4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f4c33171-597e-4ef3-9f21-3e2271d48f30-07_479_807_246_571}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A light inextensible string of length \(2 a\) has one end attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\). A second light inextensible string of length \(L\), where \(L > \frac { 12 a } { 5 }\), has one of its ends attached to \(P\) and passes over a small smooth peg fixed at a point \(B\). The line \(A B\) is horizontal and \(A B = 2 a\). The other end of the second string is attached to a particle of mass \(\frac { 7 } { 20 } m\), which hangs vertically below \(B\), as shown in Figure 2.
- Show that the potential energy of the system, when the angle \(P A B = 2 \theta\), is
$$\frac { 1 } { 5 } m g a ( 7 \sin \theta - 10 \sin 2 \theta ) + \text { constant. }$$
- Show that there is only one value of \(\cos \theta\) for which the system is in equilibrium and find this value.
- Determine the stability of the position of equilibrium.
\section*{June 2009}