A mass M is split into two parts m ad (M - m) which are then separated by a certain distance. What ratio (m/M) maximises the gravitational force between the parts ?

Explanation

If r is the distance between m and (m - m) then gravitational force will be
`F = G (m (M - m))/(r^(2)) = (G)/(r^(2)) (m M - m^(2))`
for f to be maximum `(DF)/(dm) = 0` as M and r are constants
i.e., `(d)/(dm) [(G)/( r^(2)) (m M m^(2))] = 0 `
i.e., M = 2 m = 0 `[ as (G)/( r^(2) ) ne 0]`
or `((m)/( M)) = ((1)/(2)) ` . i.e., the force will maximum when the two parts are equal