4.
\begin{figure}[h]
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\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{67a9cf74-833f-4b4a-9fde-3c62dcc08e8c-3_531_387_1226_845}
\end{figure}
A body consists of a uniform solid circular cylinder \(C\), together with a uniform solid hemisphere \(H\) which is attached to \(C\). The plane face of \(H\) coincides with the upper plane face of \(C\), as shown in Figure 2. The cylinder \(C\) has base radius \(r\), height \(h\) and mass 3M. The mass of \(H\) is \(2 M\). The point \(O\) is the centre of the base of \(C\).
- Show that the distance of the centre of mass of the body from \(O\) is
$$\frac { 14 h + 3 r } { 20 } .$$
The body is placed with its plane face on a rough plane which is inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 4 } { 3 }\). The plane is sufficiently rough to prevent slipping. Given that the body is on the point of toppling,
- find \(h\) in terms of \(r\).