4 The region bounded by the curve \(y = \sqrt { x }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution.

1. Find the \(x\)-coordinate of the centre of mass of this solid. From this solid, the cylinder with radius 1 and length 3 with its axis along the \(x\)-axis (from \(x = 1\) to \(x = 4\) ) is removed.
2. Show that the centre of mass of the remaining object, Q , has \(x\)-coordinate 3 . This object Q has weight 96 N and it is supported, with its axis of symmetry horizontal, by a string passing through the cylindrical hole and attached to fixed points A and B (see Fig. 4). AB is horizontal and the sections of the string attached to A and B are vertical. There is sufficient friction to prevent slipping. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-5_837_819_1034_628} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
3. Find the support forces, \(R\) and \(S\), acting on the string at A and B
   (A) when the string is light,
   (B) when the string is heavy and uniform with a total weight of 6 N .