8. A pendulum consists of a uniform rod $P Q$, of mass $3 m$ and length $2 a$, 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 $\frac { 33 } { 4 } m a ^ { 2 }$.

The pendulum is released from rest in the position where $P Q$ makes an angle $\alpha$ with the downward vertical. At time $t , P Q$ makes an angle $\theta$ with the downward vertical.
\item Show that the angular speed, $\dot { \theta }$, of the pendulum satisfies

$$\dot { \theta } ^ { 2 } = \frac { 40 g ( \cos \theta - \cos \alpha ) } { 33 a } .$$
\item Hence, or otherwise, find the angular acceleration of the pendulum.

Given that $\alpha = \frac { \pi } { 20 }$ and that $P Q$ has length $\frac { 8 } { 33 } \mathrm {~m}$,
\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.

\section*{Advanced Level}
\section*{Monday 25 June 2012 - Afternoon}

\section*{Materials required for examination \\
 Mathematical Formulae (Pink)}
Items included with question papers\\
Nil

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.
\end{enumerate}

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