m (10 points) You want to push a small child on a swing, keeping the oscillation amplitude small (for safety, and to make it a simple, damped harmonic oscillator). You notice that the amplitude decays very slowly, decreasing by lle after very many oscillations, so you decide you will give the child a small, short push when the swing pauses at its maximum amplitude. But, you will only do this once for every 5 cycles. You characterize the swing-oscillator with the standard parameters m, 0,=Vk/m and B =b/2m, and you characterize your push with the net FO v . you make vo while ensuring that the swing never reaches a velocity greater than Vmax · Throughout this problem, use approximations whenever possible, keeping only the lowest order terms of B/0, a) Determine the Fourier series amplitudes of the components that make up your push. Please be clear about the frequency of each term, and show that many of the lowest frequency amplitudes are all the same. b) Determine which three components create the largest responses. c) What is the ratio of " where Vmax is the maximum velocity of the steady state V Max solution? This is a ratio of velocities, so it should be unitless.
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