Question

A rod of length l and mass m, pivoted at one end, is held by a spring at its mid-point and a spring at far end. The springs have spring constant k. Find the frequency of small oscillations about the equilibrium position.

Answer 1

Restoring torgue $\tau =-\left(kl\theta \right)l-\left(k\frac{l}{2}\theta \right)\frac{l}{2}=-\frac{5}{4}k{l}^2\theta$
Now, $\left(\frac{m{l}^2}{3}\right)\alpha =-\left(\frac{5}{4}k{l}^2\right)\theta$
$f=\frac{1}{2\pi }\sqrt{\left|\frac{\alpha }{\theta }\right|}=\frac{1}{2\pi }\sqrt{\frac{15k}{4m}}$