Consider the system shown in the Figure $P1,$ where $k_t$ is torsional stiffness. Assume that the mass$-$moment of inertia about $O$ is $I_O.$

a$.$ Determine the number of degrees of freedom.

b$.$ Using energy method, find effective mass and stiffness with respect to degree of freedom $.$ Also, derive the differential equation of motion.

c$.$ Using free$-$body diagram and Newton's law of motion, derive the differential equation of motion.