Problem Set 2

Problem Set 2 - October 2, 2003 Physics 16 Problem Set 2 1....

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October 2, 2003 Physics 16 Problem Set 2 1. Form the derivation in our textbook, we know that the general solution for a damped and driven harmonic oscillator with damping force -bv and driving force F 0 cos( wt ) is: In this problem, b=mc, c=6w/5, and w d = w. We can easily solve for some of the constants in this equation: 3 2 5 b w m γ = = 2 2 2 2 ( ) (2 ) 2 R w w w w = - + = 2 2 1 1 cos ( ) cos (0) 2 w w R π φ - - - = = = 2 2 2 2 16 25 w w Ω = - = - Note that c 2 < 0, so the system is underdamped. Thus, using the solution in the textbook, we can replace the above equation with: 2 0 16 ( ) cos( ) cos( ) 25 t F w x t wt e C R ϕ - = - + + All that remains is to solve for C and c. Our initial conditions are x[0]=0 and x'[0]=0. Solving simultaneously with Mathematica, one solution is: 0 2 25 24 F C w - = - 0 ( ) cos( ) ( ) t t t F x t wt e Ae Be R - -Ω = - + +
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2-2. For each of the following, I determined whether the force could be derived from a potential by finding its gradient and seeing whether or not it was equal to the zero vector. I found the potentials by guess-and-check, not by path integrals. a.
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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 2 - October 2, 2003 Physics 16 Problem Set 2 1....

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