Question 4 - A-Level Maths

\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-5_705_501_260_863} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
\end{figure} The region \(R\), shown in Fig. 4.1, is bounded by the curve \(x ^ { 2 } - y ^ { 2 } = k ^ { 2 }\) for \(k \leqslant x \leqslant 4 k\) and the line \(x = 4 k\), where \(k\) is a positive constant. Find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when \(R\) is rotated about the \(x\)-axis.
A uniform lamina occupies the region bounded by the curve \(y = \frac { x ^ { 3 } } { a ^ { 2 } }\) for \(0 \leqslant x \leqslant 2 a\), the \(x\)-axis and the line \(x = 2 a\), where \(a\) is a positive constant. The vertices of the lamina are \(\mathrm { O } ( 0,0 ) , \mathrm { A } ( 2 a , 8 a )\) and \(\mathrm { B } ( 2 a , 0 )\), as shown in Fig. 4.2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-5_714_509_1546_858} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
\end{figure}
1. Find the coordinates of the centre of mass of the lamina.
2. The lamina is freely suspended from the point A and hangs in equilibrium. Find the angle that AB makes with the vertical.