Question 2 - A-Level Maths

A uniform solid hemisphere of volume \(\frac { 2 } { 3 } \pi a ^ { 3 }\) is formed by rotating the region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(y = \sqrt { a ^ { 2 } - x ^ { 2 } }\) for \(0 \leqslant x \leqslant a\), through \(2 \pi\) radians about the \(x\)-axis. Show that the \(x\)-coordinate of the centre of mass of the hemisphere is \(\frac { 3 } { 8 } a\).
A uniform lamina is bounded by the \(x\)-axis, the line \(x = 1\), and the curve \(y = 2 - \sqrt { x }\) for \(1 \leqslant x \leqslant 4\). Its corners are \(\mathrm { A } ( 1,1 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { C } ( 4,0 )\).
1. Find the coordinates of the centre of mass of the lamina. The lamina is suspended with AB vertical and BC horizontal by light vertical strings attached to A and C , as shown in Fig. 2. The weight of the lamina is \(W\). \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{023afdfb-21b6-40fe-9a09-e6769667ee7b-2_346_684_1672_772} \captionsetup{labelformat=empty} \caption{Fig. 2}
  \end{figure}
2. Find the tensions in the two strings in terms of \(W\).