5
\includegraphics[max width=\textwidth, alt={}, center]{3e224c82-68df-427e-a59b-7dc2bfd716a2-3_727_517_258_813} A thin uniform \(\operatorname { rod } A B\) has mass \(\frac { 3 } { 4 } m\) and length \(3 a\). The end \(A\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(C\), mass \(m\) and radius \(a\). The end \(B\) of the rod is rigidly attached to a point on the circumference of a uniform disc with centre \(D\), mass \(4 m\) and radius \(2 a\). The discs and the rod are in the same plane and \(C A B D\) is a straight line. The mid-point of \(C D\) is \(O\). The object consisting of the two discs and the rod is free to rotate about a fixed smooth horizontal axis \(l\), through \(O\) in the plane of the object and perpendicular to the rod (see diagram). Show that the moment of inertia of the object about \(l\) is \(50 m a ^ { 2 }\). The object hangs in equilibrium with \(D\) vertically below \(C\). It is displaced through a small angle and released from rest, so that it makes small oscillations about the horizontal axis \(l\). Show that it will move in approximate simple harmonic motion and state the period of the motion.