The particle P , of mass 48 kg , is now placed on the highest point of a fixed solid sphere with centre O and radius 2.5 m . The surface of the sphere is smooth. The particle P is given an initial horizontal velocity of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and it then moves in part of a vertical circle with centre O and radius 2.5 m . When OP makes an angle \(\theta\) with the upward vertical and P is still in contact with the surface of the sphere, P has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction of the sphere on P is \(R \mathrm {~N}\), as shown in Fig. 2.2.
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\includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-3_590_617_1706_804}
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\caption{Fig. 2.2}
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- Show that \(v ^ { 2 } = u ^ { 2 } + 49 - 49 \cos \theta\).
- Find an expression for \(R\) in terms of \(u\) and \(v\).
- Given that P loses contact with the surface of the sphere at the instant when its speed is \(4.15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(u\).