6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{176ae8f8-7de9-4993-825a-6067614526ae-16_739_921_299_699}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
A fixed solid sphere has centre \(O\) and radius \(r\).
A particle \(P\) of mass \(m\) is held at rest on the smooth surface of the sphere at \(A\), the highest point of the sphere.
The particle \(P\) is then projected horizontally from \(A\) with speed \(u\) and moves on the surface of the sphere.
At the instant when \(P\) reaches the point \(B\) on the sphere, where angle \(A O B = \theta , P\) is moving with speed \(v\), as shown in Figure 4.
At this instant, \(P\) loses contact with the surface of the sphere.
- Show that
$$\cos \theta = \frac { 2 g r + u ^ { 2 } } { 3 g r }$$
In the subsequent motion, the particle \(P\) crosses the horizontal through \(O\) at the point \(C\), also shown in Figure 4.
At the instant \(P\) passes through \(C , P\) is moving at an angle \(\alpha\) to the horizontal.
Given that \(u ^ { 2 } = \frac { 2 g r } { 5 }\) - find the exact value of \(\tan \alpha\).