3 Fig. 3.1 shows a thin rectangular frame ABCD , with part of it filled by a triangular lamina ABD . $\mathrm { AD } = 30 \mathrm {~cm}$ and $\mathrm { AB } = x \mathrm {~cm}$. Together they form the composite structure S .

The centre of mass of $S$ lies at a point $M , 16.5 \mathrm {~cm}$ from $A D$ and 11.7 cm from $A B$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-4_572_953_450_242}
\captionsetup{labelformat=empty}
\caption{Fig. 3.1}
\end{center}
\end{figure}

The frame and the triangular lamina are both uniform but made of different materials. The mass of the frame is 1.7 kg .
\begin{enumerate}[label=(\alph*)]
\item Show that the triangular lamina has a mass of 3.3 kg .
\item Determine the value of $x$, correct to $\mathbf { 3 }$ significant figures.

One end of a light inextensible string is attached to S at D . The other end is attached to a fixed point on a vertical wall. For S to hang in equilibrium with AD vertical, a force of magnitude $Q N$ is applied to S as shown in Fig. 3.2. The line of action of this force lies in the same plane as S . The string is taut and lies in the same plane as S at an angle $\phi$ to the downward vertical.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-4_611_994_1756_242}
\captionsetup{labelformat=empty}
\caption{Fig. 3.2}
\end{center}
\end{figure}
\item By taking moments about D , show that $Q = 50.5$, correct to 3 significant figures.
\item Determine, in degrees, the value of $\phi$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics A AS 2022 Q3 [10]}}