A drum, of mass $M,$ is rigidly fixed at the end of a circular steel shaft. The shaft is clamped at

one end supported by a plain bearing at the other. A mass, $m,$ is supported by an inextensible

string enrolled around a drum of radius $r_2.$ The spring with constant $k$ is attached to a drum

of radius $r_1$ and is in pre$-$tension with a force $F_0$ at the static equilibrium position. The radius

of gyration of the two combined concentric and attached drums is $k_g.$ Find the equation of

motion of the system in terms of the coordinate shown.