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]
\includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-4_572_953_450_242}
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\caption{Fig. 3.1}
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The frame and the triangular lamina are both uniform but made of different materials. The mass of the frame is 1.7 kg .
- Show that the triangular lamina has a mass of 3.3 kg .
- 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]
\includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-4_611_994_1756_242}
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\caption{Fig. 3.2}
\end{figure} - By taking moments about D , show that \(Q = 50.5\), correct to 3 significant figures.
- Determine, in degrees, the value of \(\phi\).