(a) Use algebraic integration to show that the centre of mass of a uniform solid hemisphere of radius $$\(a\)$$ is a distance $$\(\frac{3}{8} a\)$$ from the centre of its plane face. [You may assume that the volume of a sphere of radius $$\(r\)$$ is $$\(\frac{4}{3} \pi r^{3}\)$$ ] (5) A uniform solid hemisphere has mass $$\(m\)$$ and radius $$\(a\)$$. A particle of mass $$\(k m\)$$ is attached to a point $$\(A\)$$ on the circumference of the plane face of the hemisphere to form the loaded solid $$\(S\)$$. The centre of the plane face of the hemisphere is the point $$\(O\)$$, as shown in Figure 4. The loaded solid $$\(S\)$$ is placed on a horizontal plane. The curved surface of $$\(S\)$$ is in contact with the plane and $$\(S\)$$ rests in equilibrium with $$\(O A\)$$ making an angle $$\(\alpha\)$$ with the horizontal, where $$\(\tan \alpha=\sqrt{3}\)$$ (b) Find the exact value of $$\(k\)$$. (5)

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