The gravitational force acting on a particle, due to a solid sphere of uniform density and radius R , at a distance of 3 R from the centre of the sphere is F₁ . A spherical hole of radius (R / 2) is now made in the sphere as shown in the figure. The sphere with hole now exerts a force F₂ on the same particle. Ratio of F₁ and F₂ is

Step-by-step solution

Gravitational force due to solid sphere is F₁= G M m (3 R)^2 = G M m 9 R^2 where, M and m are mass of solid sphere and particle respectively. Gravitational force on particle due to sphere with cavity aligned F₂ & = G M m 9 R^2 - G ( M 8 ) m (5 R / 2)^2 & = G M m R^2 [ 1 9 - 4 8 25 ] & = G M m R^2 [ 41 50 9 ] F₁ F₂ & = 50 41 aligned