$4.(1\mathrm{.}5$ points$)$ A cuckoo clock pendulum consists of two pieces glued together:

$(1)$ a thin rod of mass $\backslash ($ m$\_\{2\}\backslash )$ and length $\backslash ($ I$(2)\backslash )$ a circular disk of mass $\backslash ($ m$\_\{2\}\backslash )$ and radius $\backslash ($ r $\backslash ),$ centered at the midpoint of the thin rod.

The pendulum is attached at one end to a fixed pivot $0,$ as shown below, where a time$-$varying torque, $\backslash (\backslash$bar$\{$T$\}($t$)\backslash$hat$\{$Z$\}\backslash ),$ is also applied. Note that the gravity acts on the system.

a$)$ Find the expression for the mass moment of inertia of the pendulum with with regard to O $.$

b$)$ Find an expression for the angular momentum of the pendulunwith regard to O $.$

c$)$ Draw a free body diagram for the system.