A uniform cylinder of mass `m` and radius `R` is in equilibrium on an inclined plane by the action of a light spring of stiffness `k`, gravity and reaction force acting on it. If the angle of inclination of the plane is `phi`, then angular frequency of small oscillations of the cylinder is...........

Options

A. `2sqrt((k)/(m))`
B. `2sqrt((2k)/(m))`
C. `2sqrt((2k)/(3m))`
D. `sqrt((2k)/(m))`

Answer

C

Explanation

Let, `x` be the displacement of centre of mass of cylinder from mean position.
`:.` from rolling concept, elongation in spring `=2x`
`:.` Restoring torque about point of contact is:
`tau_(rest) =- 2R (2kx) =- 4kxR`
but `x = R theta rArr tau_(rest) =- 4kR^(2)theta = I alpha`
`= (3)/(2)mR^(2).alpha` and `alpha =- omega^(2) theta` :