A solid sphere of (radius `= R`) rolls without slipping in a cylindrical vessel (radius `= 5R`). Find the time period of small of oscillations of the sphere in `s^(-1)`. Take `R = (1)/(14)m` and `g = 10 m//s^(2)`. (axis is cylinder is fixed and horizontal).

Answer

5

Explanation

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