3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0760e668-c87a-4159-a59c-f880640d14e7-3_479_444_214_815}
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\caption{Figure 1}
\end{figure}
A framework consists of two uniform rods \(A B\) and \(B C\), each of mass \(m\) and length \(2 a\), joined at \(B\). The mid-points of the rods are joined by a light rod of length \(a \sqrt { } 2\), so that angle \(A B C\) is a right angle. The framework is free to rotate in a vertical plane about a fixed smooth horizontal axis. This axis passes through the point \(A\) and is perpendicular to the plane of the framework. The angle between the \(\operatorname { rod } A B\) and the downward vertical is denoted by \(\theta\), as shown in Fig. 1.
- Show that the potential energy of the framework is
$$- m g a ( 3 \cos \theta + \sin \theta ) + \text { constant } .$$
- Find the value of \(\theta\) when the framework is in equilibrium, with \(B\) below the level of \(A\).
- Determine the stability of this position of equilibrium.