\includegraphics{figure_7}

A hollow cylinder has internal radius $a$. The cylinder is fixed with its axis horizontal. A particle $P$ of mass $m$ is at rest in contact with the smooth inner surface of the cylinder. $P$ is given a horizontal velocity $u$, in a vertical plane perpendicular to the axis of the cylinder, and begins to move in a vertical circle. While $P$ remains in contact with the surface, $OP$ makes an angle $\theta$ with the downward vertical, where $O$ is the centre of the circle. The speed of $P$ is $v$ and the magnitude of the force exerted on $P$ by the surface is $R$ (see diagram).

\begin{enumerate}[label=(\roman*)]
\item Find $v^2$ in terms of $u$, $a$, $g$ and $\theta$ and show that $R = \frac{mu^2}{a} + mg(3\cos\theta - 2)$. [7]

\item Given that $P$ just reaches the highest point of the circle, find $u^2$ in terms of $a$ and $g$, and show that in this case the least value of $v^2$ is $ag$. [4]

\item Given instead that $P$ oscillates between $\theta = \pm\frac{1}{5}\pi$ radians, find $u^2$ in terms of $a$ and $g$. [2]
\end{enumerate}

\hfill \mbox{\textit{OCR M3 2009 Q7 [13]}}