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Q.

A thin hollow spherical shell of radius R is cut symmetrically and its lower half is removed to get a hemispherical shell. Now the open base is covered with a circular plate of radius R made of same material and having the same thickness as the shell. The centre of mass of this covered hemisphere lies as a distance of nR from the centre of base, where n is _______ .

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answer is 0.33.

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Detailed Solution

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Let $σ$ be the mass per unit area (it will be same for the shell and the plate)
$m_{1} = σ (2 π R^{2}), m_{2} = σ (π R^{2})$
To find the CM, we can replace the hemispherical shell and the circular
Plate by point mass located at their CM as shown.
CM of the system from C
$= \frac{m_{1} (R / 2) + m_{2} (0)}{m_{1} + m_{2}} = \frac{σ (2 π R^{2}) (R / 2)}{σ 3 π R^{2}} = \frac{R}{3} = 0.33 R$

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