3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-05_556_576_224_687}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
A bowl \(B\) consists of a uniform solid hemisphere, of radius \(r\) and centre \(O\), from which is removed a solid hemisphere, of radius \(\frac { 2 } { 3 } r\) and centre \(O\), as shown in Figure 1.
- Show that the distance of the centre of mass of \(B\) from \(O\) is \(\frac { 65 } { 152 } r\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-05_526_1014_1292_478}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
The bowl \(B\) has mass \(M\). A particle of mass \(k M\) is attached to a point \(P\) on the outer rim of \(B\). The system is placed with a point \(C\) on its outer curved surface in contact with a horizontal plane. The system is in equilibrium with \(P , O\) and \(C\) in the same vertical plane. The line \(O P\) makes an angle \(\theta\) with the horizontal as shown in Figure 2. Given that \(\tan \theta = \frac { 4 } { 5 }\), - find the exact value of \(k\).
January 2010