A system in static equilibrium is shown in the figure, with a rigid, massless rod of length $\backslash ($ L $\backslash )$ carrying a point mass m m at its midpoint.

a$)$ Draw the free$-$body diagram of the system and, assuming small oscillations, find the equation of motion in terms of $\backslash (\backslash$theta $\backslash )($theta$)$

b$)$ Given that $\backslash ($ m$=2\backslash$mathrm$\{$~kg$\},\backslash$mathrm$\{$k$\}=4\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\},\backslash$mathrm$\{$c$\}=40\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$s$\}\backslash )$ and $\backslash (\backslash$mathrm$\{$L$\}=1\backslash$mathrm$\{$~m$\}\backslash ),$ determine the natural frequency and damping ratio of the system. What type of vibratory system is it $($underdamped$,$ critically damped, overdamped, or undamped$)?$