5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-14_565_696_219_721}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\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.
- Show that \(v ^ { 2 } = a g ( 7 + 2 \sin \theta )\)
- Verify that \(P\) will move in a complete circle.
- Find the maximum value of \(v\).