Consider the pendulum mechanism with a pinned support at point $\backslash ($ O $\backslash ).$ The bars has length $\backslash ($ L $\backslash )$ and $\backslash ($ m $\backslash ).$ A damper is attached at point $\backslash ($ C $\backslash )$ and the spring is attached at G$.\backslash ($ G $\backslash )$ is the center of mass of the beam. A force $\backslash ($ F$($t$)=$F$\_\{$o$\}\backslash$sin $\backslash$Omega t $\backslash )$ is applied at point $\backslash ($ C $\backslash ).$

a$)$ Derive the differential equation of motion $($EOM$)$ for the system.

b$)$ What is the damped and undamped natural frequency of the system?

c$)$ Derive an expression for the steady state amplitude of the system.