Mass m is split into two parts m and (M-m), which are then separated by a certain distance. What ratio (m/M) maximies the gravitation force between the parts.

Mass m is split into two parts m and (M-m), which are then separated b

1.	Mass m is split into two parts m and (M-m), which are then separated by a certain distance. What ratio (m/M) maximies the gravitation force between the parts.
Answer» Solution :Ifr is the distance m and (M-m), the GRAVITATION FORCE will be `F = G(m(M-m))/(r^(2)) = (G)/(r^(2)) (mM-m^(2))` For F to be maximum dF/dm = 0 as M and r are constants. i.e., `(d)/(dm)[(G)/(r^(2))(mM-m^(2))] = 0` i.e., `M - 2m = 0` [as `G//r^(2) != 0]` or (m/M) = (1/2), i.e., the force will be maximum when the parts are equal.