A uniform solid is formed by rotating the region enclosed between the curve with equation \(y = \sqrt { } x\), the \(x\)-axis and the line \(x = 4\), through one complete revolution about the \(x\)-axis. Find the distance of the centre of mass of the solid from the origin \(O\).
(5)
A bowl consists of a uniform solid metal hemisphere, of radius \(a\) and centre \(O\), from which is removed the solid hemisphere of radius \(\frac { 1 } { 2 } a\) with the same centre \(O\).
Show that the distance of the centre of mass of the bowl from \(O\) is \(\frac { 45 } { 112 } a\).
The bowl is fixed with its plane face uppermost and horizontal. It is now filled with liquid. The mass of the bowl is \(M\) and the mass of the liquid is \(k M\), where \(k\) is a constant. Given that the distance of the centre of mass of the bowl and liquid together from \(O\) is \(\frac { 17 } { 48 } a\),