7.
\begin{figure}[h]
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\caption{Figure 3}
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\end{figure}
A loaded plate \(L\) is modelled as a uniform rectangular lamina \(A B C D\) and three particles. The sides \(C D\) and \(A D\) of the lamina have lengths \(5 a\) and \(2 a\) respectively and the mass of the lamina is \(3 m\). The three particles have mass \(4 m , m\) and \(2 m\) and are attached at the points \(A , B\) and \(C\) respectively, as shown in Fig. 3.
- Show that the distance of the centre of mass of \(L\) from \(A D\) is \(2.25 a\).
- Find the distance of the centre of mass of \(L\) from \(A B\).
The point \(O\) is the mid-point of \(A B\). The loaded plate \(L\) is freely suspended from \(O\) and hangs at rest under gravity.
- Find, to the nearest degree, the size of the angle that \(A B\) makes with the horizontal.
A horizontal force of magnitude \(P\) is applied at \(C\) in the direction \(C D\). The loaded plate \(L\) remains suspended from \(O\) and rests in equilibrium with \(A B\) horizontal and \(C\) vertically below \(B\).
- Show that \(P = \frac { 5 } { 4 } \mathrm { mg }\).
- Find the magnitude of the force on \(L\) at \(O\).