`
`4th Edition;
`
`Volumel
`
`
`
`Resnick Halliday _ Krane
`
`‘
`|PR2020-O192
`
`A
`
`Apple EX105O Page 1
`
`IPR2020-01192
`Apple EX1050 Page 1
`
`

`

`
`
`VOLUME ONE
`
`PHYSICS
`
`FOURTH EDITION
`
`ROBERT RESNICK
`
`Professor ofPhysics
`
`Rensselaer Polytechnic Institute
`
`DAVID HALLIDAY
`
`Professor ofPhysics, Emeritus
`University ofPittsburgh
`
`KENNETH S. KRANE
`
`Professor ofPhysics
`
`Oregon State University
`
`JOHN WILEY & SONS. INC.
`Singapore
`0
`Toronto
`'
`Chichester
`.
`New York
`Brisbane
`-
`
`
`|PR2020-01192
`
`Apple EX105O Page 2
`
`IPR2020-01192
`Apple EX1050 Page 2
`
`

`

`Acquisitions Editor Clifi'ord Mills
`Marketing Manager Catherine Faduska
`Production Manager
`Joe Ford
`Production Supervisor Lucille Buonocore
`Manufacturing Manager Lorraine Fumoso
`Copy Editing Manager Deborah Herberl
`Photo Researcher
`Jennifer Atkins
`Photo Research Manager Stella Kupferberg
`Illustration John Balbalis
`Text Design Karin Gerdes Kincheloe
`Cover Design Direction Karin Gerdes Kincheloe
`Cover Design Lee Goldstem
`Cover Illustration Roy Wiemann
`
`Recognizing the importance of preserving what has been written, it is a policy of
`John Wiley & Sons‘ Inc. to have books of enduring value published in the
`United States printed on acid-free paper. and we exert our best eITorts to that end.
`
`Copyright © 1960. I962, I966, I978, I992, by John Wiley & Sons, Inc.
`
`All rights reserved. Published simultaneously in Canada.
`
`Reproduction or translation of any part of
`this work beyond that permitted by Sections
`107 and [08 of the I976 United States Copyright
`Act without the permission of the copyright
`owner is unlawful. Requests for permission
`or further information should be addressed to
`the Permissions Department, John Wiley & Sons.
`
`Library of Congress Cataloging-ln-Publlcatlon Data
`
`Halliday. David, I9I6—
`Physics / David Hnlliday, Robert Resnick. Kenneth S. Krone. — — 4th ed.
`p.
`cm.
`Includes index.
`
`: v. 1)
`ISBN 0-471-80458-4 (lib. bdg.
`I. Physics.
`I. Resnick, Robert, I923—
`III. Title.
`
`.
`
`II. Kmne, Kenneth S.
`
`QC2I.2.H355
`530-—dc20
`
`I992
`
`Printed and bound by Von Hofl'mlnn Preu. Inc.
`10
`9
`8
`7
`6
`5
`4
`3
`
`91-35885
`CIP
`
`
`
`——__
`
`|PR2020-01192
`
`Apple EX1050 Page 3
`
`IPR2020-01192
`Apple EX1050 Page 3
`
`

`

`
`
`CONTENTS
`
`
`
`
`
`CHAPTER 1
`
`W
`
`1-1 The Physical Quantities, Standards, and
`Units
`
`1-2 The International System of Units
`1-3 The Standard of Time
`
`1-4 The Standard of Length
`1-5 The Standard of Mass
`
`1-6 Precision and Significant Figures
`
`1-7 Dimensional Analysis
`
`Questions and Problems
`
`1
`
`2
`3
`
`4
`7
`
`8
`
`9
`
`10
`
`
`
`CHAPTER 2
`
`MOTION IN ONE DIMENSION 15
`
`
`2-1
`
`Particle Kinematics
`
`2-2 Descriptions of Motion
`
`2-3 Average Velocity
`
`Instantaneous Velocity
`2-4
`2-5 Accelerated Motion
`
`2-6 Motion with Constant Acceleration
`
`2-7 Freely Falling Bodies
`2-8 Galileo and Free Fall (Optional)
`
`2-9 Measuring the Free-Fall Acceleration
`(Optional)
`
`Questions and Problems
`
`15
`
`15
`
`17
`
`18
`2|
`
`23
`
`25
`26
`
`27
`
`28
`
`
`
`CHAPTER 3
`
`VECTORS
`37
`
`
`3-1 Vectors and Scalars
`3-2 Adding Vectors: Graphical Method
`3-3 Components of Vectors
`
`37
`38
`39
`
`5-10 Applications of Newton‘s Laws
`
`5-11 More Applications of Newton’s Laws
`
`Questions and Problems
`
`|PR2020-01192
`
`Apple EX105O Page 4
`
`3-4 Adding Vectors: Component Method
`
`3-5 Multiplication of Vectors
`
`3-6 Vector Laws in Physics
`
`Questions and Problems
`
`41
`
`43
`
`46
`
`48
`
`
`
`CHAPTER 4
`MOTION IN TWO AND
`
`THREE DIMENSIONS
`53
`
`
`4-1 Position, Velocity, and Acceleration
`4-2 Motion with Constant Acceleration
`
`4-3 Projectile Motion
`4-4 Uniform Circular Motion
`
`4-5 Velocity and Acceleration Vectors in
`Circular Motion (Optional)
`4-6 Relative Motion
`
`Questions and Problems
`
`53
`55
`
`57
`60
`
`62
`64
`
`67
`
`CHAPTER 5
`FORCE AND NEWTON’S
`LAWS
`77
`
`
`5-1
`
`5-2
`
`5-3
`
`Classical Mechanics
`
`Newton’s First Law
`
`Force
`
`5-4 Mass
`
`5-5
`
`5-6
`
`5-7
`
`Newton‘s Second Law
`
`Newton‘s Third Law
`
`Units of Force
`
`5-8 Weight and Mass
`
`5-9 Measuring Forces
`
`77
`
`78
`
`79
`
`80
`
`81
`
`83
`
`85
`
`86
`
`87
`
`88
`
`92
`
`94
`
`vii
`
`IPR2020-01192
`Apple EX1050 Page 4
`
`

`

`viii Contents
`
`
`
`
`
`CHAPTER 6
`PARTICLE DYNAMICS
`103
`
`
`CHAPTER 9
`
`SYSTEMS OF PARTICLES 179
`
`6-1
`
`Force Laws
`
`6-2 Frictional Forces
`
`6-3 The Dynamics of Uniform Circular Motion
`
`6-4 Equations of Motion: Constant and
`Nonconstant Forces
`
`6-5 Time—DependentForces:AnalyticalMethods
`6-6 Time-Dependent Forces: Numerical
`Methods (Optional)
`
`6-7 Drag Forces and the Motion of Projectiles
`6-8 Noninertial Frames and Pseudoforces
`(Optional)
`
`6-9 Limitations of Newton’s Laws (Optional)
`Questions and Problems
`
`103
`
`104
`
`108
`
`11 1
`
`113
`
`1 14
`
`115
`
`1 17
`
`1 19
`121
`
`
`
`9-1 Two-Particle Systems
`
`9-2 Many-Particle Systems
`9-3 Center 01‘ Mass of Solid Objects
`9-4 Linear Momentum of a Particle
`
`9-5 Linear Momentum of a System of Particles
`9-6 Conservation of Linear Momentum
`
`9-7 Work and Energy in a System of Particles
`(Optional)
`
`9-8 Systems of Variable Mass (Optional)
`
`Questions and Problems
`
`179
`
`181
`185
`188
`
`189
`189
`
`192
`
`195
`
`199
`
`CHAPTER 10
`COLLISIONS
`
`10-1 What ls a Collision?
`
`CHAPTER 7
`WORK AND ENERGY
`
`131
`
`10-2
`
`Impulse and Momentum
`
`7-1 Work Done by a Constant Force
`
`7-2 Work Done by a Variable Force: One-
`Dimensional Case
`
`131
`
`134
`
`10-3 Conservation of Momentum During
`Collisions
`
`10-4 Collisions in One Dimension
`
`10-5 Two-Dimensional Collisions
`
`10-6 Center-of-Mass Reference Frame
`
`207
`
`207
`
`209
`
`210
`
`21 1
`
`215
`
`217
`
`
`
`7-3 Work Done by a Variable Force: Two-
`Dimensional Case (Optional)
`
`7-4 Kinetic Energy and the Work—Energy
`Theorem
`7-5 Power
`
`7-6 Reference Frames (Optional)
`
`7-7 Kinetic Energy at High Speed (Optional)
`
`Questions and Problems
`
`137
`
`138
`140
`
`141
`
`143
`
`144
`
`
`
`CHAPTER 8
`CONSERVATION OF ENERGY 151
`
`8-1 Conservative Forces
`
`8-2 Potential Energy
`
`8-3 One-Dimensional Conservative Systems
`
`8-4 One—Dimensional Conservative Systems:
`The Complete Solution
`'
`8-5 Two- and Three-Dimensional Conservative
`Systems (Optional)
`
`8-6 Conservation of Energy in a System of
`Particles
`
`8-7 Mass and Energy (Optional)
`
`8-8 Quantization of Energy (Optional)
`
`Questions and Problems
`
`151
`
`[54
`
`155
`
`158
`
`161
`
`162
`
`165
`
`168
`
`169
`
`10-7 Spontaneous Decay Processes (Optional)
`Questions and Problems
`
`221
`222
`
`
`
`CHAPTER 11
`ROTATIONAL KINEMATICS
`231
`
`
`1 1-1 Rotational Motion
`
`1 1-2 The Rotational Variables
`
`l l-3 Rotation with Constant Angular
`Acceleration
`
`1 1-4 Rotational Quantities as Vectors
`
`1 1-5 Relationships Between Linear and Angular
`Variables: Scalar Form
`
`11-6 Relationships Between Linear and Angular
`Variables: Vector Form (Optional)
`
`Questions and Problems
`
`231
`
`232
`
`234
`
`235
`
`237
`
`239
`
`240
`
`
`
`CHAPTER 12
`- ROTATIONAL DYNAMICS
`245
`
`
`12-1 Rotational Dynamics: An Overview
`
`12-2 Kinetic Energy of Rotation and Rotational
`Inertia
`
`245
`
`246
`
`
`
`|PR2020-01 192
`
`Apple EX105O Page 5
`
`IPR2020-01192
`Apple EX1050 Page 5
`
`

`

`12-3 Rotational Inertia of Solid Bodies
`
`12-4 Torque Acting on a Particle
`
`12-5 Rotational Dynamics of a Rigid Body
`12-6 Combined Rotational and Translational
`Motion
`
`Questions and Problems
`
`249
`
`251
`
`253
`
`257
`
`262
`
`
`
`CHAPTER 13
`ANGULAR MOMENTUM
`271
`
`
`13-1 Angular Momentum of a Particle
`
`13-2 Systems of Particles
`
`13-3 Angular Momentum and Angular Velocity
`
`13-4 Conservation of Angular Momentum
`
`13-5 The Spinning Top
`
`13-6 Quantization of Angular Momentum
`(Optional)
`
`13-7 Rotational Dynamics: A Review
`
`Questions and Problems
`
`271
`
`273
`
`275
`
`279
`
`284
`
`285
`
`286
`
`287
`
`
`
`CHAPTER 14
`EQUILIBRIUM OF RIGID
`BODIES
`295
`
`
`14—1 Conditions of Equilibrium
`
`14-2 Center of Gravity
`14-3 Examples of Equilibrium
`14-4 Stable, Unstable, and Neutral Equilibrium
`of Rigid Bodies in a Gravitational Field
`14-5 Elasticity
`Questions and Problems
`
`295
`
`296
`298
`
`303
`304
`307
`
`_______.____———
`
`
`CHAPTER 15
`315
`OSCILLATIONS
`_______.__.__—————
`
`15-1
`15-2
`
`15-3
`15-4
`
`15-5
`15-6
`
`15-7
`
`15-8
`15-9
`
`Oscillating Systems
`The Simple Harmonic Oscillator
`
`Simple Harmonic Motion
`Energy Considerations in Simple
`Harmonic Motion
`
`315
`317
`
`318
`
`320
`
`Applications of Simple Harmonic Motion 322
`Simple Harmonic Motion and Uniform
`Circular Motion
`
`326
`
`Combinations of Harmonic Motions
`
`Damped Harmonic Motion (Optional)
`Forced Oscillations and Resonance
`(Optional)
`
`328
`
`329
`
`330
`
`Contents
`
`in
`
`15-10 Two-Body Oscillations (Optional)
`Questions and Problems
`
`332
`333
`
`____________———-_——-—-
`
`
`" CHAPTER 16
`
`GRAVITATION
`343
`__________——————-————
`
`Gravitation from the Ancients to Kepler
`
`343
`
`16-1
`
`16-2
`
`16-3
`16-4
`16-5
`
`Newton and the Law of Universal
`Gravitation
`
`The Gravitational Constant G
`Gravity Near the Earth’s Surface
`Gravitational Effect of a Spherical
`Distribution of Matter (Optional)
`
`344
`
`346
`348
`
`350
`
`352
`
`355
`356
`361
`
`363
`366
`
`16-6 Gravitational Potential Energy
`
`16-7
`
`16-8
`16-9
`
`The Gravitational Field and Potential
`(Optional)
`The Motions of Planets and Satellites
`Universal Gravitation
`
`16-10 The General Theory of Relativity
`(Optional)
`Questions and Problems
`
`
`
`CHAPTER 17
`
`FLUID STATICS
`377
`
`17-1 Fluids and Solids
`
`17-2 Pressure and Density
`17-3 Variation of Pressure in a Fluid at Rest
`
`17-4 Pascal’s Principle and Archimedes’
`Principle
`17-5 Measurement of Pressure
`
`17-6 Surface Tension (Optional)
`
`Questions and Problems
`
`377
`
`378
`380
`
`383
`386
`
`388
`
`390
`
`
`
`CHAPTER 18
`
`FLUID DYNAMICS
`397
`
`
`397
`18-1 General Concepts of Fluid Flow
`18-2 Streamlines and the Equation of Continuity 398
`18-3 Bemoulli’s Equation
`400
`
`18-4 Applications of Bernoulli’s Equation and
`the Equation of Continuity
`18-5 Fields of Flow (Optional)
`18-6 Viscosity, Turbulence, and Chaotic Flow
`(Optional)
`
`Questions and Problems
`
`403
`405
`
`407
`
`411
`
`|PR2020-01 192
`
`Apple EX105O Page 6
`
`IPR2020-01192
`Apple EX1050 Page 6
`
`

`

`X Contents
`
`L” CHAPTER 19
`.
`
`‘ WAVE MOTION
`417
`
`
`CHAPTER 22
`
`TEMPERATURE 493
`
`
`
`19-1 Mechanical Waves
`
`19-2
`
`Types of Waves
`
`Traveling Waves
`19-3
`19-4 Wave Speed
`
`19-5
`
`19-6
`
`19-7
`19-8
`
`The Wave Equation (Optional)
`
`Power and Intensity in Wave Motion
`
`The Principle of Superposition
`Interference of Waves
`
`Standing Waves
`19-9
`19-10 Resonance
`
`Questions and Problems
`
`417
`
`418
`
`419
`423
`
`425
`
`426
`
`427
`430
`
`432
`436
`
`438
`
`
`
`CHAPTER 20
`SOUND WAVES
`
`20-1 The Speed of Sound
`
`20-2 Traveling Longitudinal Waves
`20-3 Power and Intensity of Sound Waves
`
`20-4 Standing Longitudinal Waves
`
`20-5 Vibrating Systems and Sources of Sound
`20-6 Beats
`
`445
`
`445
`
`447
`449
`
`450
`
`453
`455
`
`22-1 Macroscopic and Microscopic Descriptions 493
`
`22-2 Temperature and Thermal Equilibrium
`
`22-3 Measuring Temperature
`
`22-4 The Ideal Gas Temperature Scale
`
`22-5 Thermal Expansion
`
`Questions and Problems
`
`494
`
`495
`
`498
`
`500
`
`503
`
`CHAPTER 23
`KINETIC THEORY AND
`THE IDEAL GAS
`509
`
`
`23-1 Macroscopic Properties of a Gas and the
`Ideal Gas Law
`23—2 The Ideal Gas: A Model
`
`23-3 Kinetic Calculation of the Pressure
`
`23-4 Kinetic Interpretation of the Temperature
`23-5 Work Done on an Ideal Gas
`
`23-6 The Internal Energy of an Ideal Gas
`
`Intermolecular Forces (Optional)
`23-7
`23-8 The Van der Waals Equation of State
`(Optional)
`
`509
`51 1
`
`512
`
`514
`515
`
`519
`
`521
`
`522
`
`
`
`20-7 The Doppler Effect
`
`Questions and Problems
`
`457
`
`460
`
`
`
`CHAPTER 21
`THE SPECIAL THEORY OF
`
`RELATIVITY
`467
`
`21-1
`
`21-2
`21-3
`21-4
`
`Troubles with Classical Physics
`
`The Postulates of Special Relativity
`Consequences of Einstein's Postulates
`The Lorentz Transformation
`
`21-5 Measuring the Space-Time Coordinates
`of an Event
`
`21-6
`
`21-7
`
`21-8
`
`21-9
`
`The Transformation of Velocities
`
`Consequences of the Lorentz
`Transformation
`
`Relativistic Momentum
`
`Relativistic Energy
`
`21-10 The Common Sense of Special Relativity
`
`Questions and Problems
`
`467
`
`469
`470
`473
`
`475
`
`476
`
`478
`
`482
`
`483
`
`486
`
`487
`
`Questions and Problems
`
`524
`
`
`
`CHAPTER 24
`
`STATISTICAL MECHANICS 529
`
`24-1
`
`Statistical Distributions and Mean Values
`
`24-2 Mean Free Path
`
`24-3 The Distribution of Molecular Speeds
`
`24-4 The Distribution of Energies
`24-5 Brownian Motion
`
`24-6 Quantum Statistical Distributions
`(Optional)
`
`Questions and Problems
`
`529
`
`531
`
`535
`
`538
`539
`
`541
`
`544
`
`
`
`CHAPTER 25
`HEAT AND THE FIRST LAW
`OF THERMODYNAMICS
`547
`
`
`25-1 Heat: Energy in Transit
`
`25-2 Heat Capacity and Specific Heat
`
`25-3 Heat Capacities of Solids
`
`547
`
`548
`
`550
`
`
`
`|PR2020-01 192
`
`Apple EX105O Page 7
`
`IPR2020-01192
`Apple EX1050 Page 7
`
`

`

`Contents
`
`xi
`
`_____—______—_—————
`____________________—————-
`
`APPENDICES
`
`A The lntemational System of Units (SI)
`B Some Fundamental Constants of Physics
`C Some Astronomical Data
`D Properties of the Elements
`E
`Periodic Table of the Elements
`F Elementary Particles
`G Conversion Factors
`H Mathematical Formulas
`I
`Computer Programs
`J Nobel Prizes in Physics
`
`ANSWERS TO ODD NUMBERED
`PROBLEMS
`PHOTO CREDITS
`INDEX
`
`A-l
`A-3
`A-4
`A-5
`A-7
`A-8
`A-lO
`A-l4
`A-16
`A-20
`
`A-24
`P-l
`I-l
`
`25—4 Heat Capacities of an Ideal Gas
`25-5 The First Law of Thermodynamics
`25-6 Applications of the First Law
`25-7 The Transfer of Heat
`
`Questions and Problems
`
`552
`555
`558
`561
`
`564
`
`____—___—————
`
`
`CHAPTER 26
`ENTROPY AND THE SECOND
`
`LAW OF THERMODYNAMICS 571
`____————————
`
`26-1 Reversible and Irreversible Processes
`
`26-2 Heat Engines and the Second Law
`
`26-3 Refrigerators and the Second Law
`
`26-4 The Carnot che
`
`26-5 The Thermodynamic Temperature Scale
`
`26-6 Entropy: Reversible Processes
`
`26-7 Entropy: Irreversible Processes
`
`26-8 Entropy and the Second Law
`26-9 Entropy and Probability
`Questions and Problems
`
`571
`
`573
`
`575
`
`576
`
`580
`
`581
`
`583
`
`585
`586
`588
`
`|PR2020-01192
`
`Apple EX105O Page 8
`
`IPR2020-01192
`Apple EX1050 Page 8
`
`

`

`CHAPTER 3
`
`Many of the laws ofphysics involve not only algebraic relationships
`among quantities, but geometrical relationships as well. For example, picture a
`spinning top that rotates rapidly about its axis, while the axis ofrotation itself rotates
`slowly about the vertical. This geometrical relationship is complicated to represent by
`algebraic equations. However, if we use vectors to represent the physical
`variables. a single equation is sufiicient to explain the behavior. Vectors permit such
`economy ofexpression in a great variety ofphysical laws. Sometimes the vectorform
`of a physical law permits us to see relationships or symmetries that would otherwise
`be obscured by a cumbersome algebraic equation.
`
`In this chapter we explore some ofthe properties and uses of vectors and we introduce the
`mathematical operations that involve vectors In the process you will learn that familiar
`symbols from arithmetic, such as +, —, and X, have diflerent meanings
`when applied to vectors.
`
`
`
`
`
`
`
`
`
`3-1 VECTORS AND SCALARS
`
`., A change ofposition of a particle is called a displacement. 1
`If a particle moves from position A to position B (Fig. la),
`we can represent its displacement by drawing a line from
`A to B. The direction of displacement can be shown by
`putting an arrowhead at B indicating that the displace-
`ment was from A to B. The path of the particle need not
`necessarily be a straight line from A to B; the arrow repre-
`sents only the net efl‘ect of the motion, not the actual
`motion.
`
`In Fig. lb, for example, we plot an actual path followed
`by a particle from A to B. The path is not the same as the
`displacement AB. If we were to take snapshots of the
`particle when it was at A and, later, when it was at some
`intermediate position P, we could obtain the displace-
`ment vector AP, representing the net effect of the motion
`during this interval, even though we would not know the
`actual path taken between these points. Furthermore, a
`displacement such as A’B’ (Fig. la), which is parallel to
`AB, similarly directed, and equal in length to AB, repre-
`sents the same change in position as AB. We make no
`distinction between these two displacements. A displace-
`ment is therefore characterized by a length and a direc-
`tion.
`
`(b)(a) (C)
`
`
`
`
`Figure 1 Displacement vectors. (a) Vectors AB and A’B’ are
`identical, since they have the same length and point in the
`same direction. (b) The actual path of the particle in moving
`from A to B may be the curve shown; the displacement is the
`vector AB. At the intermediate point P, the displacement is
`the vector AP. (c) After displacement AB, the particle under-
`goes another displacement BC. The net efi‘ect of the two dis-
`placements is the vector AC.
`
`In a similar way, we can represent a subsequent dis-
`placement from B to C (Fig. 1c). The net effect of the two
`displacements is the same as a displacement from A to C.
`We speak then of AC as the sum or resultant of the dis-
`placements AB and BC. Notice that this sum is not an
`algebraic sum and that a number alone cannot uniquely
`specify it.
`
`37
`
`|PR2020-O1192
`
`Apple EX105O Page 9
`
`IPR2020-01192
`Apple EX1050 Page 9
`
`

`

`38 Chapter3 Vectors
`
`Quantities that behave like displacements are called
`vectors. (The word vector means carrier in Latin. Biolo-
`gists use the term vector to mean an insect, animal, or
`other agent that carries a cause of disease from one orga-
`nism to another.) Vectors, then, are quantities that have
`both magnitude and direction and that follow certain
`mles of combination, which we describe below. The dis-
`placement vector is a convenient prototype. Some other
`physical quantities that are represented by vectors are
`force, velocity, acceleration, electric field, and magnetic
`field. Many of the laws of physics can be expressed in
`compact form by using vectors, and derivations involving
`these laws are ofien greatly simplified if we do so.
`Quantities that can be specified completely by a num-
`berand unit and that therefore have magnitude only are
`called scalars. Some physical quantities that are scalars
`are mass, length, time, density, energy, and temperature.
`Scalars can be manipulated by the rules of ordinary
`algebra.
`
`
`
`3-2 ADDING VECTORS:
`GRAPHICAL METHOD
`
`To represent a vector on a diagram we draw an arrow. We
`choose the length of the arrow to be proportional to the
`magnitude of the vector (that is, we choose a scale), and
`we choose the direction ofthe arrow to be the direction of
`
`the vector, with the arrowhead giving the sense of the
`direction. For example, a displacement of 42 m in a
`northeast direction would be represented on a scale of
`1 cm per 10 m by an arrow 4.2 cm long, drawn at an angle
`of 45 ° above a line pointing east with the arrowhead at the
`top right extreme (Fig. 2). A vector is usually represented
`in printing by a boldface symbol such as d. In handwriting
`we usually put an arrow_.above the symbol to denote a
`vector quantity, such as d.
`
`45"
`
`\
`
`\
`
`E
`
`Figure 2 The vector 1! represents a displacement of magni-
`tude 42 m (on a scale in which 10 m = 1 cm) in a direction
`45° north of east.
`
`Figure 3 The vector sum it + b = 9. Compare with Fig. lc.
`
`I
`
`Often we are interested only in the magnitude (or
`length) of the vector and not in its direction. The magni-
`tude of d is sometimes written as |d|; more frequently we
`represent the magnitude alone by the italic letter symbol
`d. The boldface symbol is meant to signify both properties
`of the vector, magnitude and direction. When handwrit-
`ten, the magnitude of the vector is represented by the
`symbol without the arrow.
`Consider now Fig. 3 in which we have redrawn and
`relabeled the vectors of Fig. 1c. The relation among these
`vectors can be written
`
`a+b=s.
`
`(1)
`
`The rules to be followed in performing this vector addi-
`tion graphically are these: (1) On a diagram drawn to scale
`lay out the vector 11 with its proper direction in the coordi-
`nate system. (2) Draw b to the same scale with its tail at the
`head of a, making sure that b has its own proper direction
`(generally different from the direction of a). (3) Draw a
`line from the tail of a to the head of b to construct the
`
`vector sum s. If the vectors were representing displace-
`ments, then s would be a displacement equivalent in
`length and direction to the successive displacements a and
`b. This procedure can be generalized to obtain the sum of
`any number of vectors.
`Since vectors differ from ordinary numbers, we expect
`different rules for their manipulation. The symbol “+” in
`Eq.
`1 has a meaning different from its meaning in arith-
`metic or scalar algebra. It tells us to carry out a different
`set of operations.
`By careful inspection of Fig. 4 we can deduce two im-
`portant properties of vector addition:
`
`a + b = b + a
`
`(commutative law)
`
`(2)
`
`and
`
`d + (e + f) = (d + e) + f
`
`(associative law).
`
`(3)
`
`
`
`(a) The commutative law for vector addition,
`Figure 4
`which states that a + h = b + a. (b) The associative law, which
`statesthatd+(e+f)=(d+e)+f.
`
`
`
`|PR2020-O1192
`
`Apple EX105O Page 10
`
`IPR2020-01192
`Apple EX1050 Page 10
`
`

`

`
`
`Figure 5 The vector difference I - b a a + (— b).
`
`These laws assert that it makes no difference in what order
`
`or in what grouping we add vectors; the sum is the same.
`In this respect, vector addition and scalar addition follow
`the same rules.
`
`By inspection of Fig. 4b, you will see how the graphical
`method is used to find the sum of more than two vectors,
`in this case (I + e + f. Each succeeding vector is placed
`with its tail at the head of the previous one. The vector
`representing the sum is then drawn from the tail of the
`first vector to the head of the last one.
`
`The operation of subtraction can be included in our
`vector algebra by defining the negative of a vector to be
`another vector ofequal magnitude but opposite direction.
`Then
`
`Section 3-3 Components of Vector: 39
`
`
`
`Figure 6 (a) The vector a has component a, in the x direc-
`tion and component a, in the y direction. (b) The vector b has
`a negative x component.
`
`a—b=a+(-b)
`
`(4)
`
`The components ax and a, in Fig. 6a are readily found
`from
`
`as shown in Fig. 5. Here — b means a vector with the same
`magnitude as b but pointing in the opposite direction. It
`follows from Eq. 4 that a - a = a + (-a) = 0.
`Remember that, although we have used displacements
`to illustrate these operations, the rules apply to all vector
`quantities, such as velocities and forces.
`
`
`
`3-3 COMPONENTS OF VECTORS
`
`Even though we defined vector addition with the graphi-
`cal method, it is not very useful for vectors in three di-
`mensions. Often it is even inconvenient for the two-di-
`
`mensional case. Another way of adding vectors is the
`analytical method, involving the resolution of a vector
`into components with respect to a particular coordinate
`system.
`Figure 6a shows a vector a whose tail has been placed at
`the origin of a rectangular coordinate system. If we draw
`perpendicular lines from the head of a to the axes, the
`quantitiesg, and ay so formed are called the (Cartesian)
`\ components of the vector 3. The process is called resolving
`a vector into its components. The vector u is completely
`and uniquely specified by these components; given ax and
`(1,, we could immediately reconstruct the vector a.
`The components of a vector can be positive, negative,
`or zero. Figure 6b shows a vector b that has b, < 0 and
`b,>o.
`
`a, = a cos d)
`
`and a, = a sin (I),
`
`(5)
`
`where (15 is the angle that the vector it makes with the
`positive xaxis, measured counterclockwise from this axis.
`As shown in Fig. 6, the algebraic signs of the components
`of a vector depend on the quadrant in which the angle (1)
`lies. For example, when ()5 is between 90° and 180°, as in
`Fig. 6b, the vector always has a negative x component and
`a positive y component. The components of a vector be-
`have like scalar quantities because, in any particular coor-
`dinate system, only a number with an algebraic sign is
`needed to specify them.
`Once a vector is resolved into its components, the com-
`ponents themselves can be used to specify the vector.
`Instead of the two numbers a (magnitude of the vector)
`and (1) (direction of the vector relative to the x axis), we
`now have the two numbers ax and a,.. We can pass back
`and forth between the description of a vector in terms of
`its components (ax and ay) and the equivalent description
`in terms of magnitude and direction (a and (15). To obtain
`a and (b from a, and (1,, we note from Fig. 6a that
`a=¢ai+a§ 5"" ‘ “A (6“)
`
`and
`
`tan d) = ay/ax.
`
`(6b)
`
`The quadrant in which d) lies is determined from the signs
`of a, and ay.
`In three dimensions the process works similarly: just
`draw perpendicular lines from the tip of the vector to the
`three coordinate axes x, y, and 2. Figure 7 shows one way
`
`
`
`
`
`|PR2020-01192
`
`Apple EX105O Page 11
`
`IPR2020-01192
`Apple EX1050 Page 11
`
`

`

`40 Chapter3 Vectors
`
`
`
`‘afifl
`
`Figure 7 A vector u in three dimensions with components
`0,, a” and a,. The x and y components are conveniently
`found by first drawing the xy projection of a. The angle 6 be-
`tween 3 and the z axis is called théjwlaWeFThe angle d)
`in the xy plane between theprojection of a and the x axis is
`called thea‘zirnuthal angle. The azimuthal angle ¢> has the
`same meaning here as it does in Fig. 6.
`
`this is ofien drawn to make the components easier to
`recognize; the component (sometimes called a projection)
`of a in the xy plane is first drawn, and then from its tip we
`can find the individual components a, and ay. We would
`obtain exactly the same x and ycomponents if we worked
`directly with the vector it instead ofwith its xy projection,
`but the drawing would not be as clear. From the geometry
`of Fig. 7, we can deduce the components of the vector :1 to
`be
`
`{a,=asin0<>os¢, a,=asin65ind>,
`l a, = 0 cos 0.
`7
`
`and
`
`(7)
`
`When resolving a vector into components it is some-
`times useful to introduce a vector of unit length in a given
`direction. Often it
`is convenient to draw unit vectors
`
`along the particular coordinate axes chosen. In the rectan-
`gular coordinate system the special symbols i, j, and k are
`usually used for unit vectors in the positive x, y, and 2
`directions, respectively (see Fig. 8). In handwritten nota-
`tion, unit vectors are often indicated with a circumflex or
`“hat," such as i, j, and ii.
`Note that i, j, and k need not be located at the origin.
`Like all vectors, they can be translated anywhere in the
`coordinate space as long as their directions with respect to
`the coordinate axes are not changed.
`In general, a vector u in a three-dimensional coordinate
`system can be written in terms of its components and the
`unit vectors as
`(\V
`H (g
`t
`t
`a = axi + ayj + azk,
`
`.
`
`‘
`
`(8a)
`
`
`
`Figure 8 The unit vectors i, j, and k are used to specify the
`positive x, y, and z axes, respectively. Each vector is dimen-
`sionless and has a length of unity.
`
`The vector relation Eq. 8b is equivalent to the scalar rela-
`tions of Eq. 6. Each equation relates the vector (a, ora and
`4)) to its components (a, and (1,). Sometimes we call
`quantities such as axi and ayj in Eq. 8b the vector compo
`nents of a. Figure 9 shows the vectors a and b of Fig. 6
`drawn in terms of their vector components. Many physi-
`cal problems involving vectors can be simplified by re-
`placing a vector by its vector components. That is, the
`
`I
`
`
`
`Figure 9 The vector components of a and b. In any ph)’Sical
`situation that involves vectors, we get the same outcome
`whether we use the vector itself, such as a, or its two vector
`components, axi and ayj. The effect of the single vector a is
`equivalent to the net effect of the two vectors axi and ayj.
`When we have replaced a vector with its vector components,
`it is helpful to draw a double line through the original vector,
`as shown; this helps us to remember not to consider the origl'
`nal vector any more.
`
`
`
`or in two dimensions as
`
`a = a,i + ayj.
`
`.
`
`(8b)
`
`
`”
`|PR2020-O1192
`
`Apple EX105O Page 12
`
`IPR2020-01192
`Apple EX1050 Page 12
`
`

`

`action of a quantity represented as a vector can be re-
`placed by the actions of its vector components. When
`necessary, we refer explicitly to vector components, while
`the word component alone continues to refer to the scalar
`quantities a, and 61,.
`
`(Optional)
`Other Coordinate Systems
`Many other varieties of coordinate systems may be appropriate
`for analyzing certain physical situations. For example, the two-
`dimensional xy coordinate system may be changed in either of
`two ways: (1) by moving the origin to another location in the xy
`plane, which is called a translation of the coordinate system, or
`(2) by pivoting the xy axes about the fixed origin, which is a
`rotation ofthe coordinate system. In both ofthese operations we
`keep the vector fixed and move the coordinate axes. Figure 10
`shows the effect of these two changes. In the case shown in Fig.
`10a, the components are unchanged, but in the case shown in
`Fig. 10b, the components do change.
`When the physical situation we are analyzing has certain sym-
`metries, it may be advantageous to choose a different coordinate
`system for resolving a vector into its components. For instance,
`we might choose the radial and tangential directions of plane
`polar coordinates, shown in Fig.
`l 1. In this case, we find the
`components on the coordinate axesjust as we did in the ordinary
`xyz system: we draw a perpendicular from the tip ofthe vector to
`each coordinate axis.
`
`
`
`
`
`Section 3-4 Adding Vectors: Component Method 4]
`
`
`
`
`
`
`Tangential
`/
`direction /
`Radial
`
`direction
`/
`a
`(r axis)
`\
`
`
`
`
`
`2:
`
`Figure l] The vector in is resolved into its radial and tangen-
`tial components. These components will have important ap-
`plications when we consider circular motion in Chapters 4
`and l l.
`
`1 1 (spherical polar
`The three-dimensional extensions of Fig.
`or cylindrical polar coordinates) in many important cases are far
`superior to Cartesian coordinate systems for the analysis ofphys-
`ical problems. For example, the gravitational force exerted by
`the Earth on distant objects has the symmetry of a sphere, and
`thus its properties are most simply described in spherical polar
`coordinates. The magnetic force exerted by a long straight
`current—carrying wire has the symmetry of a cylinder and is
`therefore most simply described in cylindrical polar coordi-
`nates. I
`
`
`
`3-4 ADDING VECTORS:
`COMPONENT METHOD
`
`Now that we have shown how to resolve vectors into their
`
`components, we can consider the addition of vectors by
`an analytic method.
`Let s be the sum of the vectors a and b, or
`
`s=a+b.
`
`(9)
`
`If two vectors, such as s and a + b, are to be equal, they
`must have the same magnitude and must point in the
`same direction. This can only happen iftheir correspond-
`ing components are equal. We stress this important con-
`clusion:
`
`(a) The origin 0 of the coordinate system of Fig.
`Figure 10
`(St: has been moved or translated to the new position 0’. The
`x and y components of a are identical to the x’ and y’ compo
`nents. (b) The x and y axes have been rotated through the
`angle [3. The x’ and y’ components are different from the x
`and y components (note that the y’ component is now smaller
`than the x' component, while in Fig. 6a the y component was
`greater than the x component), but the vector a is unchanged.
`By what angle should we rotate the coordinate axes to make
`the y’ component zero?
`
`2* Two vectors are equal to each other only iftheir corre-
`sponding components are equal.
`
`For the vectors of Eq. 9, we can write
`
`sxi + syj = axi + ayj + bxi + b,j
`= (a, + bx)i + (a, + b,)j.
`
`(10)
`
`Equating the x components on both sides of Eq. 10 gives
`
`x=ax+bx,
`
`(11a)
`
`
`
`|PR2020-01192
`
`Apple EX105O Page 13
`
`IPR2020-01192
`Apple EX1050 Page 13
`
`

`

`42 Chapter} Vectors
`
`and equating the y components gives
`
`sy=ay+b_,..
`
`(llb)
`
`These two algebraic equations. taken together. are equiva-
`lent to the single vector relation of Eq. 9.
`Instead of specifying the components ofs. we can give
`its length and direction:
`
`3 = 0.1+ 5}. = ital- + bx)2 + (ay + 1),)2
`
`(12a)
`
`and
`
`tanqS
`
`,‘l‘w‘
`=§£=_)._La'+b'
`3x
`flx'l'bx" .
`
`(121))
`
`Here is the rule for adding vectors by this method.
`(1) Resolve each vector into its components, keeping
`track ofthe algebraic sign ofeach component. (2) Add the
`components for each coordinate axis, taking the algebraic
`sign into account. (3) The sums so obtained are the com-
`ponents of the sum vector. Once we know the compo-
`nents of the sum vector, we can easily reconstruct that
`vector in space.
`The advantage of the method of breaking up vectors
`into components, rather than adding directly with the use
`of suitable trigonometric relations, is that we always deal
`with right triangles and thus simplify the calculations.
`In adding vectors by the component method, the choice
`ofcoordinate axes determines how simple the process will
`be. Sometimes the components ofthe vectors with respect
`to a particular set ofaxes are known at the start, so that the
`choice of axes is obvious. Other times a judicious choice
`of axes can greatly simplify the job of resolution of the
`vectors into components. For example, the axes can be
`oriented so that at least one ofthe vectors lies parallel to an
`axis; the components of that vector along the other axes
`will then be zero.
`
`
`
`Sample Problem I An airplane travels 209 km on a straight
`course making an angle of22.5 ° east ofdue north. How far north
`and how far east did the plane travel from its starting point?
`
`Solution We choose the positive x direction to be cast and the
`positive y direction to be nonh. Next, we draw a displacement
`vector (Fig. 12) from the origin (starting point), makin