5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-16_456_113_248_977}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at rest vertically below \(O\), with the string taut, as shown in Figure 4.
The particle is then projected horizontally with speed \(u\), where \(u > \sqrt { 2 a g }\)
Air resistance is modelled as being negligible.
At the instant when the string makes an angle \(\theta\) with the upward vertical through \(O\), the speed of \(P\) is \(v\) and the string goes slack.
- Show that \(3 v ^ { 2 } = u ^ { 2 } - 2 a g\)
From the instant when the string goes slack to the instant when \(O P\) is next horizontal, \(P\) moves as a projectile.
The time from the instant when the string goes slack to the instant when \(O P\) is next horizontal is \(T\)
Given that \(\theta = 30 ^ { \circ }\)
- show that \(T = \frac { 2 v } { g }\)
- Hence, show that the string goes taut again when it is next horizontal.