Funky Mechanics Concepts The Anti-Textbook* A work in progress. See elmichelsen.physics.ucsd.edu/ for the latest versions of the Funky Series. Please send me comments. Eric L. Michelsen constant a a (a, +d) (a+da,+d) dr (a,) constant θ (a+da,) ˆ ˆ dr = da a+d “Now in the further development of science, we want more than just a formula. First we have an observation, then we have numbers that we measure, then we have a law which summarizes all the numbers. But the real glory of science is that we can find a way of thinking such that the law is evident.” – Richard Feynman, The Feynman Lectures on Physics, p26-3. * Physical, conceptual, geometric, and pictorial physics that didn’t fit in your textbook. Please do NOT distribute this document. Instead, link to elmichelsen.physics.ucsd.edu/FunkyMechanicsConcepts.pdf. Please cite as: Michelsen, Eric L., Funky Mechanics Concepts, elmichelsen.physics.ucsd.edu/, 10/30/2018. elmichelsen.physics.ucsd.edu/ Funky Mechanics Concepts emichels at physics.ucsd.edu 2006 values from NIST. For more physical constants, see http://physics.nist.gov/cuu/Constants/ . Speed of light in vacuum c = 299 792 458 m s–1 (exact) Gravitational constant G = 6.674 28(67) x 10–11 m3 kg–1 s–2 Relative standard uncertainty ±1.0 x 10–4 Boltzmann constant k = 1.380 6504(24) x 10–23 J K–1 Stefan-Boltzmann constant σ = 5.670 400(40) x 10–8 W m–2 K–4 –6 Relative standard uncertainty ±7.0 x 10 23 –1 Avogadro constant NA, L = 6.022 141 79(30) x 10 mol Relative standard uncertainty ±5.0 x 10–8 Molar gas constant R = 8.314 472(15) J mol-1 K-1 calorie 4.184 J (exact) –31 Electron mass me = 9.109 382 15(45) x 10 kg –27 Proton mass mp = 1.672 621 637(83) x 10 kg Proton/electron mass ratio mp/me = 1836.152 672 47(80) Elementary charge e = 1.602 176 487(40) x 10–19 C Electron g-factor ge = –2.002 319 304 3622(15) Proton g-factor gp = 5.585 694 713(46) Neutron g-factor gN = –3.826 085 45(90) –28 Muon mass mμ = 1.883 531 30(11) x 10 kg Inverse fine structure constant α–1 = 137.035 999 679(94) Planck constant h = 6.626 068 96(33) x 10–34 J s Planck constant over 2π ħ = 1.054 571 628(53) x 10–34 J s –10 Bohr radius a0 = 0.529 177 208 59(36) x 10 m –26 –1 Bohr magneton μB = 927.400 915(23) x 10 J T Other values: 1 inch ≡ 0.0254 m (exact) 10/30/2018 13:29 Copyright 2002 - 2018 Eric L. Michelsen. All rights reserved. 2 of 88 elmichelsen.physics.ucsd.edu/ Funky Mechanics Concepts emichels at physics.ucsd.edu Table of Contents 1 Introduction ........................................................................................................................................... 5 Why Funky? ........................................................................................................................................ 5 How to Use This Document ................................................................................................................. 5 Notation ............................................................................................................................................... 5 Scientists of the Times ..................................................................................................................... 6 The Funky Series ................................................................................................................................. 6 Thank You ........................................................................................................................................ 7 The International System of Units (SI) ................................................................................................ 7 Possible Future Funky Mechanics Concepts .................................................................................... 8 2 Symmetries, Coordinates ...................................................................................................................... 9 Elastic Collisions ................................................................................................................................. 9 Newton’s Laws ...................................................................................................................................13 Newton’s 3rd Law Isn’t ....................................................................................................................13 It’s Got Potential: Workless Forces ....................................................................................................13 3 Rotating Stuff........................................................................................................................................15 Angular Displacements and Angular Velocity Vectors ..................................................................15 Central Forces .....................................................................................................................................15 Reduction To 1-Body ......................................................................................................................16 Effective Potential ...........................................................................................................................18 Orbits ..................................................................................................................................................19 The Easy Way to Remember Orbital Parameters ............................................................................19 Coriolis Acceleration ..........................................................................................................................20 Lagrangian for Coriolis Effect ........................................................................................................24 Mickey Mouse Physics: Parallel Axis ................................................................................................24 Moment of Inertia Tensor: 3D ............................................................................................................25 Rigid Bodies (Rotations) .................................................................................................................26 Rotating Frames ..................................................................................................................................26 Rotating Bodies vs. Rotating Axes .....................................................................................................27 4 Shorts .....................................................................................................................................................28 Friction................................................................................................................................................28 Static Friction ..................................................................................................................................28 Kinetic Friction ...............................................................................................................................28 Mathematical Formula for Static and Kinetic Friction ....................................................................28 Rolling Friction ...............................................................................................................................31 Drag .................................................................................................................................................31 Damped Harmonic Oscillator .............................................................................................................31 Critical Condition ............................................................................................................................32 Pressure ...............................................................................................................................................33 5 Intermediate Mechanics Concepts ......................................................................................................35 Generalized Coordinates .....................................................................................................................35 I Need My Space: Configuration Space, Momentum Space, and Phase Space ...............................35 Choosing Generalized Coordinates, and Finding Kinetic Energy ...................................................35 D’Alembert’s Principle .......................................................................................................................36 Lagrangian Mechanics ........................................................................................................................37 Introduction .....................................................................................................................................37 What is a Lagrangian? .....................................................................................................................38 What Is A Derivative With Respect To A Derivative? ...................................................................39 Hamilton’s Principle: A Motivated Derivation From F = ma .........................................................39 Generalizations of Hamilton’s Principle .........................................................................................41 Hamilton’s Principle of Stationary Action: A Variational Principle ...............................................43 Hamilton’s Principle: Why Isn’t “Stationary Action” an Oxymoron? ............................................44