Consider a semicircular shell of mass 'm' and radius 'r' which rolls without slipping as shown in Fig. Use the fact that maximum kinetic energy at mean position is equal to maximum potential at exterme position and find out the frequency of oscillation.

Explanation

According to problem, shell is in pure rolling, contact point `A` can be considered to be the instantaneous centre of rotation.
`(KE)_(max) = (1)/(2)I_(A) omega^(2)`
where `I_(A) = I_(CG) +m (r-a)^(2)`
`= I_(0) - ma^(2) +m (r -a)^(2)`
`=2mr (r-a)`
`(PE)_(max) = mga (1 -cos theta)` Since
`[(KE)_(max)]_(mean position) = [(PE)_(max)]_(exterme position)`
`mr (r-a) omega^(2) = mga (1-cos theta)`
and `omega = sqrt((ga(1-cos theta))/(r(r-a)))`