Question

A rigid rod of mass $m$ with a ball of mass $M$ attached to the free end is restrained to oscillate in a vertical plane. Find the natural frequency of oscillation.

Answer 1

At equilibrium position deformation of the string is $x_0$,

$k{x}_0\frac{l}{4}=Mg\left(\frac{3}{4}l\right)+mg\frac{1}{4}$

when the rod is further rotated through an angle $\theta$ from equilibrium position, the restoring torque.

$\tau =-\left[k\right(x+x_0)\frac{1}{4}\cos \theta -Mg\frac{3}{4}L\cos \theta ]-mg\frac{L}{4}\cos \theta$

$=-\left[k\right(x+x_0)\frac{1}{4}-Mg\frac{3}{4}L-mg\frac{l}{4}]\cos \theta$

For small $\theta ,\cos \theta \approx 1$

$\tau =-\frac{kl}{4}x\Rightarrow l\alpha =\frac{k{l}^2}{4}\theta$

$l=M{\left(\frac{3}{4}L\right)}^2+m{\left(\frac{L}{4}\right)}^2$

$f=\frac{1}{2\pi }\sqrt{\frac{3k}{27M+7m}}$