In the figure below, a solid cylinder attached to a horizontal massless spring r

Question

In the figure below, a solid cylinder attached to a horizontal massless spring rolls without slipping along a horizontal surface. The spring constant k is 4.00 N/m. The system is released from rest when the spring is stretched by 0.150 m.

http://www.webassign.net/hrw/16_37.gif

(a) Find the translational kinetic energy of the cylinder as it passes through the equilibrium position. J

(b) Find the rotational kinetic energy of the cylinder as it passes through the equilibrium position. J

(c) Show that under these conditions the center of mass of the cylinder executes simple harmonic motion, and that the period T can be expressed as follows in terms of M, the mass of the cylinder. T = 2 pi sqrt((3 M )/( 2 k)) (Do this on paper. Your instructor may ask you to turn in this work. Hint: Find the time derivative of the total mechanical energy.)

Explanation / Answer

c)

Friction force = f

Torque = f*R

f*R = I*alpha.......where I = Inetia, alpha = angular acceleraton = a/R

f*R = (1/2*MR^2)*a/R

f = 1/2*Ma = 1/2*Mx''

Spring force = -kx

Newton's second law: Mx'' = -kx - 1/2*Mx''

3/2*Mx'' + kx = 0

This is analogous to standard eqn mx'' + kx = 0 with m = 3M/2

So, Time period T = 2*pi*sqrt (m/k) = 2*pi*sqrt [3M / (2k)]

a)

Translational KE = 1/2*Mv^2

But v = aw = a*sqrt (2k/(3M))

So, Translational KE = 1/3 *ka^2 = (1/3)*4*0.15^2 = 0.03 J

b)

Rotational KE = 1/2*Iw^2 = 1/2*(1/2*MR^2)*(v/R)^2 = 1/4*Mv^2 = 1/2*Translational KE = 1/2*0.03 = 0.015 J

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