A classic example of a combination rigid$-$body$-$particle system is a point mass sliding on a pinned beam $($or seesaw$).$ As shown in Figure below, the system consists of a thin rigid beam, pinned to a support at its center of mass, and a particle, which slides on the beam without friction.

Assume the particle has mass $m,$ and that the beam has mass $M,$ with moment of inertia $I_0$ about $O.$

$($a$)$ How many degrees of freedom does the system have?

$($b$)$ Draw FBD for all components of the system, with clearly labeled forces. You must show all forces acting on the system.

$($c$)$ Write the equation$($s$)$ of the motion for the system. $($HINT: use polar frame pointing towards $P,$ attached to $0).$