7.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{e5b08946-7311-4cf7-9c5f-5f309a1feed7-13_449_668_221_641}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}

Diagram not drawn to scale

A uniform right circular solid cylinder has radius $3 a$ and height $2 a$. A right circular cone of height $\frac { 3 a } { 2 }$ and base radius $2 a$ is removed from the cylinder to form a solid $S$, as shown in Figure 4. The plane face of the cone coincides with the upper plane face of the cylinder and the centre $O$ of the plane face of the cone is also the centre of the upper plane face of the cylinder.
\begin{enumerate}[label=(\alph*)]
\item Show that the distance of the centre of mass of $S$ from $O$ is $\frac { 69 a } { 64 }$.

The point $A$ is on the open face of $S$ such that $O A = 3 a$, as shown in Figure 4. The solid is now suspended from $A$ and hangs freely in equilibrium.
\item Find the angle between $O A$ and the horizontal.\\
(3)

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{e5b08946-7311-4cf7-9c5f-5f309a1feed7-13_543_826_1653_557}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{center}
\end{figure}

The solid is now placed on a rough inclined plane with the face through $A$ in contact with the inclined plane, as shown in Figure 5. The solid rests in equilibrium on this plane. The coefficient of friction between the plane and $S$ is 0.6 and the plane is inclined at an angle $\phi ^ { \circ }$ to the horizontal. Given that $S$ is on the point of sliding down the plane,
\item show that $\phi = 31$ to 2 significant figures.
\end{enumerate}

\hfill \mbox{\textit{Edexcel M3 2014 Q7 [12]}}