A small solid cylinder of mass M attached to horizontal massless spring can roll without slipping along a horizontal surface. Show that if the cylinder is displaced slightly and released, it executes S.H.M. Also find its time period.

Explanation

At any instant of rolling the cylinder has rotational and translation kinetic energy
`K_("rot") = (1)/(2) l omega^(2) = (1)/(2) [(1)/(2) MR^(2)] [(v^(2))/(R^(2))] = (1)/(4) Mv^(2)`
`K_("trans") = (1)/(2) Mv^(2)`
`U_("(potential)") = (1)/(2) kx^(2)`
`rArr` Total energy `E = K_("rot") + K_("trans") + U`

`rArr E = (3)/(4) Mv^(2) +(1)/(2) kx^(2)`
Here `((dE)/(dt)) - (3)/(4) M (2v) (dV)/(dt) + (1)/(2) k2x ((dx)/(dt)) = 0`
or, `((dv)/(dt)) = - ((2k)/(3M)) x`
Hence `alpha = - omega^(2)x`
Therefore the motion is simple harmonic
`omega = sqrt((2k)/(3M))`
`rArr T = 2pi sqrt((3M)/(2k))`