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.\\
(i) 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]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{b7f8bdfd-33dc-4453-8f3a-ddd24be17372-4_511_956_1836_557}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}

(ii) 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.\\
(iii) Find the angle that PQ makes with the horizontal.

\hfill \mbox{\textit{OCR MEI M3 2006 Q4 [18]}}