3.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-05_556_576_224_687}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\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.
\begin{enumerate}[label=(\alph*)]
\item Show that the distance of the centre of mass of $B$ from $O$ is $\frac { 65 } { 152 } r$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-05_526_1014_1292_478}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\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 $\tan \theta = \frac { 4 } { 5 }$,
\item find the exact value of $k$.

January 2010
\end{enumerate}

\hfill \mbox{\textit{Edexcel M3 2010 Q3 [10]}}