3.
\begin{figure}[h]
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\caption{Figure 2}
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\end{figure}
Two uniform rods \(A B\) and \(A C\), each of mass \(2 m\) and length \(2 L\), are freely jointed at \(A\). The mid-points of the rods are \(D\) and \(E\) respectively. A light inextensible string of length \(s\) is fixed to \(E\) and passes round small, smooth light pulleys at \(D\) and \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs vertically. The points \(A , B\) and \(C\) lie in the same vertical plane with \(B\) and \(C\) on a smooth horizontal surface. The angles \(P A B\) and \(P A C\) are each equal to \(\theta ( \theta > 0 )\), as shown in Fig. 2.
- Find the length of \(A P\) in terms of \(s , L\) and \(\theta\).
- Show that the potential energy \(V\) of the system is given by
$$V = 2 m g L ( 3 \cos \theta + \sin \theta ) + \text { constant } .$$
- Hence find the value of \(\theta\) for which the system is in equilibrium.
- Determine whether this position of equilibrium is stable or unstable.