a solid sphere{radius R ) ROLLS WITHOUT SLIPPING IN A CYLINDERCAL THROUGH (RADIUS 5R). FIND THE TIME PERIOD OF SMALL OCSILLATIONS.

let the COM of the sphere has a velocity V=Rω, where ω is the angular velocity of sphere wrt the COM.Let the angular velocity of the sphere wrt the centre of the cylinder be W then V=(5R-R)W

                                                                                  W=ω/4

                                                   from conservation of energy

mg(5R-R)(1-cosθ)+(1/2)mV^2+(1/2)Iω^2=constant

differentiating the above equation wrt time we get( α/θ)=5g/28R

therefore time period =2π(k)^1/2,where k=28R/5g