7. (a) Use algebraic integration to show that the centre of mass of a uniform solid right circular cone of height \(h\) is at a distance \(\frac { 3 } { 4 } h\) from the vertex of the cone.
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[You may assume that the volume of a cone of height \(h\) and base radius \(r\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) ]
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\caption{Figure 2}
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A uniform solid \(S\) consists of a right circular cone, of radius \(r\) and height \(5 r\), fixed to a hemisphere of radius \(r\). The centre of the plane face of the hemisphere is \(O\) and this plane face coincides with the base of the cone, as shown in Figure 2.
(b) Find the distance of the centre of mass of \(S\) from \(O\).
The point \(A\) lies on the circumference of the base of the cone. The solid is suspended by a string attached at \(A\) and hangs freely in equilibrium.
(c) Find the size of the angle between \(O A\) and the vertical.
The mass of the hemisphere is \(M\). A particle of mass \(k M\) is fixed to the surface of the hemisphere on the axis of symmetry of \(S\). The solid is again suspended by the string attached at \(A\) and hangs freely in equilibrium. The axis of symmetry of \(S\) is now horizontal.
(d) Find the value of \(k\).