Question

Find the frequency of small oscillations of a thin uniform vertical rod of mass m and length l hinged at the point O (Fig.). The combined stiffness of the spring is equal to K. The spring are of negligible mass.

Answer 1

Let us locate rod at the position when it 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 oscillating system:
$\frac{1}{2}\frac{m{l}^2}{3}(\dot{\theta }{)}^2+mg\frac{1}{2}(1-\cos \theta )+\frac{1}{2}k(l\theta {)}^2=constant$
Differentiating w.r.t. time , we get :
$\frac{1}{2}\frac{m{l}^2}{3}2\dot{\theta }\ddot{\theta }+\frac{mgl}{2}\sin \theta \dot{\theta }+\frac{1}{2}k{l}^{2}2\theta \dot{\theta }=0$
Thus for very small $\theta$
$\ddot{\theta }=\frac{2g}{2l}(1+\frac{kl}{mg})\theta$
Hence, ${\omega }_{=}\sqrt{\frac{3g}{2l}(1+\frac{kl}{mg})}$