2. (a) Prove that the centre of mass of a uniform solid cone of height \(h\) and base radius \(b\) is at a height of \(\frac { 1 } { 4 } h\) above its base.
(b) A uniform solid cone \(C _ { 1 }\) has height 3 m and base radius 2 m . A smaller cone \(C _ { 2 }\) of height 2 m and base radius 1 m is contained symmetrically inside \(C _ { 1 }\). The bases of \(C _ { 1 }\) and \(C _ { 2 }\) have a common centre and the axis of \(C _ { 2 }\) is part of the axis of \(C _ { 1 }\). If \(C _ { 2 }\) is removed from \(C _ { 1 }\), show that the centre of mass of the remaining solid is at a distance of \(\frac { 11 } { 5 } \mathrm {~m}\) from the vertex of \(C _ { 1 }\).
(c) The remaining solid is suspended from a string which is attached to a point on the outer curved surface at a distance of \(\frac { 1 } { 3 } \sqrt { 13 } \mathrm {~m}\) from the vertex of \(C _ { 1 }\). Given that the axis of symmetry is inclined at an angle of \(\alpha\) to the vertical, find \(\tan \alpha\).