7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{376d12ab-022c-4070-a1e0-88eacc2fe48e-5_917_814_303_587}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
A uniform rod \(A B\), of length \(2 a\) and mass \(k M\) where \(k\) is a constant, is free to rotate in a vertical plane about the fixed point \(A\). One end of a light inextensible string of length \(6 a\) is attached to the end \(B\) of the rod and passes over a small smooth pulley which is fixed at the point \(P\). The line \(A P\) is horizontal and of length \(2 a\). The other end of the string is attached to a particle of mass \(M\) which hangs vertically below the point \(P\), as shown in Figure 3. The angle \(P A B\) is \(2 \theta\), where \(0 ^ { \circ } \leq \theta \leq 180 ^ { \circ }\).
- Show that the potential energy of the system is
$$M g a ( 4 \sin \theta - k \sin 2 \theta ) + \text { constant. }$$
The system has a position of equilibrium when \(\cos \theta = \frac { 3 } { 4 }\).
- Find the value of \(k\).
- Hence find the value of \(\cos \theta\) at the other position of equilibrium.
- Determine the stability of each of the two positions of equilibrium.