Problem Set 3

Problem Set 3 - October 9, 2003 Physics 16 Problem Set 3...

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Unformatted text preview: October 9, 2003 Physics 16 Problem Set 3 3-1. a. L = T - V 2 2 2 1 1 2 2 2 1 1 1 1 ( ) 2 2 2 L m x m x k x x = +-- & & 2 2 2 2 1 1 2 2 2 1 1 2 1 1 1 1 1 2 2 2 2 2 L m x m x kx kx kx x = +-- + & & Euler-Langrange Equations: 1 1 2 1 m x kx kx =- && 2 2 1 2 m x kx kx =- && This is indeed what we would expect from f=ma , using Hooke's Law. b. Substituting x 2 out: 2 2 2 1 1 2 1 1 1 1 ( ) 2 2 2 L ky m x m x y = - + +- & & & 2 2 2 2 1 1 2 1 2 2 1 1 1 1 1 2 2 2 2 L ky m x m x m y m x y = - + + +- & & & & & Since x 1 does not appear in the Langragian, we can find the generalized momentum conjugate: 1 1 2 1 1 ( ) ( ) L m x m x y x t = +- & & & & Substituting x 1 out: 2 2 2 2 2 1 2 1 1 1 ( ) 2 2 2 L ky m x m x y = - + + + & & & 2 2 2 2 2 2 1 2 1 1 2 1 1 1 1 2 2 2 2 L ky m x m x m y m x y = - + + + + & & & & & 2 2 1 2 2 ( ) ( ) L m x m x y x t = + + & & & & But 1 x y- & & is the velocity of m 2 , and 2 x y + & & is the velocity of m 1 , so these are both equal to each other and to the total momentum of the system. c. We know that 1 1 2 2 m x m x + = & & , so 2 2 1 1 2 1 ( [0]) [0] m x x x x m- =- + . Integrating this expression from time = 0 to t, we get that 2 2 1 1 2 1 ( [0]) [0] m x x x x m- =- + . Going back to our differential equation from part a, we can solve these two equations simultaneously by substituting the preceding expression for x 2 . L @ t D =- 1 2 k H- x 1 @ t D + x 2 @ t D L 2 + 1 2 m 1 x 1 t @ t D 2 + 1 2 m 2 x 2 t @ t D 2 L @ t D = - 1 2 k i k x 2 @ D- x 1 @ t D- m 1 H- x 1 @ D + x 1 @ t D L m 2 y { 2 + 1 2 m 1 x 1 t @ t D 2 + 1 2 m 2 x 2 t @ t D 2 Now we can use the Euler-Lagrange equation to find that: m 1 x 1 1 @ t D ==- k i k- 1- m 1 m 2 y { i k- x 1 @ t D- m 1 H- x 1 @ D + x 1 @ t D L m 2 + x 2 @ D y { The solution to this differential equation (using Mathematica) is: x 1 @ t D = km 1 x 1 @ D + km 2 x 2 @ D km 1 + km 2 + C @ 1 D Cos A D km 1 + km 2 t t m 1 t m 2 E + C @ 2 D Sin A D km 2 + km...
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This note was uploaded on 09/08/2008 for the course PHYS 16 taught by Professor Staff during the Spring '08 term at Harvard.

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Problem Set 3 - October 9, 2003 Physics 16 Problem Set 3...

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