Dynamics and Relativity

Editors’ Preface to the Manchester Physics Series xi

Author’s Preface xiii

I Introductory Dynamics 1

1 Space, Time and Motion 3

1.1 Defining Space and Time 3

1.1.1 Space and the classical particle 4

1.1.2 Unit vectors 6

1.1.3 Addition and subtraction of vectors 6

1.1.4 Multiplication of vectors 7

1.1.5 Time 8

1.1.6 Absolute space and space-time 10

1.2 Vectors and Co-ordinate Systems 11

1.3 Velocity and Acceleration 14

1.3.1 Frames of reference 16

1.3.2 Relative motion 16

1.3.3 Uniform acceleration 18

1.3.4 Velocity and acceleration in plane-polar co-ordinates: uniform circular motion 20

1.4 Standards and Units 21

2 Force, Momentum and Newton’s Laws 25

2.1 Force and Static Equilibrium 25

2.2 Force and Motion 31

2.2.1 Newton’s Third Law 35

2.2.2 Newton’s bucket and Mach’s principle 39

2.3 Applications of Newton’s Laws 41

2.3.1 Free body diagrams 41

2.3.2 Three worked examples 42

2.3.3 Normal forces and friction 46

2.3.4 Momentum conservation 49

2.3.5 Impulse 51

2.3.6 Motion in fluids 51

3 Energy 55

3.1 Work, Power and Kinetic Energy 56

3.2 Potential Energy 61

3.2.1 The stability of mechanical systems 64

3.2.2 The harmonic oscillator 65

3.2.3 Motion about a point of stable equilibrium 67

3.3 Collisions 68

3.3.1 Zero-momentum frames 68

3.3.2 Elastic and inelastic collisions 71

3.4 Energy Conservation in Complex Systems 75

4 Angular Momentum 81

4.1 Angular Momentum of a Particle 81

4.2 Conservation of Angular Momentum in Systems of Particles 83

4.3 Angular Momentum and Rotation About a Fixed Axis 86

4.3.1 The parallel-axis theorem 94

4.4 Sliding and Rolling 95

4.5 Angular Impulse and the Centre of Percussion 97

4.6 Kinetic Energy of Rotation 99

II Introductory Special Relativity 103

5 The Need for a New Theory of Space and Time 105

5.1 Space and Time Revisited 105

5.2 Experimental Evidence 108

5.2.1 The Michelson-Morley experiment 108

5.2.2 Stellar aberration 110

5.3 Einstein’s Postulates 113

6 Relativistic Kinematics 115

6.1 Time Dilation, Length Contraction and Simultaneity 115

6.1.1 Time dilation and the Doppler effect 116

6.1.2 Length contraction 121

6.1.3 Simultaneity 123

6.2 Lorentz Transformations 124

6.3 Velocity Transformations 129

6.3.1 Addition of velocities 129

6.3.2 Stellar aberration revisited 130

7 Relativistic Energy and Momentum 135

7.1 Momentum and Energy 135

7.1.1 The equivalence of mass and energy 142

7.1.2 The hint of an underlying symmetry 144

7.2 Applications in Particle Physics 145

7.2.1 When is relativity important? 146

7.2.2 Two useful relations and massless particles 149

7.2.3 Compton scattering 152

III Advanced Dynamics 157

8 Non-inertial Frames 159

8.1 Linearly Accelerating Frames 159

8.2 Rotating Frames 161

8.2.1 Motion on the earth 165

9 Gravitation 173

9.1 Newton’s Law of Gravity 174

9.2 The Gravitational Potential 177

9.3 Reduced Mass 182

9.4 Motion in a Central Force 184

9.5 Orbits 186

10 Rigid Body Motion 197

10.1 The Angular Momentum of a Rigid Body 198

10.2 The Moment of Inertia Tensor 200

10.2.1 Calculating the moment of inertia tensor 203

10.3 Principal Axes 207

10.4 Fixed-axis Rotation in the Lab Frame 212

10.5 Euler’s Equations 214

10.6 The Free Rotation of a Symmetric Top 216

10.6.1 The body-fixed frame 216

10.6.2 The lab frame 218

10.6.3 The wobbling earth 223

10.7 The Stability of Free Rotation 224

10.8 Gyroscopes 226

10.8.1 Gyroscopic precession 226

10.8.2 Nutation of a gyroscope 232

IV Advanced Special Relativity 237

11 The Symmetries of Space and Time 239

11.1 Symmetry in Physics 239

11.1.1 Rotations and translations 240

11.1.2 Translational symmetry 245

11.1.3 Galilean symmetry 246

11.2 Lorentz Symmetry 247

12 Four-vectors and Lorentz Invariants 253

12.1 The Velocity Four-vector 254

12.2 The Wave Four-vector 255

12.3 The Energy-momentum Four-vector 258

12.3.1 Further examples in relativistic kinematics 259

12.4 Electric and Magnetic Fields 262

13 Space-time Diagrams and Causality 267

13.1 Relativity Preserves Causality 270

13.2 An Alternative Approach 272

14 Acceleration and General Relativity 279

14.1 Acceleration in Special Relativity 279

14.1.1 Twins paradox 280

14.1.2 Accelerating frames of reference 282

14.2 A Glimpse of General Relativity 288

14.2.1 Gravitational fields 290

A Deriving the Geodesic Equation 295

B Solutions to Problems 297