A spherical hole is made in a lead sphere of radius R, such that its surface touches the outside surface of the lead sphere and passes through its centre. The mass of the sphere before hollowing was M. With what gravitational force will the hollowed-out lead sphere attract a small sphere of mass m, which lies at a distance d from the centre of the lead sphere on the straight line connecting the centres of the sphere and of the hollow?

Explanation

Volume of the sphere removed `v = 4/3 pi (R/2)^3`
Mass of the sphere removed
`M. = (M)/(4/3 pi R^3) xx 4/3 pi (R/2)^3 = M/8`
The force on the small sphere of mass m = Force due to sphere of radius R with the distance d -Force due to sphere of radius R/2 with the distance (d-R/2)
`i.e., F = (GMm)/(d^2) - (GM.m)/((d -R//2)^2)`
` = (GMm)/(d^2) - (G(M/8)m)/((d - R//2)^2)`
` = GMm [(1)/(d^2) - (1)/(8(d-R//2)^2) ] = (GMm)/(d^2) [1- (1)/(8(1-R//2d)^2) ]`