Table of Contents
Preface xi
Chapter 1 Mechanical Models 1
1.1 Introduction 1
1.2 Models 3
1.3 Estimates 5
1.4 Units and Dimensions 6
1.5 Equations 8
1.6 Chapter Summary 11
Chapter 2 Forces 13
2.1 Action and Reaction 14
2.2 Forces in Equilibrium 16
2.3 Horse Before Cart 20
2.4 Static Friction 22
2.5 Sliding Friction 22
2.6 A Friction Paradox 23
2.7 Rolling Friction 24
2.8 Contact Area 24
2.9 Torque: The Moment or Couple of a Force 25
2.10 Condition for Static Equilibrium 27
2.11 Center of Gravity 28
2.12 An Example 29
2.13 Problem Summary 31
2.14 Inclined Planes 33
2.15 Pulling at an Angle on a Flat Plane 37
2.16 Pulling at an Angle on an Inclined Plane 39
2.17 Solution of Problem 2 41
2.18 Tipping Point 42
2.19 Tipping on an Inclined Plane 45
2.20 Levers 47
2.21 Stress and Strain 49
2.22 Chapter Summary 50
2.23 Exercises 51
Chapter 3 Kinematics 55
3.1 Constant Speed 55
3.2 Constant Acceleration 56
3.3 A Body Projected Vertically under Gravity 59
3.4 Motion in Two Dimensions 61
3.5 Addition of Velocities 62
3.6 Projectile Motion 63
3.7 Approximate Solutions 72
3.8 Air Resistance 73
3.9 Addition of Accelerations 74
3.10 Other Forms of Acceleration 75
3.11 Chapter Summary 76
3.12 Exercises 77
Chapter 4 Energy 79
4.1 Work 79
4.2 Kinetic Energy and Work 80
4.3 Definition of Mass 82
4.4 Work and Potential Energy 83
4.5 Conservative Forces 85
4.6 Nonconservative Forces 87
4.7 Friction and "Zero Work Forces" 88
4.8 Conservation of Energy 88
4.9 Units for Energy 89
4.10 Example 90
4.11 Bound Systems 91
4.12 Virtual Work 92
4.13 Elastic Energy 92
4.14 Example - Bungee Jumping 93
4.15 Solution to the Problem 95
4.16 Chapter Summary 96
4.17 Exercises 97
Chapter 5 Motion 101
5.1 Newtonian Dynamics 102
5.2 Equations of Motion 104
5.3 An Example 106
5.4 Motion in Higher Dimensions 106
5.5 Rate of Doing Work 107
5.6 Inertial Forces 108
5.7 Systems of Particles 109
5.8 Example: Motion under Air Resistance 110
5.9 Sky Dive 112
5.10 Tower Problem 114
5.11 Model 1 114
5.12 Model 2: Terminal Speed 115
5.13 Model 3 117
5.14 The Shape of the Shot 121
5.15 Upthrust 122
5.16 Simple Harmonic Motion 124
5.17 Why SHM Is Important 127
5.18 Energy of a Harmonic Oscillator 127
5.19 Chapter Summary 128
5.20 Exercises 129
Chapter 6 Momentum 131
6.1 Conservation 131
6.2 Conservation and Invariance 133
6.3 Impulse 133
6.4 Collisions in One Dimension 134
6.5 Center of Momentum Frame 137
6.6 Inelastic Collisions 139
6.7 The Problem 140
6.8 Collisions in Two Dimensions 141
6.9 Collision Timescales 142
6.10 Rocket Equation 143
6.11 Chapter Summary 144
6.12 Exercises 144
Chapter 7 Orbital Motion 147
7.1 Angular Speed: Geometric Approach 147
7.2 Angular Speed: Algebraic Approach 149
7.3 Angular Velocity as a Vector 150
7.4 Angular Acceleration: Geometric Approach 150
7.5 Angular Acceleration: Algebraic Approach 152
7.6 Angular Momentum 152
7.7 Circular Motion: Dynamics 153
7.8 Particle in a Magnetic Field 155
7.9 Centrifugal Force 156
7.10 Rotating Frames 157
7.11 Gravity 159
7.12 Extended Bodies 160
7.13 Gravitational Potential and Potential Energy 162
7.14 Escape Speed 164
7.15 Radial Mall 165
7.16 Circular Orbits 168
7.17 Virial Theorem 170
7.18 Changing Orbits 170
7.19 Elliptical Orbits 171
7.20 Properties of the Ellipse 174
7.21 Kepler's Laws 177
7.22 Derivation of Kepler's Laws for Elliptical Orbits 178
7.23 Extended Bodies: Multipole Expansion 179
7.24 The Poisson Equation 181
7.25 Motion Inside Matter: Falling Through the Earth 182
7.26 Tidal Forces 184
7.27 Solution of the Problem: Roche Limit 185
7.28 What Is Gravity? 186
7.29 Chapter Summary 190
7.30 Exercises 191
Chapter 8 Oscillations 193
8.1 Resonance 193
8.2 Damping 198
8.3 Quality Factor 201
8.4 Forced Oscillations 202
8.5 Impedance 204
8.6 Energy and Phase 205
8.7 Power Curve 206
8.8 Complex Exponentials 208
8.9 Fourier Analysis 210
8.10 Coupled Oscillators 211
8.11 Coupled Oscillators with Dissipation 215
8.12 Forced Coupled Oscillators 218
8.13 Chapter Summary 220
8.14 Exercises 220
Chapter 9 Rigid Bodies 223
9.1 Rotational Energy 224
9.2 Moments of Inertia 224
9.3 Angular Momentum 227
9.4 The Receding Moon 229
9.5 Space Tether 230
9.6 Equation of Motion 231
9.7 Compound Pendulum 232
9.8 A Model of Running 233
9.9 Rolling and Slipping 233
9.10 Galileo's Inclined Plane 235
9.11 Spin and Precession 236
9.12 Euler Equations 238
9.13 Chapter Summary 239
9.14 Exercises 240
Chapter 10 Stability of Motion 241
10.1 Perturbations 241
10.2 Cubic Potential 242
10.3 Motion of the Planet Mercury 244
10.4 Stability: General Formulation 246
10.5 An Example of Stability: Non-Newtonian Orbits 246
10.6 A Warning 247
10.7 Solution to Problem 248
10.8 Phase Portraits: Harmonic Oscillator 254
10.9 Phase Portraits: Damped Oscillator 256
10.10 Chaos 257
10.11 Chapter Summary 258
10.12 Exercises 259
Chapter 11 Lagrangian and Hamiltonian Mechanics 261
11.1 Principle of Least Action 262
11.2 Euler-Lagrange Equations 263
11.3 Newton's Laws 265
11.4 Simple Harmonic Oscillator 265
11.5 Acceleration in Polar Coordinates 266
11.6 Rotating Coordinate System 266
11.7 Bead on a Wire 268
11.8 Cycloidal Pendulum 269
11.9 Spherical Pendulum 270
11.10 Compound Pendulum 274
11.11 Small Oscillations Revisited 277
11.12 An Example 278
11.13 Hamiltonian Mechanics 281
11.14 Conservation Laws and Noerher's Theorem 283
11.15 Energy and the Hamiltonian 285
11.16 Action Angle Variables and Integrable Systems 286
11.17 Quantum Theory 288
11.18 Chapter Summary 290
11.19 Exercises 290
Index 293
