Problem $4(20pts)$ :

The system shown in the figure below is comprised of two masses. Mass $m$ can move in the $x-$direction and is connected to a wall by a spring with spring constant $k_1.$ The rod is pinned at its center and is able to rotate about the center point $O.$ The rod has length $l$ and moment of inertia $I_O$ and is attached to a wall at one end by a spring with spring constant $k_2$ and the other end by a damper with damping coefficient $c.$ Mass $m$ and the rod are attached to each other by a rigid link. Assume small angular displacement.

Your tasks:

A$.$ Draw the free body diagram for the mass on the cart. $(4$ pts$)$

B$.$ Draw the free body diagram for the rod $($it$\text{'}$s not massless$).(4$ pts$)$

C$.$ Derive the Equation of Motion in terms of $$ and its derivatives. You should only have one equation of motion at the end. Show all your work. $(6$ pts$)$

D$.$ Derive the Equation of Motion in terms of $x$ and its derivatives. You should only have one equation of motion at the end. Show all your work. $(6$ pts$)$