Find the frequency of small oscillations of thin uniform vertical rod of mass m and length l hinged at the point O.

Explanation

Let the bar be rotated through a small angle `theta`. The restoring torque of the force mg `k_(1) x and k_(2)x` about O can be given as
`tau = - [mg (l)/(2) sin theta + k_(1)x (l cos theta) + k_(2) x (l cos theta)]`
`rArr tau = - [(k_(1) + k_(2)) l xx cos theta + mg (l)/(2) sin theta]`
since, `theta` is small `sin ~= theta, x = l theta & cos theta ~= 1`
Putting `k_(1) + k_(2) = k` we obtain
`tau = -[kl^(2) + mg (l)/(2)] theta`
`rArr l alpha = - (kl^(2) + mg (l)/(2)) theta`
`rArr omega_(osc) = sqrt(((kl^(2) + mg (l//2))/((ml^(2)//3)))) = sqrt((3k)/(m) + (3g)/(2l))`
`n = (1)/(2pi) sqrt((3k)/(m) + (3g)/(2l))`