$(8)$ A bar with mass $M$ and length $L$ is free to rotate about its mid$-$point $O.$ A mass M is hang with a massless string and the string is also pivoted at $O.$ The bar hits the mass at its bottom position and its angular velocity is ${}_0$ just before the hit. After the collision, the bar and the mass stick together. Find their angular velocity just after the collision. Do you need to consider the effect of the gravitational force in this calculation? Why?

Find the kinetic energies just before and just after the collisions.

$[10\%]$