A disk of mass M is on a surface as shown below. It is attached to a wall by a s

Question

A disk of mass M is on a surface as shown below. It is attached to a wall by a spring of disk is at rest and the spring is un-stretched. The disk is pulled a distance A and then released. Note that the disk to rolling without slipping after being released. What is the total mechanical energy of the system just before the disk is released? Express your answer in terms of the known quantities. What is the total mechanical energy of the system when the disk passes through the equilibrium position? Express your answer in terms of M, speed of the center of mass v, moment of inertia I, and the angular speed of the rolling disk, omega_rolling. What is the linear speed of the center of mass v of the disk when it passes through the equilibrium position? Express your answer in terms of k, M, and A. What is the angular frequency omega and period T of the simple harmonic motion of this spring-rolling mass system? Express your answer in terms of k and M. Assume that we have the same spring and mass, only now the disk is released from rest on a frictionless surface at a distance A from the equilibrium position. What is the period of the simple harmonic motion of the system? Express your answer in terms of k and M.

Explanation / Answer

a) Total M.E = kA^2 / 2

b) Total M.E = (Mv^2 /2) + (Iw^2 /2)

c) kA^2 / 2 = (Mv^2 /2) + (Iw^2 /2)

v = sqrt {( kA^2 - Iw^2 ) / M}

d) w = sqrt ( k/M)

T = 2*pi* sqrt ( M/k )

e) T = 2*pi* sqrt ( M/k )

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