A uniform solid hemisphere $$\(H\)$$ has radius $$\(r\)$$ and centre $$\(O\)$$ (a) Show that the centre of mass of $$\(H\)$$ is $$\(\frac{3 r}{8}\)$$ from $$\(O\)$$ $$\[ \left[\text { You may assume that the volume of } H \text { is } \frac{2 \pi r^{3}}{3}\right] \]$$ (4) A uniform solid $$\(S\)$$, shown below in Figure 3, is formed by attaching a uniform solid right circular cylinder of height $$\(h\)$$ and radius $$\(r\)$$ to $$\(H\)$$, so that one end of the cylinder coincides with the plane face of $$\(H\)$$. The point $$\(A\)$$ is the point on $$\(H\)$$ such that $$\(O A=r\)$$ and $$\(O A\)$$ is perpendicular to the plane face of $$\(H\)$$ (b) Show that the distance of the centre of mass of $$\(S\)$$ from $$\(A\)$$ is $$\[ \frac{5 r^{2}+12 r h+6 h^{2}}{8 r+12 h} \]$$ (5) The solid $$\(S\)$$ can rest in equilibrium on a horizontal plane with any point of the curved surface of the hemisphere in contact with the plane. (c) Find $$\(r\)$$ in terms of $$\(h\)$$. (2)

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