A solid cylinder of mass `m` is attached to a horizontal spring with force constant `k`. The cylinder can roll without slipping along the horizontal plane. (See the accompanying figure.) Show that the center of mass of the cylinder executes simple harmonic motion with a period `T = 2pisqrt((3m)/(2k))`, if displaced from mean position.

Explanation

(i) Time period ,T=2s
Amplitude,A=3cm
At time, t=0, the radius vector OP makes an angle `(pi)/(2)` with the positive x-asix, i.e., phase Angle `phi=+(pi)/(2)`
Therefore, the equation of simple harmonic motion for the x-projection of OP,at time t, is given by the displacement equation:
` x=Acos [(2pit)/(T)+phi]=3cos((2pit)/(2)+(pi)/(2))=-3sin((2pit)/(2))`
`x=-3sinpitcm`
(ii) Time period , T=4s
Amplitude, a=2m
At time t=0, OP makes an angle pi with the x-axis, in the anticlockwise direction. Hence phase angle ,`phi=+pi` Therefore, the equation of simple harmonic motion for the x-proection of Op,at time t,is given as:
`x=acos((2pit)/(T)+phi)=2cos((2pit)/(4)+pi)`
`x=-2cos ((pit)/(2))_(m)`