Consider the forced but undamped harmonic oscillator with mass 2 kg and a spring
ID: 3077827 • Letter: C
Question
Consider the forced but undamped harmonic oscillator with mass 2 kg and a spring whose spring constant is 32 N/m.
a) Write the second-order differential equation for y(t), the position of the mass at time t, if the external force, f(t)=8*cos(wt) [Note: w is equal to omega]
b) Find the solution such that w=2 and y(0)=y'(0)=0. If applicable, put your solution in real form. (Your final solutoin should not contain complex numbers).
c) Find the solution such that w=4 and y(0)=y'(0)=0. If applicable, put your solutoin in real form. (Your final solution shoul dnot contain complex numbers). What is the name for the phenomena the system dislays at this value of w?
I think that the phenomena is called pure resonance.
Explanation / Answer
(a) my'' = -ky + f(t) Use the info to see that m = 2 and k = 32 so that 2 y'' + 32 y = 8 cos(wt) y'' + 16y = 4 cos (wt) (b) When w = 2 y'' + 16y = 4 cos(2t) The solution is y(t) = c1 cos(4t) + c2 sin(4t) + 1/3 cos(2t) To satisfy y(0) = 0 and y'(0) = 0 it must be that c1 = -1/3 and c2 = 0. The solution is y(t) = 1/3(cos(2t) - cos(4t)) (c) If w = 4 then y'' + 16y = 4 cos(4t) The solution is y(t) = c1 cos(4t) + c2 sin(4t) + t/2 sin(4t) If y(0) = y'(0) = 0 then c1 = c2 = 0. The pure resonant solution is y(t) = t/2 sin(4t)