Question 4 - A-Level Maths

A uniform lamina occupies the region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(y = 3 - \sqrt { x }\) for \(0 \leqslant x \leqslant 9\). Find the coordinates of the centre of mass of this lamina.
Fig. 4.1 shows the region bounded by the line \(x = 2\) and the part of the circle \(y ^ { 2 } = 25 - x ^ { 2 }\) for which \(2 \leqslant x \leqslant 5\). This region is rotated about the \(x\)-axis to form a uniform solid of revolution \(S\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-5_675_659_479_705} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
\end{figure}
1. Find the \(x\)-coordinate of the centre of mass of \(S\). The solid \(S\) rests in equilibrium with its curved surface in contact with a rough plane inclined at \(25 ^ { \circ }\) to the horizontal. Fig. 4.2 shows a vertical section containing AB , which is a diameter and also a line of greatest slope of the flat surface of \(S\). This section also contains XY, which is a line of greatest slope of the plane. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-5_494_560_1615_749} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
  \end{figure}
2. Find the angle \(\theta\) that AB makes with the horizontal.