6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-4_433_282_196_726}
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\caption{Fig. 3}
\end{figure}
Figure 3 shows a uniform rectangular lamina \(A B C D\) of mass \(8 m\) in which the sides \(A B\) and \(B C\) are of length \(a\) and \(2 a\) respectively. Particles of mass \(2 m , 6 m\) and \(4 m\) are fixed to the lamina at the points \(A , B\) and \(D\) respectively.
- Write down the distance of the centre of mass from \(A D\).
- Show that the distance of the centre of mass from \(A B\) is \(\frac { 4 } { 5 } a\).
Another particle of mass \(k m\) is attached to the lamina at the point \(B\).
- Show that the distance of the centre of mass from \(A D\) is now given by \(\frac { ( 10 + k ) a } { 20 + k }\).
(4 marks)
Given that when the lamina is suspended freely from the point \(A\) the side \(A B\) makes an angle of \(45 ^ { \circ }\) with the vertical, - find the value of \(k\).