Physics - Rotation Question with Solution | TestHub

Before impact, the bullet is moving with velocity $v$. The initial impact, angular momentum of the system about $O$ the hinge point is $J=L\times mv=mvL$ After the bullet gets embedded in the rod, suppose the system attains angular velocity $\omega$. The moment of inertia of the bullet rod system about the axis through $O$ is, $I=$ (M.I Of bullet + M.I of rod) $=m{L}^2+\frac{1}{3}M{L}^2$ $I=\left(\frac{M+3m}{3}\right)L^2$ Final angular momentum of the system is $By conservation of angular momentum, $J=J^{\prime }$ $\left(\frac{M+3m}{3}\right)L^2\omega =mvL$ $\left(\frac{M+3m}{3}\right)L^2\omega =mvL$ So, the angular velocity of the system just after impact is $\omega =\frac{3mv}{(M+3m)L}$