A thin uniform right hollow cylinder, of radius $$\(2 a\)$$ and height $$\(k a\)$$, has a base but no top. A thin uniform hemispherical shell, also of radius $$\(2 a\)$$, is made of the same material as the cylinder. The hemispherical shell is attached to the end of the cylinder forming a container $$\(C\)$$. The open circular rim of the cylinder coincides with the rim of the hemispherical shell. The centre of the base of $$\(C\)$$ is $$\(O\)$$, as shown in Figure 3. (a) Show that the distance from $$\(O\)$$ to the centre of mass of $$\(C\)$$ is $$\[ \frac{\left(k^{2}+4 k+4\right)}{2(k+3)} a \]$$ (5) The container is placed with its circular base on a plane which is inclined at $$\(30^{\circ}\)$$ to the horizontal. The plane is sufficiently rough to prevent $$\(C\)$$ from sliding. The container is on the point of toppling. (b) Find the value of $$\(k\)$$. (3)

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