6 A smooth solid hemisphere of radius \(a\) is fixed with its plane face in contact with a horizontal surface.
The highest point on the hemisphere is H , and the centre of its base is O . A particle of mass \(m\) is held at a point S on the surface of the hemisphere such that angle HOS is \(30 ^ { \circ }\), as shown in Fig. 6. The particle is projected from S with speed \(0.8 \sqrt { a g }\) along the surface of the hemisphere towards H .
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\includegraphics[alt={},max width=\textwidth]{4acb019b-e630-4766-9d7f-39bc0e174ba1-5_358_1056_497_244}
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\caption{Fig. 6}
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- Show that the particle passes through H without leaving the surface of the hemisphere.
After passing through H , the particle passes through a point Q on the surface of the hemisphere, where angle \(\mathrm { HOQ } = \theta ^ { \circ }\).
- State, in terms of \(g\) and \(\theta\), the tangential component of the acceleration of the particle when it is at Q .
The particle loses contact with the hemisphere at Q and subsequently lands on the horizontal surface at a point L .
- Find the value of \(\cos \theta\) correct to 3 significant figures.
- Show that \(\mathrm { OL } = k a\), where \(k\) is to be found correct to 3 significant figures.