Physics 325 Spring 2011 Problem Session 8 Solutions

Physics 325 Spring 2011 Problem Session 8 Solutions -...

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Physics 325 Spring 2011 Discussion 8 March 28, 2011 Consider an underdamped harmonic oscillator, ¨ x + 2 β ˙ x + ω 2 0 x = F ( t ) . (a) Let α be a constant, and F ( t ) = ( αt, for t > 0 0 , for t < 0 Solve for x ( t ) given the initial conditions x (0) = 0 and ˙ x (0) = 0 . (b) Suppose that the driving force is replaced with F ( t ) = αt, for 0 < t < τ ατ, for t > τ 0 , for t < 0 Solve for x ( t ) with the same initial conditions as in part (a). Hint: First show that the driving force can be written as F 1 ( t ) - F 1 ( t - τ ) , where F 1 ( t ) is the driving force from part (a). Then, by superposition, we can write the solution as x 1 ( t ) - x 1 ( t - τ ) , where x 1 ( t ) is the solution of part (a). (c) From the solution to (b), take the limit α → ∞ and τ 0 such that ατ = a = constant, and show that the resulting x ( t ) agrees with M&T Eqn. 3.105 with t 0 = 0 . Solutions : (a) We guess a particular solution of the form A + Bt for t > 0 . x p ( t ) = A + Bt ˙ x p ( t ) = B ¨ x p ( t ) = 0 Substituting this into the equation of motion gives 2 βB + ω 2 0 ( A + Bt ) = αt Thus, we must have B = α ω 2 0 A = - 2 β ω 2 0 B = - 2 αβ ω 4 0 x p ( t ) = α ω 2 0
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