4 A smooth cylinder of radius $a \mathrm {~m}$ is fixed with its axis horizontal and $O$ is the centre of a cross-section. Particle $P$, of mass 0.4 kg , and particle $Q$, of mass 0.6 kg , are connected by a light inextensible string of length $\pi a \mathrm {~m}$. The string is held at rest with $P$ and $Q$ at opposite ends of the horizontal diameter of the crosssection through $O$ (see Fig. 1). The string is released and $Q$ begins to descend. When $O P$ has rotated through $\theta$ radians, with $P$ remaining in contact with the cylinder, the speed of each particle is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (see Fig. 2).

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-3_365_433_520_424}
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\caption{Fig. 1}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-3_396_643_484_1000}
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\caption{Fig. 2}
\end{center}
\end{figure}

(i) Show that $v ^ { 2 } = 3.92 a ( 3 \theta - 2 \sin \theta )$ and find an expression in terms of $\theta$ for the normal force of the cylinder on $P$ at this time.\\
(ii) Given that $P$ leaves the surface of the cylinder when $\theta = \alpha$, show that $\sin \alpha = k \alpha$ where $k$ is a constant to be found.

\hfill \mbox{\textit{OCR M3 2013 Q4 [11]}}