5.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{7ecb33af-c165-4b72-98f2-7574fad3fdd5-10_881_1301_173_397}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}

A uniform solid right circular cylinder has height $h$ and radius $r$. The centre of one plane face is $O$ and the centre of the other plane face is $Y$. A cylindrical hole is made by removing a solid cylinder of radius $\frac { 1 } { 4 } r$ and height $\frac { 1 } { 4 } h$ from the end with centre $O$. The axis of the cylinder removed is parallel to $O Y$ and meets the end with centre $O$ at $X$, where $O X = \frac { 1 } { 4 } r$. One plane face of the cylinder removed coincides with the plane face through $O$ of the original cylinder. The resulting solid $S$ is shown in Figure 3.
\begin{enumerate}[label=(\alph*)]
\item Show that the centre of mass of $S$ is at a distance $\frac { 85 h } { 168 }$ from the plane face\\
containing $O$. containing $O$.

The solid $S$ is freely suspended from $O$. In equilibrium the line $O Y$ is inclined at an angle $\arctan ( 17 )$ to the horizontal.
\item Find $r$ in terms of $h$.\\[0pt]
[Question 5 Continued]\\[0pt]
[Question 5 Continued]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS FM Mechanics 2025 Q5 [13]}}