4
\includegraphics[max width=\textwidth, alt={}, center]{fef8f0eb-dfed-4d2b-8a58-ca3c85b28686-06_474_631_260_756} The diagram shows a uniform lamina \(A B C D\) with \(A B = 0.75 \mathrm {~m} , A D = 0.6 \mathrm {~m}\) and \(B C = 0.9 \mathrm {~m}\). Angle \(B A D =\) angle \(A B C = 90 ^ { \circ }\).

1. Show that the distance of the centre of mass of the lamina from \(A B\) is 0.38 m , and find the distance of the centre of mass from \(B C\).
   The lamina is freely suspended at \(B\) and hangs in equilibrium.
2. Find the angle between \(B C\) and the vertical.
   \includegraphics[max width=\textwidth, alt={}, center]{fef8f0eb-dfed-4d2b-8a58-ca3c85b28686-08_428_455_260_845} Two particles \(P\) and \(Q\) have masses 0.4 kg and \(m \mathrm {~kg}\) respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string of length 0.5 m which is inclined at an angle of \(60 ^ { \circ }\) to the vertical. \(P\) and \(Q\) are joined to each other by a light inextensible vertical string. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string. The string \(B Q\) is taut and horizontal. The particles rotate in horizontal circles about an axis through \(A\) and \(B\) with constant angular speed \(\omega\) rad s \(^ { - 1 }\) (see diagram). The tension in the string joining \(P\) and \(Q\) is 1.5 N .