6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_526_620_196_598}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{figure}
Figure 1 shows a bowl formed by removing from a solid hemisphere of radius \(\frac { 3 } { 2 } r\) a smaller hemisphere of radius \(r\) having the same axis of symmetry and the same plane face.
- Show that the centre of mass of the bowl is a distance of \(\frac { 195 } { 304 } r\) from its plane face.
(7 marks)
The bowl has mass \(M\) and is placed with its curved surface on a smooth horizontal plane. A stud of mass \(\frac { 1 } { 2 } M\) is attached to the outer rim of the bowl.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_517_729_1318_539}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{figure}
When the bowl is in equilibrium its plane surface is inclined at an angle \(\alpha\) to the horizontal as shown in Figure 2. - Find tan \(\alpha\).
(6 marks)