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

Answer

B

Explanation

For pure rolling to take place,
`v = R omega`
`omega' =` angular velocity of COM of sphere C about O
` = (v)/(4R) = (Romega)/(4R) = (omega)/(4)`
`:. (d omega')/(dt) = (1)/(4) (d omega)/(dt)`
or `alpha' = (alpha)/(4)`
`alpha = (a)/(R)` for pure rolling
where, `a = (g sin theta)/(1 + (I)/(mR^(2))`
`= (5 g sin theta)/(7)`
as, `I = (2)/(5) mR^(2)`
`:. alpha' = (5 g sin theta)/(28R)`
For small `theta, sin theta ~~ theta`, being restoring in nature,
`alpha = - (5g)/(28R) theta`
`:. T = 2pi sqrt(|(theta)/(alpha')|`
` = 2pi sqrt((28R)/(5g))`
.