7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{85d8fc7d-8863-419e-8eef-8751a6fb6315-11_412_1054_260_447}
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\caption{Figure 4}
\end{figure}
A hollow sphere has internal radius \(r\) and centre \(O\). A bowl with a plane circular rim is formed by removing part of the sphere. The bowl is fixed to a horizontal floor with the rim uppermost and horizontal. The point \(B\) is the lowest point of the inner surface of the bowl. The point \(A\), where angle \(A O B = 120 ^ { \circ }\), lies on the rim of the bowl, as shown in Figure 4. A particle \(P\) of mass \(m\) is projected from \(A\), with speed \(U\) at \(90 ^ { \circ }\) to \(O A\), and moves on the smooth inner surface of the bowl. The motion of \(P\) takes place in the vertical plane \(O A B\).
- Find, in terms of \(m , g , U\) and \(r\), the magnitude of the force exerted on \(P\) by the bowl at the instant when \(P\) passes through \(B\).
- Find, in terms of \(g , U\) and \(r\), the greatest height above the floor reached by \(P\).
Given that \(U > \sqrt { 2 g r }\)
- show that, after leaving the surface of the bowl, \(P\) does not fall back into the bowl.