LIST OF EXAMPLES ............................................... xi
PREFACE ........................................................ xv
TO THE TEACHER ................................................ xix
1 VECTORS AND KINEMATICS - A FEW MATHEMATICAL PRELIMINARIES
1.1 INTRODUCTION ............................................ 2
1.2 VECTORS ................................................. 2
Definition of a Vector, The Algebra of Vectors .......... 3
1.3 COMPONENTS OF A VECTOR .................................. 8
1.4 BASE VECTORS ........................................... 10
1.5 DISPLACEMENT AND THE POSITION VECTOR ................... 11
1.6 VELOCITY AND ACCELERATION .............................. 13
Motion in One Dimension ................................ 14
Motion in Several Dimensions ........................... 14
A Word about Dimensions and Units ...................... 18
1.7 FORMAL SOLUTION OF KINEMATICAL EQUATIONS ............... 19
1.8 MORE ABOUT THE DERIVATIVE OF A VECTOR .................. 23
1.9 MOTION IN PLANE POLAR COORDINATES ...................... 27
Polar Coordinates ...................................... 27
Velocity in Polar Coordinates .......................... 27
Evaluating dr/dt ....................................... 31
Acceleration in Polar Coordinates ...................... 36
Note 1.1 MATHEMATICAL APPROXIMATION METHODS ................. 39
The Binomial Series .................................... 41
Taylor's Series ........................................ 42
Differentials .......................................... 45
Some References to Calculus Texts ...................... 47
PROBLEMS .................................................... 47
2 NEWTON'S LAWS - THE FOUNDATIONS OF NEWTONIAN MECHANICS
2.1 INTRODUCTION ........................................... 52
2.2 NEWTON'S LAWS .......................................... 53
Newton's First Law .......................................... 55
Newton's Second Law .................................... 56
Newton's Third Law ..................................... 59
2.3 STANDARDS AND UNITS .................................... 64
The Fundamental Standards .............................. 64
Systems of Units ....................................... 67
2.4 SOME APPLICATIONS OF NEWTON'S LAWS ..................... 68
2.5 THE EVERYDAY FORCES OF PHYSICS ......................... 79
Gravity, Weight, and the Gravitational Field ........... 80
The Electrostatic Force ................................ 86
Contact Forces ......................................... 87
Tension—The Force of a String .......................... 87
Tension and Atomic Forces .............................. 91
The Normal Force ....................................... 92
Friction ............................................... 92
Viscosity .............................................. 95
The Linear Restoring Force: Hooke's Law, the Spring,
and Simple Harmonic Motion ............................. 97
Note 2.1 THE GRAVITATIONAL ATTRACTION OF A SPHERICAL
SHELL ................................................. 101
PROBLEMS ................................................... 103
3 MOMENTUM
3.1 INTRODUCTION .......................................... 112
3.2 DYNAMICS OF A SYSTEM OF PARTICLES ..................... 113
Center of Mass ........................................ 116
3.3 CONSERVATION OF MOMENTUM .............................. 122
Center of Mass Coordinates ............................ 127
3.4 IMPULSE AND A RESTATEMENT OF THE MOMENTUM RELATION .... 130
3.5 MOMENTUM AND THE FLOW OF MASS ......................... 133
3.6 MOMENTUM TRANSPORT .................................... 139
Note 3.1 CENTER OF MASS .................................... 145
PROBLEMS ................................................... 147
4 WORK AND ENERGY
4.1 INTRODUCTION .......................................... 152
4.2 INTEGRATING THE EQUATION OF MOTION IN ONE DIMENSION ... 153
4.3 THE WORK-ENERGY THEOREM IN ONE DIMENSION .............. 156
4.4 INTEGRATING THE EQUATION OF MOTION IN SEVERAL
DIMENSIONS ............................................ 158
4.5 THE WORK-ENERGY THEOREM ............................... 160
4.6 APPLYING THE WORK-ENERGY THEOREM ...................... 162
4.7 POTENTIAL ENERGY 168 Illustrations of Potential
Energy ................................................ 170
4.8 WHAT POTENTIAL ENERGY TELLS US ABOUT FORCE ............ 173
Stability ............................................. 174
4.9 ENERGY DIAGRAMS ....................................... 176
4.10 SMALL OSCILLATIONS IN A BOUND SYSTEM .................. 178
4.11 NONCONSERVATIVE FORCES ................................ 182
4.12 THE GENERAL LAW OF CONSERVATION OF ENERGY ............. 184
4.13 POWER ................................................. 186
4.14 CONSERVATION LAWS AND PARTICLE COLLISIONS ............. 187
Collisions and Conservation Laws ...................... 188
Elastic and Inelastic Collisions ...................... 188
Collisions in One Dimension ........................... 189
Collisions and Center of Mass Coordinates ............. 190
PROBLEMS ................................................... 194
5 SOME MATHEMATICAL ASPECTS OF FORCE AND ENERGY
5.1 INTRODUCTION .......................................... 202
5.2 PARTIAL DERIVATIVES ................................... 202
5.3 HOW TO FIND THE FORCE IF YOU KNOW THE POTENTIAL
ENERGY ................................................ 206
5.4 THE GRADIENT OPERATOR ................................. 207
5.5 THE PHYSICAL MEANING OF THE GRADIENT .................. 210
Constant Energy Surfaces and Contour Lines ............ 211
5.6 HOW TO FIND OUT IF A FORCE IS CONSERVATIVE ............ 215
5.7 STOKES' THEOREM ....................................... 225
PROBLEMS ................................................... 228
6 ANGULAR MOMENTUM AND FIXED AXIS ROTATION
6.1 INTRODUCTION .......................................... 232
6.2 ANGULAR MOMENTUM OF A PARTICLE ........................ 233
6.3 TORQUE ................................................ 238
6.4 ANGULAR MOMENTUM AND FIXED AXIS ROTATION .............. 248
6.5 DYNAMICS OF PURE ROTATION ABOUT AN AXIS ............... 253
6.6 THE PHYSICAL PENDULUM ................................. 255
The Simple Pendulum ................................... 253
The Physical Pendulum ................................. 257
6.7 MOTION INVOLVING BOTH TRANSLATION AND ROTATION ........ 260
The Work-energy Theorem ............................... 267
6.8 THE BOHR ATOM ......................................... 270
Note 6.1 CHASLES'THEOREM ................................... 274
Note 6.2 PENDULUM MOTION ................................... 276
PROBLEMS ................................................... 279
7 RIGID BODY MOTION AND THE CONSERVATION OF ANGULAR MOMENTUM
7.1 INTRODUCTION .......................................... 288
7.2 THE VECTOR NATURE OF ANGULAR VELOCITY AND ANGULAR
MOMENTUM .............................................. 288
7.3 THE GYROSCOPE ......................................... 295
7.4 SOME APPLICATIONS OF GYROSCOPE MOTION ................. 300
7.5 CONSERVATION OF ANGULAR MOMENTUM ...................... 305
7.6 ANGULAR MOMENTUM OF A ROTATING RIGID BODY ............. 308
Angular Momentum and the Tensor of Inertia ............ 308
Principal Axes ........................................ 313
Rotational Kinetic Energy ............................. 313
Rotation about a Fixed Point .......................... 315
7.7 ADVANCED TOPICS IN THE DYNAMICS OF RIGID BODY
ROTATION .............................................. 316
Introduction .......................................... 316
Torque-free Precession: Why the Earth Wobbles ......... 317
Euler's Equations ..................................... 320
Note 7.1 FINITE AND INFINITESIMAL ROTATIONS ............... 326
Note 7.2 MORE ABOUT GYROSCOPES ............................ 328
Case 1 Uniform Precession ............................. 331
Case 2 Torque-free Precession ......................... 331
Case 3 Nutation ....................................... 331
PROBLEMS .................................................. 334
8 NONINERTIAL SYSTEMS AND FICTITIOUS FORCES
8.1 INTRODUCTION .......................................... 340
8.2 THE GALILEAN TRANSFORMATIONS .......................... 340
8.3 UNIFORMLY ACCELERATING SYSTEMS ........................ 343
8.4 THE PRINCIPLE OF EQUIVALENCE .......................... 346
8.5 PHYSICS IN A ROTATING COORDINATE SYSTEM ............... 355
Time Derivatives and Rotating Coordinates ............. 356
Acceleration Relative to Rotating Coordinates ......... 358
The Apparent Force in a Rotating Coordinate System .... 359
Note 8.1 THE EQUIVALENCE PRINCIPLE AND THE
GRAVITATIONAL RED SHIFT ............................... 369
Note 8.2 ROTATING COORDINATE TRANSFORMATION ............... 371
PROBLEMS ................................................... 372
9 CENTRAL FORCE MOTION
9.1 INTRODUCTION .......................................... 378
9.2 CENTRAL FORCE MOTION AS A ONE BODY PROBLEM ............ 378
9.3 GENERAL PROPERTIES OF CENTRAL FORCE MOTION ............ 380
The Motion Is Confined to a Plane ..................... 380
The Energy and Angular Momentum Are Constants of the
Motion ................................................ 380
The Law of Equal Areas ................................ 382
9.4 FINDING THE MOTION IN REAL PROBLEMS ................... 382
9.5 THE ENERGY EQUATION AND ENERGY DIAGRAMS ............... 383
9.6 PLANETARY MOTION ...................................... 390
9.7 KEPLER'S LAWS ......................................... 400
Note 9.1 PROPERTIES OF THE ELLIPSE ......................... 403
PROBLEMS ................................................... 406
10 THE HARMONIC OSCILLATOR
10.1 INTRODUCTION AND REVIEW ............................... 410
Standard Form of the Solution ......................... 410
Nomenclature .......................................... 411
Energy Considerations ................................. 412
Time Average Values ................................... 413
Average Energy ........................................ 413
10.2 THE DAMPED HARMONIC OSCILLATOR ........................ 414
Energy ................................................ 416
The Q of an Oscillator ................................ 418
10.3 THE FORCED HARMONIC OSCILLATOR ........................ 421
The Undamped Forced Oscillator ........................ 421
Resonance ............................................. 423
The Forced Damped Harmonic Oscillator ................. 424
Resonance in a Lightly Damped System: The Quality
Factor Q .............................................. 426
10.4 RESPONSE IN TIME VERSUS RESPONSE IN FREQUENCY ......... 432
Note 10.1 SOLUTION OF THE EQUATION OF MOTION FOR
THE UNDRIVEN DAMPED OSCILLATOR ........................ 433
The Use of Complex Variables .......................... 433
The Damped Oscillator ................................. 435
Note 10.2 SOLUTION OF THE EQUATION OF MOTION FOR
THE FORCED OSCILLATOR ................................. 437
PROBLEMS ................................................... 438
11 THE SPECIAL THEORY OF RELATIVITY
11.1 THE NEED FOR A NEW MODE OF THOUGHT .................... 442
11.2 THE MICHELSON-MORLEY EXPERIMENT ....................... 445
11.3 THE POSTULATES OF SPECIAL RELATIVITY .................. 450
The Universal Velocity ................................ 451
The Principle of Relativity ........................... 451
The Postulates of special Relativity .................. 452
11.4 THE GALILEAN TRANSFORMATIONS .......................... 453
11.5 THE LORENTZ TRANSFORMATIONS ........................... 455
PROBLEMS ................................................... 459
12 RELATIVISTIC KINEMATICS
12.1 INTRODUCTION .......................................... 462
12.2 SIMULTANEITY AND THE ORDER OF EVENTS .................. 463
12.3 THE LORENTZ CONTRACTION AND TIME DILATION ............. 466
The Lorentz Contraction ............................... 466
Time Dilation ......................................... 468
12.4 THE RELATIVISTIC TRANSFORMATION OF VELOCITY ........... 472
12.5 THE DOPPLER EFFECT .................................... 475
The Doppler Shift in Sound ............................ 475
Relativistic Doppler Effect ........................... 477
The Doppler Effect for an Observer off the Line of
Motion ................................................ 478
12.6 THE TWIN PARADOX ...................................... 480
PROBLEMS ................................................... 484
13 RELATIVISTIC MOMENTUM AND ENERGY
13.1 MOMENTUM .............................................. 490
13.2 ENERGY ................................................ 493
13.3 MASSLESS PARTICLES .................................... 500
13.4 DOES LIGHT TRAVEL AT THE VELOCITY OF LIGHT? ........... 508
PROBLEMS ................................................... 512
14 FOUR-VECTORS AND RELATIVISTIC INVARIANCE
14.1 INTRODUCTION .......................................... 516
14.2 VECTORS AND TRANSFORMATIONS ........................... 516
Rotation about the z Axis ............................. 517
Invariants of a Transformation ........................ 520
The Trans- formation Properties of Physical Laws .... 520
Scalar Invariants ..................................... 521
14.3 MINIKOWSKI SPACE AND FOUR-VECTORS ..................... 521
14.4 THE MOMENTUM-ENERGY FOUR-VECTOR ....................... 527
14.5 CONCLUDING REMARKS .................................... 534
PROBLEMS ................................................... 536
INDEX ......................................................... 539
LIST OF EXAMPLES
1 VECTORS AND KINEMATICS - A FEW MATHEMATICAL PRELIMINARIES
EXAMPLES, CHAPTER 1
1.1 Law of Cosines .......................................... 5
1.2 Work and the Dot Product ................................ 5
1.3 Examples of the Vector Product in Physics ............... 7
1.4 Area as a Vector ........................................ 7
1.5 Vector Algebra .......................................... 9
1.6 Construction of a Perpendicular Vector ................. 10
1.7 Finding v from r ....................................... 16
1.8 Uniform Circular Motion ................................ 17
1.9 Finding Velocity from Acceleration ..................... 20
1.10 Motion in a Uniform Gravitational Field ................ 21
1.11 Nonuniform Acceleration—The Effect of a Radio Wave on
an Ionospheric Electron ................................ 22
1.12 Circular Motion and Rotating Vectors ................... 25
1.13 Circular Motion and Straight Line Motion in Polar
Coordinates ............................................ 34
1.14 Velocity of a Bead on a Spoke .......................... 35
1.15 Off-center Circle ...................................... 35
1.16 Acceleration of a Bead on a Spoke ...................... 37
1.17 Radial Motion without Acceleration ..................... 38
2 NEWTON'S LAWS—THE FOUNDATIONS OF NEWTONIAN MECHANICS
EXAMPLES, CHAPTER 2
2.1 Astronauts in Space—Inertial Systems and Fictitious
Force .................................................. 60
2.2 The Astronauts' Tug-of-war ............................. 70
2.3 Freight Train .......................................... 72
2.4 Constraints ............................................ 74
2.5 Block on String 1 ...................................... 75
2.6 Block on String 2 ...................................... 76
2.7 The Whirling Block ..................................... 76
2.8 The Conical Pendulum ................................... 77
2.9 Turtle in an Elevator .................................. 84
2.10 Block and String 3 ..................................... 87
2.11 Dangling Rope .......................................... 88
2.12 Whirling Rope .......................................... 89
2.13 Pulleys ................................................ 90
2.14 Block and Wedge with Friction .......................... 93
2.15 The Spinning Terror .................................... 94
2.16 Free Motion in a Viscous Medium ........................ 96
2.17 Spring and Block—The Equation for Simple Harmonic
Motion ................................................. 98
2.18 The Spring Gun - An Example Illustrating Initial
Conditions ............................................. 99
3 MOMENTUM
EXAMPLES, CHAPTER 3
3.1 The Bola .............................................. 115
3.2 Drum Major's Baton .................................... 117
3.3 Center of Mass of a Nonuniform Rod .................... 119
3.4 Center of Mass of a Triangular Sheet .................. 120
3.5 Center of Mass Motion ................................. 122
3.6 Spring Gun Recoil ..................................... 123
3.7 Earth, Moon, and Sun - A Three Body System ............ 125
3.8 The Push Me-Pull You .................................. 128
3.9 Rubber Ball Rebound ................................... 131
3.10 How to Avoid Broken Ankles ............................ 132
3.11 Mass Flow and Momentum ................................ 134
3.12 Freight Car and Hopper ................................ 135
3.13 Leaky Freight Car ..................................... 136
3.14 Rocket in Free Space .................................. 138
3.15 Rocket in a Gravitational Field ....................... 139
3.16 Momentum Transport to a Surface ....................... 141
3.17 A Dike at the Bend of a River ......................... 143
3.18 Pressure of a Gas ..................................... 144
4 WORK AND ENERGY
EXAMPLES, CHAPTER 4
4.1 Mass Thrown Upward in a Uniform Gravitational Field ... 154
4.2 Solving the Equation of Simple Harmonic Motion ........ 154
4.3 Vertical Motion in an Inverse Square Field ............ 156
4.4 The Conical Pendulum .................................. 161
4.5 Escape Velocity—The General Case ...................... 162
4.6 The Inverted Pendulum ................................. 164
4.7 Work Done by a Uniform Force .......................... 165
4.8 Work Done by a Central Force .......................... 167
4.9 A Path-dependent Line Integral ........................ 167
4.10 Parametric Evaluation of a Line Integral .............. 168
4.11 Potential Energy of a Uniform Force Field ............. 170
4.12 Potential Energy of an Inverse Square Force ........... 171
4.13 Bead, Hoop, and Spring ................................ 172
4.14 Energy and Stability - The Teeter Toy ................. 175
4.15 Molecular Vibrations .................................. 179
4.16 Small Oscillations .................................... 181
4.17 Block Sliding down Inclined Plane ..................... 183
4.18 Elastic Collision of Two Balls ........................ 190
4.19 Limitations on Laboratory Scattering Angle ............ 193
5 SOME MATHEMATICAL ASPECTS OF FORCE AND ENERGY
EXAMPLES, CHAPTER 5
5.1 Partial Derivatives ................................... 203
5.2 Applications of the Partial Derivative ................ 205
5.3 Gravitational Attraction by a Particle ................ 208
5.4 Uniform Gravitational Field ........................... 209
5.5 Gravitational Attraction by Two Point Masses .......... 209
5.6 Energy Contours for a Binary Star System .............. 212
5.7 The Curl of the Gravitational Force ................... 219
5.8 A Nonconservative Force ............................... 220
5.9 A Most Unusual Force Field ............................ 221
5.10 Construction of the Potential Energy Function ......... 222
5.11 How the Curl Got Its Name ............................. 224
5.12 Using Stokes' Theorem ................................. 227
6 ANGULAR MOMENTUM AND FIXED AXIS ROTATION
EXAMPLES, CHAPTER 6
6.1 Angular Momentum of a Sliding Block ................... 236
6.2 Angular Momentum of the Conical Pendulum .............. 237
6.3 Central Force Motion and the Law of Equal Areas ....... 240
6.4 Capture Cross Section of a Planet ..................... 241
6.5 Torque on a Sliding Block ............................. 244
6.6 Torque on the Conical Pendulum ........................ 245
6.7 Torque due to Gravity ................................. 247
6.8 Moments of Inertia of Some Simple Objects ............. 250
6.9 The Parallel Axis Theorem ............................. 252
6.10 Atwood's Machine with a Massive Pulley ................ 254
6.11 Grandfather's Clock ................................... 256
6.12 Kater's Pendulum ...................................... 258
6.13 The Doorstep .......................................... 259
6.14 Angular Momentum of a Rolling Wheel ................... 262
6.15 Disk on Ice ........................................... 264
6.16 Drum Rolling down a Plane ............................. 265
6.17 Drum Rolling down a Plane: Energy Method .............. 268
6.18 The Falling Stick ..................................... 269
7 RIGID BODY L MOTION AND THE CONSERVATION OF ANGULAR
MOMENTUM
EXAMPLES, CHAPTER 7
7.1 Rotations through Finite Angles ....................... 289
7.2 Rotation in the xy Plane .............................. 291
7.3 Vector Nature of Angular Velocity ..................... 291
7.4 Angular Momentum of a Rotating Skew Rod ............... 292
7.5 Torque on the Rotating Skew Rod ....................... 293
7.6 Torque on the Rotating Skew Rod (Geometric Method) .... 294
7.7 Gyroscope Precession .................................. 298
7.8 Why a Gyroscope Precesses ............................. 299
7.9 Precession of the Equinoxes ........................... 300
7.10 The Gyrocompass Effect ................................ 301
7.11 Gyrocompass Motion .................................... 302
7.12 The Stability of Rotating Objects ..................... 304
7.13 Rotating Dumbbell ..................................... 310
7.14 The Tensor of Inertia for a Rotating Skew Rod ......... 312
7.15 Why Flying Saucers Make Better Spacecraft than Do
Flying Cigars ......................................... 314
7.16 Stability of Rotational Motion ........................ 322
7.17 The Rotating Rod ...................................... 323
7.18 Euler's Equations and Torque-free Precession .......... 324
8 NONINERTIAL SYSTEMS AND FICTITIOUS FORCES
EXAMPLES, CHAPTER 8
8.1 The Apparent Force of Gravity ......................... 346
8.2 Cylinder on an Accelerating Plank ..................... 347
8.3 Pendulum in an Accelerating Car ....................... 347
8.4 The Driving Force of the Tides ........................ 350
8.5 Equilibrium Height of the Tide ........................ 352
8.6 Surface of a Rotating Liquid .......................... 362
8.7 The Coriolis Force .................................... 363
8.8 Deflection of a Falling Mass .......................... 364
8.9 Motion on the Rotating Earth .......................... 366
8.10 Weather Systems ....................................... 366
8.11 The Foucault Pendulum ................................. 369
9 CENTRAL FORCE MOTION
EXAMPLES, CHAPTER 9
9.1 Noninteracting Particles .............................. 384
9.2 The Capture of Comets ................................. 387
9.3 Perturbed Circular Orbit .............................. 388
9.4 Hyperbolic Orbits ..................................... 393
9.5 Satellite Orbit ....................................... 396
9.6 Satellite Maneuver .................................... 398
9.7 The Law of Periods .................................... 403
10 THE HARMONIC OSCILLATOR EXAMPLES, CHAPTER 10
10.1 Initial Conditions and the Frictionless Harmonic
Oscillator ............................................ 411
10.2 The Q of Two Simple Oscillators ....................... 419
10.3 Graphical Analysis of a Damped Oscillator ............. 420
10.4 Forced Harmonic Oscillator Demonstration .............. 424
10.5 Vibration Eliminator .................................. 428
11 THE SPECIAL THEORY OF RELATIVITY
EXAMPLES, CHAPTER 11
11.1 The Galilean Transformations .......................... 453
11.2 A Light Pulse as Described by the Galilean
Transformations ....................................... 455
12 RELATIVISTIC KINEMATICS
EXAMPLES, CHAPTER 12
12.1 Simultaneity .......................................... 463
12.2 An Application of the Lorentz Transformations ......... 464
12.3 The Order of Events: Timelike and Spacelike
Intervals ............................................. 465
12.4 The Orientation of a Moving Rod ....................... 467
12.5 Time Dilation and Meson Decay ......................... 468
12.6 The Role of Time Dilation in an Atomic Clock .......... 470
12.7 The Speed of Light in a Moving Medium ................. 474
12.8 Doppler Navigation .................................... 479
13 RELATIVISTIC MOMENTUM AND ENERGY
EXAMPLES, CHAPTER 13
13.1 Velocity Dependence of the Electron's Mass ............ 492
13.2 Relativistic Energy and Momentum in an Inelastic
Collision ............................................. 496
13.3 The Equivalence of Mass and Energy .................... 498
13.4 The Photoelectric Effect .............................. 502
13.5 Radiation Pressure of Light ........................... 502
13.6 The Compton Effect .................................... 503
13.7 Pair Production ....................................... 505
13.8 The Photon Picture of the Doppler Effect .............. 507
13.9 The Rest Mass of the Photon ........................... 510
13.10 Light from a Pulsar .................................. 510
14 FOUR-VECTORS AND RELATIVISTIC INVARIANCE
EXAMPLES, CHAPTER 14
14.1 Transformation Properties of the Vector Product ....... 518
14.2 A Non-vector .......................................... 519
14.3 Time Dilation ......................................... 524
14.4 Construction of a Four-vector: The Four-velocity ...... 525
14.5 The Relativistic Addition of Velocities ............... 526
14.6 The Doppler Effect, Once More ......................... 530
14.7 Relativistic Center of Mass Systems ................... 531
14.8 Pair Production in Electron-electron Collisions ....... 533
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