2 A fixed hollow sphere with centre O has an inside radius of 2.7 m . A particle P of mass 0.4 kg moves on the smooth inside surface of the sphere.
At first, P is moving in a horizontal circle with constant speed, and OP makes a constant angle of \(60 ^ { \circ }\) with the vertical (see Fig. 2.1).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-3_655_666_488_696}
\captionsetup{labelformat=empty}
\caption{Fig. 2.1}
\end{figure}
- Find the normal reaction acting on P .
- Find the speed of P .
The particle P is now placed at the lowest point of the sphere and is given an initial horizontal speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then moves in part of a vertical circle. When OP makes an angle \(\theta\) with the upward vertical and P is still in contact with the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\) (see Fig. 2.2).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-3_716_778_1653_696}
\captionsetup{labelformat=empty}
\caption{Fig. 2.2}
\end{figure} - Find \(v ^ { 2 }\) in terms of \(\theta\).
- Show that \(R = 4.16 - 11.76 \cos \theta\).
- Find the speed of P at the instant when it leaves the surface of the sphere.