Mass m is divided into two parts X\\ m and (1-X)\\ m . For a given separation, the value of X for which the gravitational force of attraction between the two pieces becomes maximum is :
Gravitational Force between 2 bodies is given by F = \\dfrac{Gm_1 m_2}{R^2} where , m_1 and m_2 are the masses of the bodies , G is the universal Gravitational constant and R is the distance between them.For a given distance , F = \\dfrac{Gm(1-X) m(X)}{R^2} is maximum when X(1-X) is maximum . By differentiation, we get, \\dfrac{dF}{dX} = \\dfrac{Gm^2}{R^2}\\dfrac{d}{dX}(X(1-X)) \\dfrac{dF}{dX} = \\dfrac{Gm^2}{R^2}\\dfrac{d}{dX}(X-X^2) \\dfrac{dF}{dX} = \\dfrac{Gm^2}{R^2}(1-2X)=0 1-2X =0 X = \\dfrac{1}{2} The gravitational force of attraction has a maximum value at X =1/2