4 A fixed smooth sphere has centre O and radius \(a\). A particle P of mass \(m\) is placed at the highest point of the sphere and given an initial horizontal speed \(u\).
For the first part of its motion, P remains in contact with the sphere and has speed \(v\) when OP makes an angle \(\theta\) with the upward vertical. This is shown in Fig. 4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{76875226-2e6c-4571-9318-ecce51ba8b9f-03_663_679_557_740}
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\caption{Fig. 4}
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- By considering the energy of P , show that \(v ^ { 2 } = u ^ { 2 } + 2 g a ( 1 - \cos \theta )\).
- Show that the magnitude of the normal contact force between the sphere and particle P is
$$m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a } .$$
The particle loses contact with the sphere when \(\cos \theta = \frac { 3 } { 4 }\).
- Find an expression for \(u\) in terms of \(a\) and \(g\).