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Question

A cavity of radius R/2 is made inside a solid sphere of radius R. The centre of the cavity is located at a distance R/2 from the centre of the sphere. The gravitational force on a particle of mass m at a distance R/2 from the centre of the sphere on the line joining both the centres of the sphere and the cavity is (opposite to the centre of the cavity)
[Here g=(GM)/R2, where M is the mass of the sphere]

A
mg2
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B
3mg8
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C
mg16
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D
none of these
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Solution

The correct option is B 3mg8
The case can be considered as a sphere of density ρ and another sphere of density ρ of the exact dimensions and position as the cavity.
Attractive force due to the sphere of density ρ=G(ρ43π(R/2)3)m(R/2)2=mg2(Since M=ρ43πR3)
Force due to sphere of density ρ=G(ρ43π(R/2)3)R2=mg8
Hence net force =mg2mg8=3mg8

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