5. A uniform solid hemisphere has radius \(r\). The centre of the plane face of the hemisphere is \(O\).
- Use algebraic integration to show that the distance from \(O\) to the centre of mass of the hemisphere is \(\frac { 3 } { 8 } r\).
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[You may assume that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ]
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\caption{Figure 1}
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A solid \(S\) is formed by joining a uniform solid hemisphere of radius \(a\) to a uniform solid hemisphere of radius \(\frac { 1 } { 2 } a\). The plane faces of the hemispheres are joined together so that their centres coincide at \(O\), as shown in Figure 1. The mass per unit volume of the smaller hemisphere is \(k\) times the mass per unit volume of the larger hemisphere. - Find the distance from \(O\) to the centre of mass of \(S\).
When \(S\) is placed on a horizontal plane with any point on the curved surface of the larger hemisphere in contact with the plane, \(S\) remains in equilibrium.
- Find the value of \(k\).