Assignment $2$

Consider a uniform slender rod $($Fig $1).$ Assume the spring remains vertical, the bar is in equilibrium when it is Horizontal, and the small angel assumption applies. Using $x,$ the displacement of mass center of the bar and $,$ the clockwise angular displacement of the bar from equilibrium, as the generalized coordinates, derive the differential equations governing the forced vibrations of this $2-$DOF system.

Fig. $1$