A thin uniform vertical rod of mass `m` and length `l` pivoted at point `O` is shown in Fig. The combined stiffness of the springs is equal to `k`. The mass of the springs is negligible. The angular frequency of small oscillation is

Options

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

Answer

C

Explanation

Let the diagram in disturbed position is as shown.

`:.` Restoring torque about 'O' is
`tau_(rest) = mg ((l)/(2)) sin theta +k_(1)x (l cos theta) +k_(2)x (l cos theta)`
`= (mgl)/(2) sin theta +kxl cos theta ( :' k_(1)+k_(2)=k)`
`1 ( :' theta rarr small , sin theta rarr theta, cos theta rarr 1, x = l theta ) = [(mgl)/(2)+kl^(2)]theta`,
`= I.alpha = (ml^(2))/(3).alpha rArr alpha = omega^(2) theta rArr omega = ?`