UPI_text_7-31-19-2.pdf - University Physics I Classical Mechanics Julio Gea-Banacloche First revision Fall 2019 This work is licensed under a Creative

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Unformatted text preview: University Physics I: Classical Mechanics Julio Gea-Banacloche First revision, Fall 2019 This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. Developed thanks to a grant from the University of Arkansas Libraries ii Contents Preface i 1 Reference frames, displacement, and velocity 1 1.1 1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Particles in classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Aside: the atomic perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Position, displacement, velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Reference frame changes and relative motion . . . . . . . . . . . . . . . . . . . . . . 14 1.4 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5.1 Motion with (piecewise) constant velocity . . . . . . . . . . . . . . . . . . . . 21 1.5.2 Addition of velocities, relative motion . . . . . . . . . . . . . . . . . . . . . . 24 iii iv CONTENTS 1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 Acceleration 2.1 The law of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.1 2.2 31 Inertial reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.1 Average and instantaneous acceleration . . . . . . . . . . . . . . . . . . . . . 35 2.2.2 Motion with constant acceleration . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2.3 Acceleration as a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2.4 Acceleration in different reference frames . . . . . . . . . . . . . . . . . . . . 41 2.3 Free fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5.1 2.6 Motion with piecewise constant acceleration . . . . . . . . . . . . . . . . . . . 46 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3 Momentum and Inertia 3.1 3.2 51 Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.1.1 Relative inertia and collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.1.2 Inertial mass: definition and properties . . . . . . . . . . . . . . . . . . . . . 55 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.1 Conservation of momentum; isolated systems . . . . . . . . . . . . . . . . . . 56 CONTENTS 3.3 v Extended systems and center of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.1 Center of mass motion for an isolated system . . . . . . . . . . . . . . . . . . 59 3.3.2 Recoil and rocket propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.6 3.5.1 Reading a collision graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.5.2 Collision in different reference frames, center of mass, and recoil . . . . . . . 65 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4 Kinetic Energy 4.1 4.2 69 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1.1 Kinetic energy in collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.1.2 Relative velocity and coefficient of restitution . . . . . . . . . . . . . . . . . . 74 “Convertible” and “translational” kinetic energy . . . . . . . . . . . . . . . . . . . . 76 4.2.1 Kinetic energy and momentum in different reference frames . . . . . . . . . . 79 4.3 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5 4.4.1 Collision graph revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4.2 Inelastic collision and explosive separation . . . . . . . . . . . . . . . . . . . . 83 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5 Interactions and energy 89 vi CONTENTS 5.1 Conservative interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1.1 Potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.1.2 Potential energy functions and “energy landscapes” . . . . . . . . . . . . . . 94 5.2 Dissipation of energy and thermal energy . . . . . . . . . . . . . . . . . . . . . . . . 96 5.3 Fundamental interactions, and other forms of energy . . . . . . . . . . . . . . . . . . 98 5.4 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.5 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.7 5.8 5.6.1 Inelastic collision in the middle of a swing . . . . . . . . . . . . . . . . . . . . 103 5.6.2 Kinetic energy to spring potential energy: block collides with spring . . . . . 104 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.7.1 Two carts colliding and compressing a spring . . . . . . . . . . . . . . . . . . 106 5.7.2 Getting the potential energy function from collision data . . . . . . . . . . . . 107 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6 Interactions, part 2: Forces 6.1 113 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.1.1 Forces and systems of particles . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 Forces and potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.3 Forces not derived from a potential energy . . . . . . . . . . . . . . . . . . . . . . . . 120 6.3.1 Tensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 CONTENTS vii 6.3.2 Normal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.3.3 Static and kinetic friction forces . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.3.4 Air resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.4 Free-body diagrams 6.5 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.6.1 Dropping an object on a weighing scale . . . . . . . . . . . . . . . . . . . . . 129 6.6.2 Speeding up and slowing down . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 7 Impulse, Work and Power 137 7.1 Introduction: work and impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.2 Work on a single particle 7.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Work done by the net force, and the Work-Energy Theorem . . . . . . . . . . 140 7.3 The “center of mass work” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.4 Work done on a system by all the external forces . . . . . . . . . . . . . . . . . . . . 142 7.4.1 The no-dissipation case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.4.2 The general case: systems with dissipation 7.4.3 Energy dissipated by kinetic friction . . . . . . . . . . . . . . . . . . . . . . . 150 . . . . . . . . . . . . . . . . . . . 147 7.5 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.6 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 viii CONTENTS 7.7 7.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.7.1 Braking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.7.2 Work, energy and the choice of system: dissipative case . . . . . . . . . . . . 154 7.7.3 Work, energy and the choice of system: non-dissipative case . . . . . . . . . . 155 7.7.4 Jumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 8 Motion in two dimensions 161 8.1 Dealing with forces in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8.2 Projectile motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.3 Inclined planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.4 Motion on a circle (or part of a circle) . . . . . . . . . . . . . . . . . . . . . . . . . . 169 8.4.1 Centripetal acceleration and centripetal force . . . . . . . . . . . . . . . . . . 169 8.4.2 Kinematic angular variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 8.5 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 8.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 8.6.1 8.7 The penny on the turntable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 8.7.1 Staying on track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 8.7.2 Going around a banked curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 8.7.3 Rotating frames of reference: Centrifugal force and Coriolis force . . . . . . . 184 CONTENTS 8.8 ix Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 9 Rotational dynamics 191 9.1 Rotational kinetic energy, and moment of inertia . . . . . . . . . . . . . . . . . . . . 191 9.2 Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.3 The cross product and rotational quantities . . . . . . . . . . . . . . . . . . . . . . . 197 9.4 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 9.5 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 9.6 Rolling motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 9.7 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 9.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 9.9 9.8.1 Torques and forces on the wheels of an accelerating bicycle . . . . . . . . . . 213 9.8.2 Blocks connected by rope over a pulley with non-zero mass . . . . . . . . . . 214 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 10 Gravity 221 10.1 The inverse-square law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 10.1.1 Gravitational potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . 224 10.1.2 Types of orbits under an inverse-square force . . . . . . . . . . . . . . . . . . 227 10.1.3 Kepler’s laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 10.2 Weight, acceleration, and the equivalence principle . . . . . . . . . . . . . . . . . . . 236 10.3 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 x CONTENTS 10.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 10.4.1 Orbital dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 10.4.2 Orbital data from observations: Halley’s comet . . . . . . . . . . . . . . . . . 245 10.5 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 10.5.1 Tidal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 10.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 11 Simple harmonic motion 253 11.1 Introduction: the physics of oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 253 11.2 Simple harmonic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 11.2.1 Energy in simple harmonic motion . . . . . . . . . . . . . . . . . . . . . . . . 259 11.2.2 Harmonic oscillator subject to an external, constant force . . . . . . . . . . . 260 11.3 Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 11.3.1 The simple pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 11.3.2 The “physical pendulum” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 11.4 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 11.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 11.5.1 Oscillator in a box (a basic accelerometer!) . . . . . . . . . . . . . . . . . . . 268 11.5.2 Meter stick as a physical pendulum . . . . . . . . . . . . . . . . . . . . . . . . 270 11.6 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 11.6.1 Mass on a spring damped by friction with a surface . . . . . . . . . . . . . . 272 CONTENTS xi 11.6.2 The Cavendish experiment: how to measure G with a torsion balance . . . . 273 11.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 12 Waves in one dimension 279 12.1 Traveling waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 12.1.1 The “wave shape” function: displacement and velocity of the medium. . . . . 281 12.1.2 Harmonic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 12.1.3 The wave velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 12.1.4 Reflection and transmission of waves at a medium boundary . . . . . . . . . 286 12.2 Standing waves and resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 12.3 Conclusion, and further resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 12.4 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 12.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 12.5.1 Displacement and density/pressure in a longitudinal wave . . . . . . . . . . . 295 12.5.2 Violin sounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 12.6 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 12.6.1 Chain of masses coupled with springs: dispersion, and long-wavelength limit. 299 12.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 13 Thermodynamics 303 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 13.2 Introducing temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 xii CONTENTS 13.2.1 Temperature and heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 304 13.2.2 The gas thermometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 13.2.3 The zero-th law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 13.3 Heat and the first law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 13.3.1 “Direct exchange of thermal energy” and early theories of heat . . . . . . . . 309 13.3.2 The first law of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 311 13.4 The second law and entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 13.4.1 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 13.4.2 The efficiency of heat engines . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 13.4.3 But what IS entropy, anyway? . . . . . . . . . . . . . . . . . . . . . . . . . . 317 13.5 In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 13.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 13.6.1 Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 13.6.2 Equipartition of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 13.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Preface Students: if this is too long, at the very least read the last four paragraphs. Thank you! For many years Eric Mazur’s Principles and Practice of Physics was the required textbook for University Physics I at the University of Arkansas. In writing this open-source replacement I have tried to preserve some of its best features, while at the same time condensing much of the presentation, and reworking several sections that did not quite fit the needs of our curriculum: primarily, the chapters on Thermodynamics, Waves, and Work. I have also skipped entirely the chapter on the “Principle of Relativity,” and instead distributed its contents among other chapters: in particular, the Galilean reference frame transformations are now introduced at the very beginning of the book, as are the law of inertia and the concept of inertial reference frames. Over the past few decades, there has been a trend to increase the size of introductory physics textbooks, by including more and more visual aids (pictures, diagrams, boxes. . .), as well as lengthier and more detailed explanations, perhaps in an attempt to reach as many students as possible, and maybe even to take the place of the instructor altogether. It seems to me that the result is rather the opposite: a massive (and expensive) tome that no student could reasonably be expected to read all the way through, at a time when “TL;DR” has become a popular acronym, and visual learning aids (videotaped lectures, demonstrations, and computer simulations) are freely available everywhere. Our approach at the University of Arkansas, developed as a result of the work of Physics Education Research experts John and Gay Stewart, is based instead on two essential facts. First, that different students learn differently: some will learn best from a textbook, others will learn best from a lecture, and most will only really learn from a hands-on approach, by working out the answers to questions themselves. Second, just about everybody will benefit from repeated presentations of the material to be learned, in different environments and even from slightly different points of view. In keeping with this, we start by requiring the students to read the textbook material before coming to lecture, and also take an online “reading quiz” where they can check their unde...
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