Let, M is the mass of the sphere of uniform density ρ and m is the mass of the particle.
The gravitational force of attraction F1 is given by,
F1=GMm(2R)2 ........................(1)
The mass of sphere of volume V, is given by
M=ρV
M=ρ43πR3 ......................(2)
From equations (1) and (2), we get
F1=G[43πR3]m(2R)2 ...................(3)
When the spherical cavity is made in sphere, the gravitational force F2 will be
F2=G[M−ΔM]m(2R)2
F2=G[43πR3−43π(R2)3]m(2R)2
F2=G[43πR3(1−(12)3)]m(2R)2 .......................(4)
Now,
F2F1=G[43πR3(1−(12)3)]m(2R)2G[43πR3]m(2R)2 ...............from equations (3) and (4)
F2F1=1−(12)3
F2F1=1−(18)
F2F1=78