4 Fig. 4.1 shows a uniform lamina in the shape of a sector of a circle of radius $r$ and angle $2 \theta$ where $\theta$ is in radians. The sector consists of a triangle $O A B$ and a segment bounded by the chord $A B$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{8859baf3-f8e8-4fbf-b54f-34f550b02c26-03_358_545_543_255}
\captionsetup{labelformat=empty}
\caption{Fig. 4.1}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Explain why the centre of mass of the segment lies on the radius through the midpoint of $A B$.
\item Show that the distance of the centre of mass of the segment from $O$ is $\frac { 2 r \sin ^ { 3 } \theta } { 3 ( \theta - \sin \theta \cos \theta ) }$.

A uniform circular lamina of radius 5 units is placed with its centre at the origin, $O$, of an $x - y$ coordinate system. A component for a machine is made by removing and discarding a segment from the lamina. The radius of the circle from which the segment is formed is 3 units and the centre of this circle is $O$. The centre of the straight edge of the segment has coordinates $( 0,2 )$ and this edge is perpendicular to the $y$-axis (see Fig. 4.2).

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{8859baf3-f8e8-4fbf-b54f-34f550b02c26-03_748_743_1594_251}
\captionsetup{labelformat=empty}
\caption{Fig. 4.2}
\end{center}
\end{figure}
\item Find the $y$-coordinate of the centre of mass of the component, giving your answer correct to 3 significant figures.\\
$C$ is the point on the component with coordinates $( 0,5 )$. The component is now placed horizontally and supported only at $O$. A particle of mass $m \mathrm {~kg}$ is placed on the component at $C$ and the component and particle are in equilibrium.
\item Find the mass of the component in terms of $m$.

Total Marks for Question Set 3: 40
\end{enumerate}

\hfill \mbox{\textit{OCR Further Mechanics 2021 Q4 [12]}}