A pendulum consists of a uniform rod $PQ$, of mass $3m$ and length $2a$, which is rigidly fixed at its end $Q$ to the centre of a uniform circular disc of mass $m$ and radius $a$. The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis $L$ which passes through the end $P$ of the rod and is perpendicular to the rod.

\begin{enumerate}[label=(\alph*)]
\item Show that the moment of inertia of the pendulum about $L$ is $\frac{33}{4}ma^2$.
[5]
\end{enumerate}

The pendulum is released from rest in the position where $PQ$ makes an angle $\alpha$ with the downward vertical. At time $t$, $PQ$ makes an angle $\theta$ with the downward vertical.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that the angular speed, $\dot{\theta}$, of the pendulum satisfies
$$\dot{\theta}^2 = \frac{40g(\cos \theta - \cos \alpha)}{33a}$$
[4]
\end{enumerate}

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Hence, or otherwise, find the angular acceleration of the pendulum.
[3]
\end{enumerate}

Given that $\alpha = \frac{\pi}{20}$ and that $PQ$ has length $\frac{8}{33}$ m,

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item find, to 3 significant figures, an approximate value for the angular speed of the pendulum 0.2 s after it has been released from rest.
[5]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M5  Q8 [17]}}