Six Ideas That Shaped Physics: Unit CConservation Laws Constrain, 4th Edition
Thomas A. Moore
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Some Physical Constants
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Prefix Symbol 1018 exa E 1015 peta P 1012 tera T 109 giga G 106 mega M 103 kilo k 10−2 centi c 10−3 milli m 10−6 micro µ 10−9 nano n 10−12 pico p 10−15 femto f
Standard Metric Prefixes (for powers of 10) Power
Power Prefix Symbol
10−18 atto a
Commonly Used Physical Data
Gravitational field strength (near the earth’s surface)
Mass
Radius
N/kg = 9.80 m/s2
× 1024 kg
km (equatorial) Mass
Radius
× 1030 kg
km
× 1022 kg
km Distance
9.80
5.98
of the earth Me
6380
of the
M⊙ 1.99
of the earth Re
sun
of the sun R⊙ 696,000
the
7.36
Mass of
moon
1740
to the moon 3.84
to the sun 1.50
1011
Density of water† 1000 kg/m3 = 1 g/cm3 Density of air† 1.2 kg/m3 Absolute zero 0 K = −273.15°C = −459.67°F Freezing point of water‡ 273.15 K = 0°C = 32°F Boiling point of water‡ 373.15 K = 100°C = 212°F Normal atmospheric pressure 101.3 kPa
At normal atmospheric pressure and 20°C.
At normal atmospheric pressure.
Radius of the moon
× 108 m Distance
×
m
†
‡
Useful Conversion Factors
1 meter = 1 m = 100 cm = 39.4 in = 3.28 ft
1 mile = 1 mi = 1609 m = 1.609 km = 5280 ft
1 inch = 1 in = 2.54 cm
1 light-year = 1 ly = 9.46 Pm = 0.946 × 1016 m
1 minute = 1 min = 60 s
1 hour = 1 h = 60 min = 3600 s
1 day = 1 d = 24 h = 86.4 ks = 86,400 s
1 year = 1 y = 365.25 d = 31.6 Ms = 3.16 × 107 s
1 newton = 1 N = 1 kg·m/s2 = 0.225 lb
1 joule = 1 J = 1 N·m = 1 kg·m2/s2 = 0.239 cal
1 watt = 1 W = 1 J/s
1 pascal = 1 Pa = 1 N/m2 = 1.45 × 10−4 psi
1 kelvin (temperature difference) = 1 K = 1°C = 1.8°F
1 radian = 1 rad = 57.3° = 0.1592 rev
1 revolution = 1 rev = 2πrad = 360°
1 cycle = 2πrad
1 hertz = 1 Hz = 1 cycle/s
1 m/s = 2.24 mi/h = 3.28 ft/s
1 mi/h = 1.61 km/h = 0.447 m/s = 1.47 ft/s
1 liter = 1 l = (10 cm)3 = 10 3 m3 = 0.0353 ft3
1 ft3 = 1728 in3 = 0.0283 m3
1 gallon = 1 gal = 0.00379 m3 = 3.79 l ≈ 3.8 kg H2O
Weight of 1-kg object near the earth = 9.8 N = 2.2 lb
1 pound = 1 lb = 4.45 N
1 calorie = energy needed to raise the temperature of 1 g of H2O by
1 K = 4.186 J
1 horsepower = 1 hp = 746 W
1 pound per square inch = 6895 Pa
1 food calorie = 1 Cal = 1 kcal = 1000 cal = 4186 J
1 electron volt = 1 eV = 1.602 × 10 19 J
Six Ideas That Shaped Physics
Unit C: Conservation Laws Constrain Interactions
Fourth Edition
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Thomas A. Moore
SIX IDEAS THAT SHAPED PHYSICS UNIT C
Published by McGraw Hill LLC, 1325 Avenue of the Americas, New York, NY 10019. Copyright © 2023 by McGraw Hill LLC. All rights reserved. Printed in the United States of America. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw Hill LLC, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning.
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1 2 3 4 5 6 7 8 9 LWI 27 26 25 24 23 22
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About the Author viii
Preface ix
Introduction for Students xvi
Chapter C1 2 The
Chapter C2 18
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C1.1 C1.2 C1.3 C1.4 C1.5 C1.6 Contents: Unit C Conservation Laws Constrain Interactions
Art of Model Building 2
Overview 2 The Nature of Science 4 The Development and Structure of Physics 5 A Model-Building Example 7 Trick Bag: Unit Awareness 10 Trick Bag: Unit Conversions 11 Trick Bag: Dimensional Analysis 12 TWO-MINUTE PROBLEMS 14 HOMEWORK PROBLEMS 14 ANSWERS TO EXERCISES 17
Chapter
C2.1 C2.2 C2.3 C2.4 C2.5 C2.6 C2.7 C3.1 C3.2 C3.3 C3.4 C3.5 C3.6 C3.7 Particles and Interactions 18 Chapter Overview 18 The Principles of Modern Mechanics 20 Describing an Object’s Motion 21 Vector Operations 23 Momentum and Impulse 24 Force and Weight 26 Interaction Categories 28 Momentum Transfer 30 TWO-MINUTE PROBLEMS 32 HOMEWORK PROBLEMS 33 ANSWERS TO EXERCISES 37 Chapter C3 38 Vectors 38 Chapter Overview 38 Introduction 40 Reference Frames 40 Displacement Vectors 41 Arbitrary Vectors 44 Seven Rules to Remember 45 Vectors in Two Dimensions 47 Vectors in One Dimension 48 TWO-MINUTE PROBLEMS 49 HOMEWORK PROBLEMS 49 ANSWERS TO EXERCISES 51
C4.1 C4.2 C4.3 C4.4 C4.5 C4.6 C5.1 C5.2 C5.3 C5.4 C5.5 Page vi
C4 52 Systems and Frames 52 Chapter Overview 52 Systems of Particles 54 A System’s Center of Mass 54 How the Center of Mass Moves 57 Inertial Reference Frames 59 Freely Floating Frames 60 Interactions with the Earth 62 TWO-MINUTE PROBLEMS 64 HOMEWORK PROBLEMS 65 ANSWERS TO EXERCISES 67
C5 68 Conservation of Momentum 68 Chapter Overview 68 Degrees of Isolation 70 How to Solve Physics Problems 71 Conservation of Momentum Problems 75 Examples 76 Airplanes and Rockets 79 TWO-MINUTE PROBLEMS 81 HOMEWORK PROBLEMS 82 ANSWERS TO EXERCISES 83
Chapter
Chapter
Chapter C6 84
C6.1 C6.2 C6.3 C6.4 C6.5 C6.6 C6.7 C7.1 C7.2 C7.3 C7.4 C7.5 C7.6 C7.7 Conservation of Angular Momentum 84 Chapter Overview 84 Introduction 86 Quantifying Orientation 86 Angular Velocity 87 The Angular Momentum of a Rigid Object 88 Twirl and Torque 91 Gyroscopic Precession 92 Conservation of Angular Momentum 93 TWO-MINUTE PROBLEMS 96 HOMEWORK PROBLEMS 97 ANSWERS TO EXERCISES 99 Chapter C7 100 More About Angular Momentum 100 Chapter Overview 100 First Steps 102 The Cross Product 103 The Angular Momentum of a Moving Particle 105 Rotating Objects 106 Rotating and Moving Objects 107 Torque and Force 108 Why Angular Momentum Is Conserved 110 TWO-MINUTE PROBLEMS 112 HOMEWORK PROBLEMS 113 ANSWERS TO EXERCISES 117
C8.1 C8.2 C8.3 C8.4 C8.5 C8.6 C8.7 C9.1 C9.2 C9.3 C9.4 C9.5 C9.6 C9.7 Chapter C8 118 Conservation of Energy 118 Chapter Overview 118 Introduction to Energy 120 Kinetic Energy 121 Potential Energy 122 Fundamental Potential Energy Formulas 124 Internal Energy and Power 126 Isolation 126 Solving Conservation-of-Energy Problems 127 TWO-MINUTE PROBLEMS 130 HOMEWORK PROBLEMS 131 ANSWERS TO EXERCISES 134 Chapter C9 136 Potential Energy Graphs 136 Chapter Overview 136 Interactions Between Macroscopic Objects 138 Interactions Between Two Atoms 138 One-Dimensional Potential Energy Diagrams 139 Relaxing the Mass Limitation 144 The Spring Approximation 145 The Potential Energy “of an Object” 146 An Example 146 TWO-MINUTE PROBLEMS 148 HOMEWORK PROBLEMS 149 ANSWERS TO EXERCISES 152
C10.1 C10.2 C10.3 C10.4 C10.5 C11.1 C11.2 C11.3 C11.4 C11.5 Chapter C10 154 Work 154 Chapter Overview 154 The Momentum Requirement 156 The Dot Product 157 The Definition of Work 158 Recognizing When Internal Energy is Involved 160 Contact Forces Perpendicular to Motion 162 TWO-MINUTE PROBLEMS 165 HOMEWORK PROBLEMS 166 ANSWERS TO EXERCISES 169 Chapter C11 170 Rotational Energy 170 Chapter Overview 170 Introduction to Rotational Energy 172 Rotational Energy of an Object at Rest 172 Calculating Moments of Inertia 174 When an Object Both Moves and Rotates 176 Rolling Without Slipping 177 TWO-MINUTE PROBLEMS 182 HOMEWORK PROBLEMS 183 ANSWERS TO EXERCISES 185 Chapter C12 186 Thermal Energy 186
C12.1 C12.2 C12.3 C12.4 C12.5 C12.6 C12.7 Page vii C13.1 C13.2 C13.3 C13.4 C13.5 C13.6 C13.7 Chapter Overview 186 The Case of the Disappearing Energy 188 Caloric Is Energy 188 Thermal Energy as Microscopic Energy 190 Friction and Thermal Energy 192 Heat, Work, and Energy Transfer 193 Specific “Heat” 194 Problems Involving Thermal Energies 196 TWO-MINUTE PROBLEMS 199 HOMEWORK PROBLEMS 200 ANSWERS TO EXERCISES 202 Chapter C13 204 Other Forms of Internal Energy 204 Chapter Overview 204 Bonds 206 Latent “Heat” 208 Chemical Energy 211 Nuclear Energy 212 Modes of Energy Transfer 213 Mechanisms of Heat Transfer 214 A Comprehensive Energy Master Equation 216 TWO-MINUTE PROBLEMS 218 HOMEWORK PROBLEMS 219 ANSWERS TO EXERCISES 221 Chapter C14 222
C14.1 C14.2 C14.3 C14.4 C14.5 C14.6 CA.2 CA.3 CA.4 CA.5 CA.6 Collisions 222 Chapter Overview 222 Types of Collisions 224 One-Dimensional Collisions 224 Two-Dimensional Collisions 227 The Slingshot Effect 230 Using All Three Conservation Laws 232 Asteroid Impacts 234 TWO-MINUTE PROBLEMS 236 HOMEWORK PROBLEMS 237 ANSWERS TO EXERCISES 239 Appendix CA 240 The Standard Model 240 CA.1 Introduction 240 Matter Particles 240 Fundamental Interactions 240 The Importance of Color-Neutrality 242 Stability and the Weak Interaction 243 Conclusion 245 TWO-MINUTE PROBLEMS 246 HOMEWORK PROBLEMS 246 ANSWERS TO EXERCISES 246 Index 247 Short Answers to Selected Problems 255 Periodic table 256
About the Author
Thomas A. Moore graduated from Carleton College (magna cum laude with Distinction in Physics) in 1976. He won a Danforth Fellowship that year that supported his graduate education at Yale University, where he earned a Ph.D. in 1981. He taught at Carleton College and Luther College before taking his current position at Pomona College in 1987, where he won a Wig Award for Distinguished Teaching in 1991. He served as an active member of the steering committee for the national Introductory University Physics Project (IUPP) from 1987 through 1995. This textbook grew out of a model curriculum that he developed for that project in 1989, which was one of only four selected for further development and testing by IUPP.
He has published a number of articles about astrophysical sources of gravitational waves, detection of gravitational waves, and new
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(Credit: Courtesy of Thomas Moore)
approaches to teaching physics, as well as a book on general relativity entitled AGeneralRelativityWorkbook(University Science Books, 2013). He has also served as a reviewer and as an associate editor for AmericanJournalofPhysics. He currently lives in Claremont, California, with his wife Joyce, a retired pastor. When he is not teaching, doing research, or writing, he enjoys reading, hiking, calling contradances, and playing Irish traditional music.
Preface
Introduction
This volume is one of six that together comprise the text materials for SixIdeasThatShapedPhysics, a unique approach to the two- or three-semester calculus-based introductory physics course. I have designed this curriculum (for which these volumes only serve as the text component) to support an introductory course that combines three elements:
Inclusion of 20th-century physics topics, A thoroughly 21st-century perspective on even classical topics, and
Support for a student-centered and active-learning-based classroom.
This course is based on the premises that innovative metaphors for teaching basic concepts, explicitly instructing students in the processes of constructing physical models, and active learning can help students learn the subject much more effectively. Physics education research has guided the presentation of all topics. Moreover, because such research has consistently underlined the importance of active learning, I have sought to provide tools for professors (both in the text and online) to make creating a coherent and self-consistent course structure based on a student-centered classroom as easy and practical as possible. All of the materials have been tested, evaluated, and rewritten multiple times: the result is the culmination of more than 30 years of continual testing and revision.
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Rather than oversimplifying the material, I have sought to make physics more accessible by providing students with the tools and guidance to become more sophisticated in their thinking. This book helps students to step beyond rote thinking patterns to develop flexible and powerful, conceptual reasoning and model-building skills. My experience and that of other users is that normal students in a wide range of institutional settings can (with appropriate support and practice) successfully learn these skills.
Each of six volumes in the text portion of this course is focused on a single core concept that has been crucial in making physics what it is today. The six volumes and their corresponding ideas are as follows:
Unit C: Conservation laws constrain interactions
Unit N: The laws of physics are universal (Newtonian mechanics)
Unit R: The laws of physics are frame-independent (Relativity)
Unit E: Electricity and magnetism are unified
Unit Q: Matter behaves like waves (Quantum physics)
Unit T: Some processes are irreversible (Thermal physics)
I have listed the units in the order that I recommendthat they be taught, but I have also constructed units R, E, Q, and T to be sufficiently independent so that they can be taught in any order after units C and N. (This is why the units are lettered as opposed to numbered.) There are sixunits (as opposed to five or seven) to make it possible to easily divide the course into two semesters, three quarters, or three semesters. This unit organization therefore not only makes it possible to dole out the text in small, easily-handled pieces and provide a great deal of flexibility in fitting the course to a given schedule, but also carries its own important pedagogical message: physicsisorganizedhierarchically, structured around only a handful of core ideas and metaphors.
An important feature of all of the volumes is that each chapter represents a logical unit that one might hope to
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handle in a single 50-minute class session, providing guidance about pacing based on decades of experience. This organization also gives instructors increased flexibility in designing a well-paced course in any particular institutional setting, since a number of chapters have been designed so that they can be omitted without loss of continuity. The preface to each unit, the chapter headers, and the instructor’s manual all provide guidance about chapter dependencies.
Finally, let me emphasize again that the text materials are just one part of the comprehensive SixIdeascurriculum. On the SixIdeas website, at www.physics.pomona.edu/sixideas/ you will find a wealth of supporting resources. The most important of these is a detailed instructor’s manual that provides guidance (based on SixIdeasusers’ experiences over more than 30 years) about how to construct an effective course. This manual exposes the important issues and raises the questions that an instructor should consider in creating an effective SixIdeascourse at a particular institution. The site also provides software that allows instructors to post selected problem solutions online where only their students can access them and assign each solution a time window for viewing. Web-based computer applets on the site provide experiences that support student learning in important ways. The site also provides a (steadily increasing) number of other resources that instructors and students may find valuable.
There is a preface for students appearing just before the first chapter of each unit that explains some important features of the text and assumptions behind the course. I recommend that everyoneread it.
Comments About the Fourth Edition
Our main goals in this edition have been to simplify certain difficult sections in units T, Q, E, and C, make it easier for instructors to drop certain chapters, reduce the pace in unit E, add some new problems (and cull a few less-effective ones), and add more end-of-chapter problems to Connect for each unit.
In addition (under the chapter number on each chapter’s first page), I have noted how crucial that chapter is for what follows. The categories are
Core (essential for understanding the unit)
Extension (extends core material in valuable ways, and may be useful for future chapters in this category, but not essential)
Optional (interesting but not needed for any future chapter).
Information clarifying the chapter’s designation and chapter dependencies typically appears for chapters in the non-Core categories.
Specifically About Unit C
This unit provides the foundation on which a SixIdeascourse rests. Unit C contains core material used in all of the other units as well as providing an introduction to the process of model-building that is central to the course.
Why study conservation laws before Newtonian mechanics? The most important reasons are the following:
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The concepts of energy and momentum are more fundamental than Newtonian mechanics and are crucial for units R, E, Q, and T.
Conservation is a simple idea whose mastery builds student confidence.
Conservation of momentum and angular momentum provides a good context for building student’s familiarity with vectors.
Starting with conservation laws delays having to use vector calculus.
The model of interactions presented helps students better understand force and avoid classic difficulties with Newton’s third law.
See the SixIdeaswebsite for more about the logic behind this approach.
Note that this unit implements the approach to teaching energy proposed by John Jewitt in a series of influential articles in The PhysicsTeacherin 2008.
This unit has not been changed much for the fourth edition. The main change is that I have refined Chapter C10 to make it simpler and clearer.
Most of the chapters in this unit are crucial and should be discussed in order. Chapter C7 on the more difficult aspects of angular momentum can be delayed or even omitted, and Chapter C14 is also optional. Our “dessert first” course for potential majors at Pomona starts with chapters C1–C6, C8, and C9 before moving on to the contemporary physics topics in units R, Q, and T: these chapters provide a satisfactory minimal background for those units. (We circle back to the rest of unit C and unit N in a later half-course for those who need to strengthen their background in classical physics.)
Appreciation
Thanking everyone who has offered important and greatly appreciated help with this project over the past three decades would be much too long to provide here. So, as in previous editions, I will focus on thanking those who have helped with this particular edition.
Thanks to my colleagues David Tanenbaum and Dwight Whitaker who offered good ideas and thoughtful advice for this edition. I’d like to thank Marisa Dobbeleare and especially Megan Platt and Beth Bettcher at McGraw-Hill for having faith in the SixIdeasproject and starting the push for this edition. Theresa Collins has been superb at guiding the project at the detail level. Many others at McGraw-Hill and its contractors, including Jeni McAtee, Sarita Yadav, Ashish Vyas, and Anand Singh, were instrumental in producing this particular edition. Finally, very special thanks to my wife Joyce, who (as always) has sacrificed, supported me, and loved me during my work on this edition. I am very grateful to you all!
Thomas A. Moore Claremont, California
Digital Learning Tools
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