An object moves with simple harmonic motion of period T and amplitude A. During

Question

An object moves with simple harmonic motion of period T and amplitude A. During one complete cycle, for what length of time is the speed of the object greater than vmax/2? t = _________ Express your answer in terms of T

Explanation / Answer

x(t) = A*sin(2pt/T) x(t) > A/2 when A*sin(2pt/T) > A/2 , or sin(2pt/T) > 1/2 But sin(X) = 1/2 when X = 30 degrees or (180 - 30) = 150 degrees; in between, sin(X) > 1/2 Therefore, between X = 30 and X = 150 degrees, sin(X) > 1/2 That spans 150 - 30 = 120 degrees = 2p/3 Hence: 2pdt/T = 2p/3 dt = T/3 So x > A/2 for 1/3 of the period T

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