Question

A uniform rod of mass $m$ and length $l$ can rotate freely on a smooth horizontal plane about a vertical axis hinged at point $H$ . A point mass having the same mass $m$ coming with an initial speed u perpendicular to the rod, strikes the rod and sticks to it at a distance of $\frac{3 l}{4}$ from hinge point. The angular velocity of the rod just after collision is $\frac{36 u}{n l}$ . Find the value of $n$ .

Answer 1

Angular Momentum conservation about hinge,
$L_{i}=L_{f}$
$\Rightarrow mu\left(\frac{3 l}{4}\right)=\left(\frac{m l^{2}}{3} + m \left(\frac{3 l}{4}\right)^{2}\right)\omega \Rightarrow u\left(\frac{3}{4}\right)=\omega l\times \frac{43}{48}$
$\Rightarrow \omega =\frac{36 u}{43 l}$