4. The diagram shows three light rods $A B , B C$ and $C A$ rigidly joined together so that $A B C$ is a right-angled triangle with $A B = 45 \mathrm {~cm} , A C = 28 \mathrm {~cm}$ and $\widehat { A B } = 90 ^ { \circ }$. The rods support a uniform lamina, of density $2 m \mathrm {~kg} / \mathrm { cm } ^ { 2 }$, in the shape of a quarter circle $A D E$ with radius 12 cm and centre at the vertex $A$. Three particles are attached to $B C$ : one at $B$, one at $C$ and one at $F$, the midpoint of $B C$.

The masses at $C , F$ and $B$ are $50 m \mathrm {~kg} , 30 m \mathrm {~kg}$ and $20 m \mathrm {~kg}$ respectively.\\
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-5_604_908_756_575}
\begin{enumerate}[label=(\alph*)]
\item Calculate the distance of the centre of mass of the system from
\begin{enumerate}[label=(\roman*)]
\item $A C$,
\item $A B$.
\end{enumerate}\item When the system is freely suspended from a point $P$ on $A C$, it hangs in equilibrium with $A B$ vertical. Write down the length $A P$.
\item When the system is freely suspended from a point $Q$ on $A D$, it hangs in equilibrium with $Q B$ making an angle of $60 ^ { \circ }$ with the vertical. Find the distance $A Q$.
\end{enumerate}

\hfill \mbox{\textit{WJEC Further Unit 6 2023 Q4 [15]}}