Physics 201 Analytical Mechanics J Kiefer May 2006 © 2006 I. INTRODUCTION......................................................................................................3 A. Fundamental Concepts and Assumptions ....................................................................................................... 3 1. Concepts ......................................................................................................................................................... 3 2. Assumptions ................................................................................................................................................... 3 B. Kinematics – Describing Motion...................................................................................................................... 3 1. Coordinate Systems ........................................................................................................................................ 3 2. Variables of Motion........................................................................................................................................ 4 3. Galilean Relativity.......................................................................................................................................... 5 C. Newton’s “Laws” of Motion ............................................................................................................................. 6 1. First “Law” or “Law” of Inertia...................................................................................................................... 6 2. Second “Law”................................................................................................................................................. 7 3. Third “Law”.................................................................................................................................................... 7 II. DYNAMICS IN ONE DIMENSION............................................................................8 A. Constant Force................................................................................................................................................... 8 1. Equation of Motion......................................................................................................................................... 8 2. Examples ........................................................................................................................................................ 9 B. Force as an Explicit Function of Time............................................................................................................. 9 1. Equation of Motion......................................................................................................................................... 9 2. Examples ...................................................................................................................................................... 10 3. Impulse, a Vector.......................................................................................................................................... 10 C. Force as a Function of Position ...................................................................................................................... 11 1. Equation of Motion in One Dimension......................................................................................................... 11 2. Potential Energy Function ............................................................................................................................ 12 D. Force as a Function of Velocity ...................................................................................................................... 14 1. Equations of Motion—Two Ways to Go...................................................................................................... 14 2. Examples ...................................................................................................................................................... 15 E. Harmonic Oscillator........................................................................................................................................ 17 1. Simple Harmonic Oscillator in One Dimension ........................................................................................... 17 2. Damped Harmonic Oscillator ....................................................................................................................... 18 3. Driven Harmonic Oscillator.......................................................................................................................... 20 III. DYNAMICS OF A POINT IN THREE DIMENSIONS..........................................24 A. Extension of the Concepts to Three Dimensions........................................................................................... 24 1. Impulse ......................................................................................................................................................... 24 2. Work-Energy Theorem................................................................................................................................. 24 3. Work Integrals .............................................................................................................................................. 25 4. Potential Energy Functions........................................................................................................................... 26 5. Angular Momentum...................................................................................................................................... 27 6. Examples ...................................................................................................................................................... 28 B. Separable Forces.............................................................................................................................................. 29 1. Projectile Motion in a Uniform Gravitational Field...................................................................................... 29 1 2. Harmonic Oscillator.......................................................................................Error! Bookmark not defined. C. Constrained Motion of a Particle ................................................................................................................... 34 1. Smooth Constraints....................................................................................................................................... 34 2. Motion on a Curve........................................................................................................................................ 35 IV. ACCELERATED REFERENCE FRAMES..........................................................37 A. Galilean Transformation ................................................................................................................................ 37 1. Transformation Equations ............................................................................................................................ 37 2. Translating Reference Frames ...................................................................................................................... 37 B. Rotating Reference Frames ................................................................................................................................. 38 1. Equations of Motion ..................................................................................................................................... 38 2. Rotating Earth............................................................................................................................................... 41 V. POTPOURRI ..........................................................................................................45 A. Systems of Particles ......................................................................................................................................... 45 1. N-particles..................................................................................................................................................... 45 2. Rocket........................................................................................................................................................... 46 3. Collisions...................................................................................................................................................... 48 B. Rigid Body........................................................................................................................................................ 49 1. Equations of motion...................................................................................................................................... 49 2. Computing moments of inertia ..................................................................................................................... 51 3. Laminar Motion of a Rigid Body ................................................................................................................. 53 C. Central Forces.................................................................................................................................................. 56 1. General Properties .......................................................................................................................................