Physics in Proportion Mark A. Peterson °c 2005 M.A. Peterson Contents 1 What is physics? 3 1.1 Proportionality . 4 1.2 Two Kinds of Physics . 7 1.3 Learning Physics . 9 1.4 A Capsule History of Physics . 11 2 Mathematical Tools 21 2.1 Proportion . 22 2.2 Units . 24 2.3 Data: Straight Line Plots . 26 2.4 Uncertainty in Data . 30 2.5 Dimension . 32 2.6 Position, Time, and (constant) Velocity . 33 2.7 The speed of light, and SI units . 38 2.8 Dimension and Scaling . 39 2.9 Power Laws, and the Logarithm . 40 2.10 Numbers in Geometry are Ratios . 43 2.11 The Trigonometric Functions . 44 iii iv CONTENTS 2.12 Angular Measures . 47 2.13 Trigonometric functions of special angles . 50 2.14 Small angle approximations . 51 3 Geometrical Optics 61 3.1 Angular Size . 62 3.2 The Eye . 65 3.3 Binocular Vision and Parallax . 66 3.4 Wide Open Pupils . 68 3.5 The Lens of the Eye . 70 3.6 Refraction . 71 3.7 Focal Length . 74 3.8 Interpreting Relationships . 76 3.9 The focal length of the eye . 78 3.10 Virtual Images . 80 3.11 Thin Lenses . 83 3.12 Object and Image . 86 3.13 Optical Systems . 89 3.13.1 The Magnifying Glass . 90 3.13.2 The Microscope . 93 3.13.3 Two lenses together . 95 3.13.4 The Astronomical Telescope . 96 3.13.5 Galilean Telescope . 98 3.14 Mirrors . 99 3.15 Spherical Aberrations . 102 3.16 Reflection and Refraction . 104 3.17 Fermat's Principle . 106 3.18 Wavefronts: A Dual Theory of Light . 109 CONTENTS v 4 Time and Oscillation 121 4.1 Angular Clocks . 122 4.1.1 The Solar Clock . 123 4.1.2 The Sidereal Clock . 124 4.1.3 Solar vs. Sidereal . 124 4.1.4 Aside on Kepler's Laws . 128 4.2 Atomic Clocks . 130 4.3 GPS: Global Positioning System . 131 4.4 Longitude . 134 4.5 The Moons of Jupiter . 137 4.6 Period, Frequency and Amplitude . 141 4.7 Velocity in Orbit, Projected . 143 4.8 Pendulums . 144 4.8.1 The Period of a Pendulum . 146 4.9 The Binomial Approximation for Perturbations . 149 4.10 Pendulums and the Rotation of the Earth . 152 4.11 Simple Harmonic Oscillators . 154 4.12 Exponential Decay . 158 4.13 Dating by Radioactive Decay . 163 5 Mass, Weight, and Equilibrium 173 5.1 Archimedes . 173 5.2 Torque and Force . 177 5.3 Spring Forces: Hooke's Law . 182 5.4 Weight and Mass . 183 5.5 Springs in Parallel and Series . 186 vi CONTENTS 5.6 Newton's Third Law . 189 5.7 Young's Modulus . 191 5.8 The Force Between Atoms . 193 6 Mechanical Energy and Motion 201 6.1 Gravitational Potential Energy . 201 6.2 Spring Potential Energy . 206 6.3 The Potential Energy of a Pendulum . 207 6.4 Falling, and Kinetic Energy . 209 6.5 Velocity v in falling . 213 6.6 Universal Gravitation . 214 6.7 Energy of an Oscillator . 217 6.8 Oscillators Losing Energy . 219 6.9 A Chemical Bond . 222 7 Vector Quantities 231 7.1 Projectile Motion . 231 7.2 Vector Addition . 233 7.3 Velocity and Speed . 236 7.4 Galilean Relativity . 237 7.5 Falling and Relativity . 238 7.6 Falling and Impulse . 240 7.7 More on Projectile Motion . 241 7.8 Impulse and Conservation of Momentum . 243 7.9 Impulse and Circular Motion . 244 CONTENTS vii 8 Density and Fluids 253 8.1 Mass Density . 253 8.2 Archimedes' Principle . 255 8.3 Galileo's Balance . 260 8.4 Galileo's Proof of Archimedes' Principle . 261 8.5 Buoyancy and Pressure . 263 8.6 More on Hydrostatic Pressure . 266 8.7 Atmospheric Pressure . 268 8.8 The Barometer . 270 8.9 Bernoulli's Principle . 272 8.10 Applications of Bernoulli's Principle . 273 8.10.1 Force of the wind . 273 8.10.2 Flow Past an Airfoil . 275 8.11 Flow in Pipes . 276 8.11.1 Venturi Flow Meter . 277 8.11.2 Poisseuille Flow . 278 8.11.3 Current Density . 278 8.12 Shear Stress and Viscosity . 280 8.13 Stokes Flow . 282 8.14 Poisseuille Flow Revisited . 283 8.15 The Reynolds Number . 286 8.16 Resistance in Series and Parallel . 288 8.17 The Human Circulatory System . 290 8.18 A Fractal Model of Circulation . 295 viii CONTENTS 9 Temperature, Heat, and Internal Energy 311 9.1 Temperature . 314 9.2 Thermometers . 317 9.3 The Gas Thermometer . 319 9.4 Avogadro's Hypothesis . 321 9.5 Heat Capacity . 323 9.6 Molar Heat Capacities . 327 9.7 Statistical Model for Molar Heat Capacity . 328 9.8 Phase Transitions . 331 9.9 Entropy . 333 10 Thermodynamics 339 10.1 Work . 340 10.2 P ¢V Work . 342 10.3 Various Processes . 343 10.3.1 Adiabatic Process: ¢Q = 0 . 343 10.3.2 Isothermal Processes, ¢T = 0 . 345 10.3.3 A Constant Pressure Process . 347 10.3.4 Reversible and Irreversible Processes . 348 10.4 Heat Engines . 351 10.4.1 The Carnot Cycle . 354 10.4.2 Refrigerators, Heat Pumps . 355 10.5 Life at Fixed Temperature . 356 10.6 Life at Fixed Temperature and Pressure . 358 CONTENTS ix 11 Statistical Physics 361 11.1 Ideal Solutions as Ideal Gases . 361 11.2 Statistical Mechanics . 364 11.3 Randomness . ..