6. The points \(O , A , B\) and \(C\) lie in a straight line, in that order, where \(O A = 0.6 \mathrm {~m}\), \(O B = 0.8 \mathrm {~m}\) and \(O C = 1.2 \mathrm {~m}\). A particle \(P\), moving along this straight line, has a speed of \(\left( \frac { 3 } { 10 } \sqrt { 3 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\) at \(A , \left( \frac { 1 } { 5 } \sqrt { 5 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\) at \(B\) and is instantaneously at rest at \(C\).

1. Show that this information is consistent with \(P\) performing simple harmonic motion with centre \(O\). Given that \(P\) is performing simple harmonic motion with centre \(O\),
2. show that the speed of \(P\) at \(O\) is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
3. find the magnitude of the acceleration of \(P\) as it passes \(A\),
4. find, to 3 significant figures, the time taken for \(P\) to move directly from \(A\) to \(B\). \section*{7.} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{a46d3d34-2381-4e73-837f-a60663fb1419-6_694_690_270_758}
   \end{figure} Figure 3 shows a fixed hollow sphere of internal radius a and centre \(O\). A particle \(P\) of mass \(m\) is projected horizontally from the lowest point \(A\) of a sphere with speed \(\sqrt { } \left( \frac { 7 } { 2 } a g \right)\). It moves in a vertical circle, centre \(O\), on the smooth inner surface of the sphere. The particle passes through the point \(B\), which is in the same horizontal plane as \(O\). It leaves the surface of the sphere at the point \(C\), where \(O C\) makes an angle \(\theta\) with the upward vertical.
5. Find, in terms of \(m\) and \(g\), the normal reaction between \(P\) and the surface of the sphere at \(B\).
6. Show that \(\theta = 60 ^ { \circ }\). After leaving the surface of the sphere, \(P\) meets it again at the point \(A\).
7. Find, in terms of \(a\) and \(g\), the time \(P\) takes to travel from \(C\) to \(A\).