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It is desired to construct a toy by attaching a solid cone on the top of solid hemisphere as shown in the figure. The radius of the hemisphere and the base of cone is $R$ . If the toy is to be in a stable equilibrium in the shown position then (Given that both the cone and hemisphere are made of same material)

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$H < \sqrt{3} R$

$H > \sqrt{5} R$

$\sqrt{3} R < H < 2 R$

Centre of mass of the combined system must not be above O

answer is A, D.

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Detailed Solution

The figure shown that for stable equilibrium centre of mass must be below O in critical case COM can coincide with O.
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$(- \frac{3 R}{8}) (\frac{2}{3} π R^{3}) + \frac{H}{4} . \frac{1}{3} π R^{2} H = 0 \Rightarrow H = \sqrt{3} R$

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