Two identical spheres placed at a fixed distance exert a gravitational force F on each other. A certain fraction of the mass of the first sphere is transferred to the second sphere, while keeping the distance between their centres unchanged. If the new gravitational force between them becomes 0.75 F , the fraction of the initial mass of the first sphere that was transferred is

Step-by-step solution

Let the initial mass of each sphere be m and the distance between them be r . The initial gravitational force is F = G m^2 r^2 . Let x be the fraction of the mass transferred. The new masses are m₁ = m(1-x) and m₂ = m(1+x) . The new gravitational force is F' = G m(1-x) m(1+x) r^2 = G m^2 (1-x^2) r^2 . Given that F' = 0.75 F = 3 4 F , we have: G m^2 (1-x^2) r^2 = 3 4 G m^2 r^2 1 - x^2 = 3 4 x^2 = 1 4 x = 1 2 Answer: 1 2