7.
\begin{figure}[h]
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\caption{Figure 4}
\includegraphics[alt={},max width=\textwidth]{044c5866-0a12-4309-8ced-b463e1615fb0-5_604_596_391_760}
\end{figure}
Part of a hollow spherical shell, centre \(O\) and radius \(a\), is removed to form a bowl with a plane circular rim. The bowl is fixed with the circular rim uppermost and horizontal. The point \(A\) is the lowest point of the bowl. The point \(B\) is on the rim of the bowl and \(\angle A O B = 120 ^ { \circ }\), as shown in Fig. 4. A smooth small marble of mass \(m\) is placed inside the bowl at \(A\) and given an initial horizontal speed \(u\). The direction of motion of the marble lies in the vertical plane \(A O B\). The marble stays in contact with the bowl until it reaches \(B\). When the marble reaches \(B\), its speed is \(v\).
- Find an expression for \(v ^ { 2 }\).
- For the case when \(u ^ { 2 } = 6 g a\), find the normal reaction of the bowl on the marble as the marble reaches \(B\).
- Find the least possible value of \(u\) for the marble to reach \(B\).
The point \(C\) is the other point on the rim of the bowl lying in the vertical plane \(O A B\).
- Find the value of \(u\) which will enable the marble to leave the bowl at \(B\) and meet it again at the point \(C\).