A uniform right solid cone $$\(C\)$$ has diameter $$\(6 a\)$$ and height $$\(8 a\)$$, as shown in Figure 3. The solid $$\(S\)$$ is formed by removing a cone of height $$\(4 a\)$$ from the top of $$\(C\)$$ and then removing an identical, inverted cone. The vertex of the removed cone is at the point $$\(O\)$$ in the centre of the base of $$\(C\)$$, as shown in Figure 4. (a) Find the distance of the centre of mass of $$\(S\)$$ from $$\(O\)$$. (5) The point $$\(A\)$$ lies on the circumference of the base of $$\(S\)$$ and the point $$\(B\)$$ lies on the circumference of the top of $$\(S\)$$. The points $$\(O, A\)$$ and $$\(B\)$$ all lie in the same vertical plane, as shown in Figure 5. The solid $$\(S\)$$ is freely suspended from the point $$\(B\)$$ and hangs in equilibrium. (b) Find the size of the angle that $$\(A B\)$$ makes with the downward vertical. (4)

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