A uniform rod of mass, `m`, and length `l` remains in equilibrium inside a smooth hemisphere of radius `R` as shown. The period of small oscillations of the rod is….. [`:' r^(2) =R^(2) -l^(2)//4]`

Options

A. `2pi sqrt(((l^(2))/(12)+r^(2))/(gr))`
B. `2pi sqrt(((l^(2))/(4)+r^(2))/(gr))`
C. `2pi sqrt((R )/(g))`
D. `2pi sqrt((l+R)/(g))`

Answer

A

Explanation

Restoring torque: `tau = mgr theta`

`rArr I omega^(2) theta = mgr theta rArr omega^(2) = (mgr)/(I)`and
`I = (ml^(2))/(12) +mr^(2), r^(2) +(l^(2))/(4) =R^(2) :. T = (2pi)/(omega)`