7.
\begin{figure}[h]
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\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{c4b453e7-8a32-458b-8041-58c9e4ef9533-6_710_729_172_672}
\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 fixed at a point \(O\). The particle is held with the string taut and \(O P\) horizontal. It is then projected vertically downwards with speed \(u\), where \(u ^ { 2 } = \frac { 3 } { 2 } g a\). When \(O P\) has turned through an angle \(\theta\) and the string is still taut, the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Fig. 3.
- Find an expression for \(v ^ { 2 }\) in terms of \(a , g\) and \(\theta\).
- Find an expression for \(T\) in terms of \(m , g\) and \(\theta\).
- Prove that the string becomes slack when \(\theta = 210 ^ { \circ }\).
- State, with a reason, whether \(P\) would complete a vertical circle if the string were replaced by a light rod.
After the string becomes slack, \(P\) moves freely under gravity and is at the same level as \(O\) when it is at the point \(A\).
- Explain briefly why the speed of \(P\) at \(A\) is \(\sqrt { } \left( \frac { 3 } { 2 } g a \right)\).
The direction of motion of \(P\) at \(A\) makes an angle \(\varphi\) with the horizontal.
- Find \(\varphi\).