the bottom of the Marianas Trench, a depth of over 10,500 m. The crewed submersible Alvin, shown below, can safely dive to a depth of 4500 m.

Pressure of Water on a Body P hg The pressure that a column of water exerts on a body is equal to the density of water times the height of the column times the acceleration due to gravity.

This formula works for all fluids, not just water. The pressure of a fluid on a body depends on the density of the fluid, its depth, and g. If there were water on the Moon, the pressure of the water at any depth would be one-sixth as great as on Earth. As illustrated in Figure 13-12, submersibles, both crewed and robotic, have explored the deepest ocean trenches and encountered pressures in excess of 1000 times standard air pressure. Section 13.3 Fluids at Rest and in Motion

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h hl Ftop

Fbottom ■ Figure 13-13 A fluid exerts a greater upward force on the bottom of an immersed object than the downward force on the top of the object. The net upward force is called the buoyant force.

Buoyancy What produces the upward force that allows you to swim? The increase in pressure with increasing depth creates an upward force called the buoyant force. By comparing the buoyant force on an object with its weight, you can predict whether the object will sink or float. Suppose that a box is immersed in water. It has a height of l and its top and bottom each have a surface area of A. Its volume, then, is V lA. Water pressure exerts forces on all sides, as shown in Figure 13-13. Will the box sink or float? As you know, the pressure on the box depends on its depth, h. To find out whether the box will float in water, you will need to determine how the pressure on the top of the box compares with the pressure from below the box. Compare these two equations: Ftop Ptop A hgA Fbottom Pbottom A (l h)gA

On the four vertical sides, the forces are equal in all directions, so there is no net horizontal force. The upward force on the bottom is larger than the downward force on the top, so there is a net upward force. The buoyant force can now be determined. Fbuoyant Fbottom Ftop (l h)gA hgA lgA Vg These calculations show the net upward force to be proportional to the volume of the box. This volume equals the volume of the fluid displaced, or pushed out of the way, by the box. Therefore, the magnitude of the buoyant force, Vg, equals the weight of the fluid displaced by the object. Buoyant Force

Fbuoyant fluidVg

The buoyant force on an object is equal to the weight of the fluid displaced by the object, which is equal to the density of the fluid in which the object is immersed multiplied by the object’s volume and the acceleration due to gravity.

This relationship was discovered in the third century B.C. by Greek scientist Archimedes. Archimedes’ principle states that an object immersed in a fluid has an upward force on it that is equal to the weight of the fluid displaced by the object. The force does not depend on the weight of the object, only on the weight of the displaced fluid. Sink or float? If you want to know whether an object sinks or floats, you have to take into account all of the forces acting on the object. The buoyant force pushes up, but the weight of the object pulls it down. The difference between the buoyant force and the object’s weight determines whether an object sinks or floats. Suppose that you submerge three objects in a tank filled with water (water 1.00103 kg/m3). Each of the objects has a volume of 100 cm3, or 1.00104 m3. The first object is a steel block with a mass of 0.90 kg. 354

Chapter 13 States of Matter

The second is an aluminum soda can with a mass of 0.10 kg. The third is an ice cube with a mass of 0.090 kg. How will each item move when it is immersed in water? The upward force on all three objects, as shown in Figure 13-14, is the same, because all displace the same weight of water. This buoyant force can be calculated as follows.

a Fbuoyant

Fbuoyant waterVg

Fnet

(1.00103 kg/m3)(1.00104 m3)(9.80 m/s2) 0.980 N

Fg

The weight of the block of steel is 8.8 N, much greater than the buoyant force. There is a net downward force, so the block will sink to the bottom of the tank. The net downward force, its apparent weight, is less than its real weight. All objects in a liquid, even those that sink, have an apparent weight that is less than when the object is in air. The apparent weight can be expressed by the equation Fapparent Fg Fbuoyant. For the block of steel, the apparent weight is 8.8 N 0.98 N, or 7.8 N. The weight of the soda can is 0.98 N, the same as the weight of the water displaced. There is, therefore, no net force, and the can will remain wherever it is placed in the water. It has neutral buoyancy. Objects with neutral buoyancy are described as being weightless; their apparent weight is zero. This property is similar to that experienced by astronauts in orbit, which is why astronaut training sometimes takes place in swimming pools. The weight of the ice cube is 0.88 N, less than the buoyant force, so there is a net upward force, and the ice cube will rise. At the surface, the net upward force will lift part of the ice cube out of the water. As a result, less water will be displaced, and the upward force will be reduced. The ice cube will float with enough volume in the water so that the weight of water displaced equals the weight of the ice cube. An object will float if its density is less than the density of the fluid in which it is immersed. Ships Archimedes’ principle explains why ships can be made of steel and still float; if the hull is hollow and large enough so that the average density of the ship is less than the density of water, the ship will float. You may have noticed that a ship loaded with cargo rides lower in the water than a ship with an empty cargo hold. You can demonstrate this effect by fashioning a small boat out of folded aluminum foil. The boat should float easily, and it will ride lower in the water if you add a cargo of paper clips. If the foil is crumpled into a tight ball, the boat will sink because of its increased density. Similarly, the continents of Earth float upon a denser material below the surface. The drifting motion of these continental plates is responsible for the present shapes and locations of the continents. Other examples of Archimedes’ principle in action include submarines and fishes. Submarines take advantage of Archimedes’ principle as water is pumped into or out of a number of different chambers to change the submarine’s average density, causing it to rise or sink. Fishes that have swim bladders also use Archimedes’ principle to control their depths. Such a fish can expand or contract its swim bladder, just like you can puff up your cheeks. To move upward in the water, the fish expands its swim bladder to displace more water and increase the buoyant force. The fish moves downward by contracting the volume of its swim bladder.

b

Fbuoyant Fnet 0

Fg

c Fbuoy