6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2226b09b-a19d-4253-baaa-3c3d602d0d6d-10_442_871_264_525}
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\caption{Figure 2}
\end{figure}
A uniform triangular lamina \(A B C\) of mass \(M\) is such that \(A B = A C , B C = 2 a\) and the distance of \(A\) from \(B C\) is \(h\). A line, parallel to \(B C\) and at a distance \(\frac { 2 h } { 3 }\) from \(A\), cuts \(A B\) at \(D\) and cuts \(A C\) at \(E\), as shown in Figure 2.
It is given that the mass of the trapezium \(B C E D\) is \(\frac { 5 M } { 9 }\).
- Show that the centre of mass of the trapezium \(B C E D\) is \(\frac { 7 h } { 45 }\) from \(B C\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2226b09b-a19d-4253-baaa-3c3d602d0d6d-10_357_679_1354_630}
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\caption{Figure 3}
\end{figure}
The portion \(A D E\) of the lamina is folded through \(180 ^ { \circ }\) about \(D E\) to form the folded lamina shown in Figure 3. - Find the distance of the centre of mass of the folded lamina from \(B C\).
The folded lamina is freely suspended from \(D\) and hangs in equilibrium. The angle between \(D E\) and the downward vertical is \(\alpha\).
- Find \(\tan \alpha\) in terms of \(a\) and \(h\).