$($Part A: $10$ points$)$ Particle A $($having a mass of $\backslash ($ m $\backslash ))$ slides on a smooth horizontal surface with a thin homogeneous bar $($having a length of $\backslash ($ L $\backslash ),$ mass $\backslash ($ M $\backslash ),$ and moment of inertia $\backslash ($ I$\_\{$G$\}\backslash ))$ being attached to A with a smooth pin, as shown. A spring and dashpot connect A to ground. The coordinate $\backslash ($ x $\backslash )$ describes the motion of A $(\backslash ($ x $\backslash )$ is measured positive to the right and $\backslash ($ x$=0\backslash )$ when the spring is unstretched$).$ The coordinate $\backslash (\backslash$theta $\backslash )$ describes the orientation of the bar, with $\backslash (\backslash$theta $\backslash )$ being positive for a counterclockwise rotation from the vertical.

Find:

$($a$)$ The kinetic energy associated with the SYSTEM;

$($b$)$ The potential energy associated with the SYSTEM; and

$($c$)$ The Rayleigh dissipation function associated with the SYSTEM.