4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-12_490_1177_219_507}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
A smooth hemisphere of radius \(a\) is fixed on a horizontal surface with its plane face in contact with the surface. The centre of the plane face of the hemisphere is \(O\).
A particle \(P\) of mass \(M\) is disturbed from rest at the highest point of the hemisphere.
When \(P\) is still on the surface of the hemisphere and the radius from \(O\) to \(P\) is at an angle \(\theta\) to the vertical,
- the speed of \(P\) is \(v\)
- the normal reaction between the hemisphere and the particle is \(R\), as shown in Figure 2.
- Show that \(\mathrm { R } = \mathrm { Mg } ( 3 \cos \theta - 2 )\)
- Find, in terms of \(a\) and \(g\), the speed of the particle at the instant when the particle leaves the surface of the hemisphere.