help please! The uniform solid cylinder of mass m and radius r rolls without sli

Question

help please!

The uniform solid cylinder of mass m and radius r rolls without slipping during its oscillation on the circular surface of radius R. If the motion is confined to small amplitudes theta = theta 0, determine the period tau of the oscillations. Also determine the angular velocity omega of the cylinder as it crosses the vertical centerline. (Caution: Do not confuse omega with theta or with omega n as used in the defining equations. Note also that theta is not the angular displacement of the cylinder.)

Explanation / Answer

Let the friction force be F

mg Sin(theta) - F = ma

F*r =I*alpha

& a= r*Alpha

So,Â

mgSin(theta) - I*(alpha)/r = m*(r*alpha)

=> mgrSin(theta) = (mr^2 + I)alpha

alpha = -d2(theta)/dt2

=> d2(theta)/dt2 = - mgr/(mr^2 + I) (theta)

( Since theta is small so sin(theta) is theta)

Simple haronic equation with w^2 = mgr/(mr^2 + I)

(w is the angular frequency  not velocity)

=> T = 2*pi / wÂ

=> T = 2*pi* Sqrt [( mr^2+I)/mgr]

I= 0.5 mr^2

=> T = 2*pi* Sqrt [( mr^2+0.5mr^2)/mgr]

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