4 The region between the curve \(y = 4 - x ^ { 2 }\) and the \(x\)-axis, from \(x = 0\) to \(x = 2\), is occupied by a uniform lamina. The units of the axes are metres.
- Show that the coordinates of the centre of mass of this lamina are \(( 0.75,1.6 )\).
This lamina and another exactly like it are attached to a uniform rod PQ , of mass 12 kg and length 8 m , to form a rigid body as shown in Fig. 4. Each lamina has mass 6.5 kg . The ends of the rod are at \(\mathrm { P } ( - 4,0 )\) and \(\mathrm { Q } ( 4,0 )\). The rigid body lies entirely in the \(( x , y )\) plane.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b7f8bdfd-33dc-4453-8f3a-ddd24be17372-4_511_956_1836_557}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{figure} - Find the coordinates of the centre of mass of the rigid body.
The rigid body is freely suspended from the point \(\mathrm { A } ( 2,4 )\) and hangs in equilibrium.
- Find the angle that PQ makes with the horizontal.