Four indentical masses m are kept at the corners of a square of side a. Find the gravitational force exerted on one of the masses by the other masses.

Explanation

Given all masses are equal
`:.m_1 =m_2=m_3=m_4`
Force between `m_1,m_4=F_1=(G*m^2)/(a^2)"……….."(1)`
Force between `m_4,m_3=F_2=(G*m^2)/(a^2)"………"(2)`
Forces `F_1" and " F_2` act perpendicularly.

Their magnitudes are equal .
`:. F_R=sqrt(2F)=sqrt(2)*(Gm^2)/(a^2)" " ".........."(3)`
Force between `m_4,m_2=(Gm^2)/(sqrt(2a))^2=(Gm^2)/(2a^2)`.
Now forces `F_R" and "F_3` are like parallel.
So resultant is sum of these forces.
Total force at `m_4` due to other masses `=sqrt(2)(Gm^2)/(a^2)+(Gm^2)/(2a^2)=(Gm^2)/(a^2)(sqrt(2)+(1)/(2))`.