As shown in figure. When a spherical cavity (centred at $O$ ) of radius $1$ is cut out of a uniform sphere of radius $R$ (centred at $C$ ), the centre of mass of remaining (shaded part of sphere is at $G,$ i.e., on the surface of the cavity. $R$ can be determined by the equation:

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M 1 = 4 3 π R 3 ρ M 2 = 4 3 π 1 3 - ρ X c o m = M 1 X 1 + M 2 X 2 M 1 + M 2 ⇒ 4 3 π R 3 ρ 0 + 4 3 π 1 3 - ρ R - 1 4 3 π R 3 ρ + 4 3 π 1 3 - ρ - 2 - R ⇒ R - 1 ( R 3 - 1 ) = 2 - R R ≠ 1 R - 1 R - 1 R 2 + R + 1 = 2 - 1 R 2 + R + 1 2 - R = 1 Alternative: M r e m a i n i n g 2 - R = M c a v i t y 1 - R ⇒ R 3 - 1 3 2 - R = 1 3 R - 1 ⇒ R 2 + R + 1 2 - R = 1

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