3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f06704e5-454c-41c1-9577-b1210f60480d-06_608_924_226_541}
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\caption{Figure 2}
\end{figure}
A uniform solid cylinder has radius \(2 a\) and height \(h ( h > a )\).
A solid hemisphere of radius \(a\) is removed from the cylinder to form the vessel \(V\).
The plane face of the hemisphere coincides with the upper plane face of the cylinder.
The centre \(O\) of the hemisphere is also the centre of the upper plane face of the cylinder, as shown in Figure 2.
- Show that the centre of mass of \(V\) is \(\frac { 3 \left( 8 h ^ { 2 } - a ^ { 2 } \right) } { 8 ( 6 h - a ) }\) from \(O\).
The vessel \(V\) is placed on a rough plane which is inclined at an angle \(\phi\) to the horizontal. The lower plane circular face of \(V\) is in contact with the inclined plane.
Given that \(h = 5 a\), the plane is sufficiently rough to prevent \(V\) from slipping and \(V\) is on the point of toppling,
- find, to three significant figures, the size of the angle \(\phi\).