CLASSICAL MECHANICS
1 Physical Quantities and Units of Measurement 1 11 Physics and the Scientific Method 1 12 Measurable Quantities and the International System 3 121 Distances 5 122 Times 6 123 Masses 6 13 Measuring Physical Quantities 8 131 Direct and Indirect Measurements 8 132 The Distance Ladder 9 133 Mass Measurements Out of Range for Scales 9 134 From Pico to Tera 10 14 The Concept of Space and Time 11 141 The Euclidean Space 11 142 Newtonian Space and Time 12 143 Four-Dimensional Spacetime (*) 13 144 Cosmological Time 14 15 Time Synchronization and Dissemination 15 16 Time Measurements Without Periodic Phenomena (*) 17 161 The Law of Radioactive Decay 18 162 Radiocarbon Dating 20 17 Dimensional Analysis 22 18 Questions and Exercises 24 2 Vectors and Operations with Vectors 27 21 Introduction: Scalars, Vectors and Tensors 27 22 Cartesian Coordinate Systems 28 221 Right-Handed Coordinate System and Right-Hand Rule 29 23 Representation of Vectors 30 231 Vectors in Intrinsic Representation 30 232 Vectors in Cartesian Representation 31 233 Unit Vectors in Cartesian Representation 32 24 Product of a Scalar and a Vector 34 25 Sum and Difference of Vectors 34 26 Scalar Product 36 261 Scalar Product in Intrinsic Representation 37 262 Scalar Product in Cartesian Representation 37 27 Vector Product Between Vectors 38 271 Vector Product in Intrinsic Representation 38 272 Vector Product in Cartesian Representation 40 28 Area and Volume in Vector Spaces (*) 40 29 The Equals Are Not All Equal 43 210 Laws and Principles, Physics and Mathematics 44 211 Questions 46 3 Kinematics of the Particle 47 31 Introduction and Definitions 47 32 Uniform and Uniformly Accelerated Motion 49 33 Equation of Motion, Velocity and Acceleration 56 331 Equation of Motion 57 332 Velocity, Definition 59 333 Acceleration, Definition 60 34 The Direct Kinematics Problem 60 341 Infinitesimal Displacement and Infinitesimal Path 61 342 Composition of Motions and Trajectory 61 35 The Inverse Kinematics Problem 62 36 Uniform and Non-uniform Circular Motion 64 361 Uniform Circular Motion 65 362 The Cartesian Representation 65 363 Non-uniform Circular Motion 67 37 Polar Plane Coordinates 68 371 Circular Motion with Co-moving Unit Vectors 68 372 Definition of Polar and Cylindrical Coordinates 69 38 The Intrinsic Coordinates 70 39 Poisson’s Rules for Moving Unit Vectors 72 310 Motion on any Trajectory (*) 73 311 Questions and Exercises 74 4 Forces and the Dynamics of the Particle 79 41 Introduction 79 42 Measuring Forces 81 421 Weight and Anthropomorphic “effort” 81 422 The Dynamometer 82 423 The Weight is a Force 84 43 The Vector Nature of Forces 86 44 Constraint Forces 88 45 Contact Friction Between Solids 89 46 Newton’s First Law of Motion 92 47 Newton’s Second Law of Motion 93 48 Kinematic Effects of Some Forces 94 481 Weight 94 482 Fall with the Presence of Viscous Friction 95 49 Harmonic Oscillators 98 491 The Simple Pendulum 98 492 The Elastic Force and the Harmonic Oscillator 103 410 What Do We Know Today About Forces 104 4101 Fundamental Interactions 104 4102 Force Fields 106 4103 The Arrow of Time 106 411 Questions and Exercises 108 5 Frames of Reference in Relative Motion 113 51 Galilean Relativity Principle 113 52 Inertial Frames of Reference 116 53 Non-inertial Frames of Reference 119 54 Pseudo-forces in Non-inertial Frames 123 541 Dynamics in an Accelerated Vehicle 123 542 Dynamics in a Rotating Frame 125 543 Earth’s Drag Acceleration 126 55 Coriolis Force in Guglielmini’s Experiment (*) 128 56 The Coriolis Force on the Motion of Terrestrial Fluids 132 57 Generalized Principles of Relativity 133 58 Questions and Exercises 134 6 Work and Energy 137 61 Introduction 137 62 Definition of Work 138 621 Work in Vectors’ Intrinsic Representation 138 622 Work in Cartesian Coordinates 139 623 Measurement Units of Work 140 63 The Work-Energy Theorem 140 64 Examples of Calculating the Work of a Force 142 641 Work of the Weight Force 142 642 Work of the Elastic Force 143 643 Work of the Dynamic Friction 143 644 Anthropomorphic Work (“work” in Common Parlance) 145 65 Conservative and Non-conservative Forces 146 66 Potential Energy 148 67 Mechanical Energy and Its Conservation 149 671 Definition of Mechanical Energy 149 672 Mechanical Energy with Non-conservative Forces 150 68 Exact Differentials and Potential Energy 151 681 Force is the Negative Gradient of Potential Energy 151 69 Differential Operators: Divergence, Curl and Laplacian (*) 152 691 Divergence 152 692 Curl 153 693 Laplacian 153 694 Schwarz’s Theorem 153 610 Conservative Forces: Null Curl 154 611 Central Forces are Conservative 155 612 Power 157 613 Towards the Principle of Energy Conservation 158 614 Questions and Exercises 159 7 Dynamics of Mechanical Systems 165 71 Mass Distribution: Discrete and Continuous Systems 165 711 Systems of Material Points 165 712 Continuous Systems 166 72 Degrees of Freedom 168 73 Center of Mass 170 731 The Center of Mass of a System of Points 170 732 The Center of Mass of a Continuous Body 172 74 Torque 173 75 Linear Momentum and Angular Momentum 177 751 Linear Momentum of a System 177 752 Angular Momentum of a System 178 76 Conservation of Linear Momentum 179 761 Euler’s First Law 180 77 Conservation of Angular Momentum 182 771 Euler’s Second Law 183 78 Newton’s Third Law of Motion 183 781 Comment on Newton’s Laws 185 79 Properties of the Center of Mass Frame 186 791 Linear Momentum P’ in the CM Frame 187 792 Intrinsic Angular Momentum (Spin) L’ in the CM Frame 187 793 Euler’s Second Law in the CM Frame 188 794 Kinetic Energy T’ in the CM Frame 189 710 Equilibrium of a Rigid Body 190 711 Questions and Exercises 191 8 Collisions and Decays 195 81 Introduction 195 82 Impulsive Forces 197 83 Elastic Collision Between Two Bodies 199 831 The One-Dimensional Case 200 832 Two-Dimensional Case and Constraining Forces 202 833 Let Us Reflect on the Nature of the Constraining Forces 203 84 Collisions in the Center-of-Mass Frame 204 841 One-Dimensional Case in the Center-of-Mass Frame 205 85 Motion of a Rocket 207 86 Energy-Mass Conservation in Collisions and Decays 210 87 Conservation Laws in Two or Three Body Decays 213 871 Alpha Decay (Two-Body Process) (*) 215 872 Beta Decay (Three-Body Process) (*) 216 88 Partially Elastic Collisions 219 89 Questions and Exercises 221 9 Considerations on Vectors 225 91 Isotropy and Homogeneity of the Universe 225 92 Translation of Reference Frames 227 93 Rotation of Reference Frames 229 94 Reflexion of Reference Frames 232 95 Not all Triplets are Vectors 234 96 Polar and Axial Vectors 235 961 How Can Axial Vectors be Distinguished 236 962 Scalar and Pseudoscalar Quantities 238 963 Pseudoscalar Quantities in Nuclear Phenomena 239 10 Newton’s Law of Gravitation 241 101 Introduction 241 102 Pre-Galilean Astronomical Measurements 242 1021 Earth’s Radius 242 1022 Earth-Moon Distance 244 1023 Earth-Sun Distance 244 103 The Apple and the Moon 245 104 Law of Universal Gravitation 247 1041 The Inverse of the Square of the Distance 248 105 Inertial Mass and Gravitational Mass 250 106 Gravitational Potential Energy 252 1061 Potential Energy of the Weight Force 253 1062 Classical Limits of Relativity and Quantum Mechanics 254 107 Escape Speed From a Celestial Body of Mass M 255 1071 Event Horizon 257 108 Measurement of G: The Torsion Balance 259 109 Spherical Coordinates (*) 262 1091 Line, Surface and Volume Elements 263 1092 Gravitational Potential Energy: Second Method 266 1010 Mass in the Center of the Sphere (*) 266 1011 Questions 270 11 Motions in Gravitational Fields 273 111 Historical Introduction 273 112 Kepler’s Empirical Laws 276 113 The Two-Body System 278 114 Angular Momentum, Kepler’s 1st and 2nd Laws 280 1141 Kepler’s First Law, Plane Orbits 281 1142 Kepler’s Second Law 282 115 Kepler’s Third Law 283 116 Mechanical Energy of the Two-Body System 284 1161 Qualitative Solutions for the Effective Potential 285 117 Kepler’s First Law, Elliptic Orbits (*) 288 1171 Conic Functions 289 1172 First Integral 290 1173 Eccentricity Versus Energy and Angular Momentum 292 1174 Elliptic Solutions: Semi-axes as a Function of E, L 294 1175 Degeneracy in Classical and Quantum Physics 295 118 Kepler’s Third Law, Revised 296 1181 The Black Hole in the Center of the Galaxy 297 119 Two Neutron Stars 299 1110 Triumphs and Falls of Newtonian Theory 302 1111 Gravitational Indications for Dark Matter 304 1112 Questions and Exercises 307 12 Dynamics of Rigid Bodies 311 121 Introduction 311 122 Angular Momentum and Angular Velocity 312 123 Moment of Inertia of Rigid Bodies 315 1231 Moment of Inertia of a Cylinder and Hollow Cylinder 315 1232 Moment of Inertia of a Rod 316 1233 Moment of Inertia of a Rod on a Disk 317 124 Conservation of Angular Momentum and Angular Velocity 318 125 An Application of the II Euler’s Law 319 126 Huygens-Steiner Theorem for Moments of Inertia 321 127 Center of Gravity 323 1271 Physical Pendulum 325 1272 Torsion Pendulum 326 128 Moment of Inertia Tensor (*) 327 129 Rotational Kinetic Energy 330 1291 Body Rolling Without Sliding 330 1292 The Motion of the Wheel 332 1210 The Motion of the Spinning Top (*) 333 1211 Questions and Summary Exercise 335 13 Considerations on Energy 345 131 Work Needed to Build a System 345 132 Potential Energy of a Bound Spherical System 348 1321 Age of the Sun 350 1322 Energy Conservation in Stellar Gravitational Collapse (*) 352 133 Gravitational Field and Potential 355 134 First Integral from Energy Conservation 357 1341 Application to a Falling Particle 358 135 Motion in a Potential Energy Field 359 1351 Stable and Unstable Equilibrium 359 1352 Permitted and Forbidden Regions 361 136 More on Harmonic Motion (*) 362 1361 Complex Numbers 363 1362 Harmonic Oscillator with Complex Numbers 364 1363 Mechanical Energy of the Harmonic Oscillator 366 137 Damped Harmonic Oscillator (*) 366 1371 Overdamped Motion 368 1372 Discussion on Mechanical Energy and Developments 371 138 Developments and Problems of Classical Mechanics 372 1381 Lagrangian and Hamiltonian Formalisms 372 1382 Determinism in Newtonian Mechanics 373 139 Epilogue 374 1310 Questions and Exercises 375