Phys 325 HW 7A due Thursday March 19 , 2015 by 1pm Name:________________1. 20 points Find ω and all the coefficients am and bm for the Fourier seriesF(t) = ao/2 + Σ an cos(n ωt) + Σ bn sin(n ωt)of a saw-tooth wave of period 1. F(t) = Fo (1-t) for 0 < t < 1, and then periodic(You may find it helpful to realize that F(t) is Fo/2 plus an odd function.)2. 20 points The Fourier series for a square wave was worked out in classFor the case T = 2π, it is F(t) = (4/π) [ sin t + (1/3) sin 3t + (1/5) sin(5t) + . . . ]Find the steady state response (you may truncate the series at frequency 5) of the damped massspring system to this F(t).d2x/dt2 + 0.10 dx/dt + 10 x = F(t)What frequency dominates the steady state response?Phys 325 HW 7B due Thursday March 19 , 2015 by 1pm Name:________________1. 10 points Evaluate the integral δ(2t3− 54)exp(2t)dt−∞∞∫Hint: make a variable substitution u = 2 t 3 that puts the delta-function into a morestandard form.2. 20 points A force acts on an initially quiescent undamped mass spring systemm d2x/dt2 + k x = F(t)where F is piece-wise analyticF = 0 for t < 0F = Fo t/T for 0 < t < TF = Fo for t > TLike F(t), the response x(t) will be piece-wise analytic, so it has a different analytic form,depending on whether t is greater than or less than T.• Use the convolution to compose an expression for the response x(t) valid at times0 < t < T. You needn't evaluate your integral, but you should conclude with anexpression a person could find in a table of integrals.• Use the convolution to compose an expression for the response x(t) valid at times t > T.You needn't evaluate your integral, but you should conclude with an expression a personcould find in a table of integrals.3. 10 points Take the Laplace transform of the differential equationm d2x/dt2 + k x = F(t)forF = 0 for t < 0F = Fo t/T for 0 < t < TF = 0 for t > Tand construct the Laplace transform x (s) of the response x(t) if x is initially quiescent.[ you are NOT asked for x(t)
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