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

Explanation

The rod will perform small angular oscillations about O. Let `theta` be the angular displacement, `x=ltheta`

Restoring torque
`tau=-k_1xl-k_2xl-mgx/2`
`lalpha=-[(k_1+k_2)l+(mg)/(2)]x`
`(ml^2)/(3)alpha=-[(k_1+k_2)l+(mg)/(2)]ltheta`
`alpha=-[(3(k_1+k_2))/(m)+(3g)/(2l)]theta=-omega^2theta`
`T=2pisqrt((1)/((3(k_1+k_2))/(m)+(3g)/(2l))`
`f=1/T=(1)/(2pi)sqrt((3(k_1+k_2))/(m)+(3g)/(2l))`