A solid sphere `(radius = R)` rolls without slipping in a cylindrical throuh`(radius = 5R)`. Findthe time period of small oscillations.

Explanation

Using conservation of mechanical energy
`E_(A)=E_(B)`
`mg 4R(1-cos theta)=7/10 mv_(0)^(2)rArr 8 mg R sin^(2) ((theta)/2)=7/10 mv_(0)^(2)`
since `theta` is very small
`7/10 v_(0)^(2)=2 g R theta_(0)^(2) v_(0)^(2)=(20 gR)/7 theta_(0)^(2)`
Linear amplitude of SHM `a=4R theta_(0)rArr theta_(0)=a/(4R)`
`v_(0)^(2) =(20 gR)/7 (a^(2))/(16 R^(2))=5/28g/R a^(2)`
comparing `v_(0)^(2)=omega^(2)a^(2)`
`omega=sqrt((5g)/(28R)), T=2pisqrt((28R)/(5g))`
Alternate solution

`mg sin theta-f malpha' R.`
`fR=2/5 mR^(2)(alpha')`
acceleration of centre of the mass of solid sphere `=alpha'R=alpha4R`
solving above 3 equations `rArr mg sin theta=28/5 malphaR`
For small `theta`
`alpha=(5 g theta)/(28 R)`
`omega=sqrt((5g)/(28R)),T=2pisqrt((28R)/(5g))`