5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-16_492_442_237_744}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The region \(R\), shown shaded in Figure 3, is bounded by the circle with centre \(O\) and radius \(r\), the line with equation \(x = \frac { 3 } { 5 } r\) and the \(x\)-axis. The region is rotated through one complete revolution about the \(x\)-axis to form a uniform solid \(S\).
- Use algebraic integration to show that the \(x\) coordinate of the centre of mass of \(S\) is \(\frac { 48 } { 65 } r\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-16_394_643_1311_653}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
A bowl is made from a uniform solid hemisphere of radius 6 cm by removing a hemisphere of radius 5 cm . Both hemispheres have the same centre \(A\) and the same axis of symmetry. The bowl is fixed with its open plane face uppermost and horizontal. Liquid is poured into the bowl. The depth of the liquid is 2 cm , as shown in Figure 4. The mass of the empty bowl is \(5 M \mathrm {~kg}\) and the mass of the liquid is \(2 M \mathrm {~kg}\). - Find, to 3 significant figures, the distance from \(A\) to the centre of mass of the bowl with its liquid.