A pendulum consists of a uniform rod $AB$, of length $4a$ and mass $2m$, whose end $A$ is rigidly attached to the centre $O$ of a uniform square lamina $PQRS$, of mass $4m$ and side $a$. The rod $AB$ is perpendicular to the plane of the lamina. The pendulum is free to rotate about a fixed smooth horizontal axis $L$ which passes through $B$. The axis $L$ is perpendicular to $AB$ and parallel to the edge $PQ$ of the square.

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\item Show that the moment of inertia of the pendulum about $L$ is $75ma^2$.
[4]
\end{enumerate}

The pendulum is released from rest when $BA$ makes an angle $\alpha$ with the downward vertical through $B$, where $\tan \alpha = \frac{3}{4}$. When $BA$ makes an angle $\theta$ with the downward vertical through $B$, the magnitude of the component, in the direction $AB$, of the force exerted by the axis $L$ on the pendulum is $X$.

\begin{enumerate}[label=(\alph*)]
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\item Find an expression for $X$ in terms of $m$, $g$ and $\theta$.
[9]
\end{enumerate}

Using the approximation $\theta \approx \sin \theta$,

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find an estimate of the time for the pendulum to rotate through an angle $\alpha$ from its initial rest position.
[6]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M5  Q6 [19]}}