4 A smooth cylinder of radius \(a \mathrm {~m}\) is fixed with its axis horizontal and \(O\) is the centre of a cross-section. Particle \(P\), of mass 0.4 kg , and particle \(Q\), of mass 0.6 kg , are connected by a light inextensible string of length \(\pi a \mathrm {~m}\). The string is held at rest with \(P\) and \(Q\) at opposite ends of the horizontal diameter of the crosssection through \(O\) (see Fig. 1). The string is released and \(Q\) begins to descend. When \(O P\) has rotated through \(\theta\) radians, with \(P\) remaining in contact with the cylinder, the speed of each particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-3_365_433_520_424}
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\caption{Fig. 1}
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\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-3_396_643_484_1000}
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\caption{Fig. 2}
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- Show that \(v ^ { 2 } = 3.92 a ( 3 \theta - 2 \sin \theta )\) and find an expression in terms of \(\theta\) for the normal force of the cylinder on \(P\) at this time.
- Given that \(P\) leaves the surface of the cylinder when \(\theta = \alpha\), show that \(\sin \alpha = k \alpha\) where \(k\) is a constant to be found.