Find the frequency of small oscillatinos of a thin uniform vertical rod of mass ``m and length `l` hinged at the point `O` (figure). The combined stiffness fo the springs is equal to `x`. The mass of the springs is negligible.

Explanation

Let us locate the rod at the position when it makes an angle `theta` from the vertical . In this problem both, the gravity and spring forces are restoring conservative forces, thus from the conservation of mechanical energy of oscillation of the oscillation system `:`
`(1)/(2)(ml^(2))/(3)(theta)^(2)+mg(l)/(2)(1-cos theta)+(1)/(2)k(l theta)^(2)=` constant
Differentiating w.r.t. time, we get `: `
`(1)/(2) (ml^(2))/(3)2 dot (theta)ddot(theta)+(mgl)/(2)sin thetadot(theta)+(1)/(2)k l^(2)2 theta dot (theta)=0`
Thus for very small `theta`
`ddot(theta)=-(3g)/(2l)(1+(kl)/(mg))theta`
Hence, `omega_(0)=sqrt((3g)/(2l)(1+(kl)/(mg)))`