4 Particles $A , B$ and $C$ of masses $2 \mathrm {~kg} , 3 \mathrm {~kg}$ and 5 kg respectively are joined by light rigid rods to form a triangular frame. The frame is placed at rest on a horizontal plane with $A$ at the point $( 0,0 )$, $B$ at the point ( $0.6,0$ ) and $C$ at the point ( $0.4,0.2$ ), where distances in the coordinate system are measured in metres (see Fig. 1).

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{709f3a7a-d857-4813-98ab-de6b41a3a8dc-03_311_661_338_258}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}

$G$, which is the centre of mass of the frame, is at the point $( \bar { x } , \bar { y } )$.
\begin{enumerate}[label=(\alph*)]
\item - Show that $\bar { x } = 0.38$.

\begin{itemize}
  \item Find $\bar { y }$.
\item Explain why it would be impossible for the frame to be in equilibrium in a horizontal plane supported at only one point.
\end{itemize}

A rough plane, $\Pi$, is inclined at an angle $\theta$ to the horizontal where $\sin \theta = \frac { 3 } { 5 }$. The frame is placed on $\Pi$ with $A B$ vertical and $B$ in contact with $\Pi . C$ is in the same vertical plane as $A B$ and a line of greatest slope of $\Pi . C$ is on the down-slope side of $A B$.

The frame is kept in equilibrium by a horizontal light elastic string whose natural length is $l \mathrm {~m}$ and whose modulus of elasticity is $g \mathrm {~N}$. The string is attached to $A$ at one end and to a fixed point on $\Pi$ at the other end (see Fig. 2).

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{709f3a7a-d857-4813-98ab-de6b41a3a8dc-03_605_828_1525_248}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}

The coefficient of friction between $B$ and $\Pi$ is $\mu$.
\item Show that $l = 0.3$.
\item Show that $\mu \geqslant \frac { 14 } { 27 }$.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Mechanics 2021 Q4 [13]}}