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CAF123
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Homework Statement
A damped oscillator is subjected to a simple harmonic force, satisfying $$\ddot{x}(t) + 2k\dot{x}(t) + \omega^2x(t) = g \cos (nt), $$where ##g, k, \omega, n +ve.##
1) Show that for ##t >>1/k## the position x(t) has the form ##A \cos (nt - \phi)##, and find A and ##\phi## in terms of k,n,\omega, g.
2) Find the resonance frequency. Under what conditions is there a maximum?
3) For ##k^2 << \omega^2## find the width of the resonance, defined by the distance between the two frequencies ##n_{1,2}## where the squared amplitude goes down to half its orginal value.
The Attempt at a Solution
1)No condition on k and ω is given so I think I have three separate homogeneous solns, each for the conditions ##k^2 = \omega^2, k^2 < \omega^2 ## and ## k^2 > \omega^2##. In order to find the particular solution, I made the trial function ##x_p = C \cos (nt) + D \sin (nt)##, differentaited twice and subbed into the eqn given. This give me very ugly and messy expressions for C and D. Am I correct in saying that in the case t >>1/k, then the expressions for the homogeneous solutions (in all three cases outlined above) simply vanish? If so, this would just leave the particular soln ##x_p = C \cos (nt) + D \sin(nt)##. The expressions I got for C and D are too messy to put up here , so I might scan my work instead.
In order to find A and phi in terms of those quantities, should I just sub ##A \cos (nt - \phi)## into the eqn and get A and ##\phi## that way? This would make 2) easier and I am yet to attempt 3).
Many thanks.