5.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-14_565_696_219_721}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}

A hollow cylinder is fixed with its axis horizontal. A particle $P$ moves in a vertical circle, with centre $O$ and radius $a$, on the smooth inner surface of the cylinder. The particle moves in a vertical plane which is perpendicular to the axis of the cylinder. The particle is projected vertically downwards with speed $\sqrt { 7 a g }$ from the point $A$, where $O A$ is horizontal and $O A = a$. When angle $A O P = \theta$, the speed of $P$ is $v$, as shown in Figure 4.
\begin{enumerate}[label=(\alph*)]
\item Show that $v ^ { 2 } = a g ( 7 + 2 \sin \theta )$
\item Verify that $P$ will move in a complete circle.
\item Find the maximum value of $v$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel M3 2017 Q5 [12]}}