Question $5\mathrm{.}4$

A point mass $m$ collides inelastically with a uniform rigid rod of mass $M$ and center$-$of$-$mass moment of inertia I. Suppose that the mass $m$ initially travels at a velocity $u$ perpendicular to the rod, and sticks to the rod at a distance $d$ from the center.

$($a$)$ Find the translational velocity $v,$ and the angular velocity $$ of the rod about its centre immediately after the impact.

$($b$)$ Verify your answer in the case where the point mass hits the center of the