6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{962c2b40-3c45-4eed-a0af-a59068bda0e1-20_533_543_242_760}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(O\). The particle is held with the string taut and \(O P\) horizontal. The particle is then projected vertically downwards with speed \(u\), where \(u ^ { 2 } = \frac { 9 } { 5 } \mathrm { gl }\). When \(O P\) has turned through an angle \(\alpha\) and the string is still taut, the speed of \(P\) is \(v\), as shown in Figure 5. At this instant the tension in the string is \(T\).
- Show that \(T = 3 m g \sin \alpha + \frac { 9 } { 5 } m g\)
- Find, in terms of \(g\) and \(l\), the speed of \(P\) at the instant when the string goes slack.
- Find, in terms of \(l\), the greatest vertical height reached by \(P\) above the level of \(O\).