A thin uniform vertical rod of mass `m` and length l pivoted at point O is shown is Fig. The combined stiffness of the springs is equal to `k`. The mass of the spring is negligible. The 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 bar be rotated through a small angle `theta`.
The restoring torque of the forces `mg`,`k_1x` and `k_2x` about O can be given as
`tau=-[mg((1)/(2))sintheta+k_1x(lcostheta)+k_2x(lcostheta)]`
Since `theta` is small, `sinthetacongtheta`,`x=ltheta` and `costheta=cong1` Putting `k_1+k_2=k`, we obtain
`tau=-[kl^2+mg((l)/(2))]theta`
or `Ialpha=-[kl^2+mg((l)/(2))]theta`
`impliesomega_(OSC)=sqrt(((kl^2+mg((l)/(2)))/((ml^2)/(3))))`
`=sqrt((3k)/(m)+(3g)/(2l))`