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\includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-4_524_611_255_703} A hollow cylinder is fixed with its axis horizontal. The inner surface of the cylinder is smooth and has radius 0.6 m . A particle \(P\) of mass 0.45 kg is projected horizontally with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the lowest point of a vertical cross-section of the cylinder and moves in the plane of the cross-section, which is perpendicular to the axis of the cylinder. While \(P\) remains in contact with the surface, its speed is \(v \mathrm {~ms} ^ { - 1 }\) when \(O P\) makes an angle \(\theta\) with the downward vertical at \(O\), where \(O\) is the centre of the cross-section (see diagram). The force exerted on \(P\) by the surface is \(R \mathrm {~N}\).

1. Show that \(v ^ { 2 } = 4.24 + 11.76 \cos \theta\) and find an expression for \(R\) in terms of \(\theta\).
2. Find the speed of \(P\) at the instant when it leaves the surface.