1.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{e0f141c7-ecd0-4f62-bfad-76c81c2d6396-02_538_881_278_534}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}

A uniform rod $A B$ has mass $m$ and length 4a. The end $A$ of the rod is freely hinged to a fixed point. One end of a light elastic string, of natural length $a$ and modulus $\frac { 1 } { 4 } m g$, is attached to the end $B$ of the rod. The other end of the string is attached to a small light smooth ring $R$. The ring can move freely on a smooth horizontal wire which is fixed at a height $a$ above $A$, and in a vertical plane through $A$. The angle between the rod and the horizontal is $\theta$, where $0 < \theta < \frac { \pi } { 2 }$, as shown in Figure 1. Given that the elastic string is vertical,
\begin{enumerate}[label=(\alph*)]
\item show that the potential energy of the system is

$$2 m g a \left( \sin ^ { 2 } \theta - \sin \theta \right) + \text { constant }$$
\item Show that when $\theta = \frac { \pi } { 6 }$ the rod is in stable equilibrium.
\end{enumerate}

\hfill \mbox{\textit{Edexcel M4 2018 Q1 [11]}}