Pontential energy in a 4-particle system

[Oldblood]

Homework Statement

Find the potential energy in a system being 4 particles, each with mass M forming a square with sides d.

Homework Equations

Newtons law of gravity

The Attempt at a Solution

First I was trying to find the gravitatinal field as a function of d. I soon realized that this is a field in 1 dimension, along the diagonal of the square. 1 particle will feel a force from the 3 others and the force will be parallel to the diogonal of the square. From the furthest particle a particle will feel the force GM^2/(sqrt 2 *d)^2. From the sum of the 2 others it will feel (sqrt 2 *GM^2)/d^2 in the same direction. So we have that g (d)= GM^2/(sqrt 2 *d)^2+(sqrt 2 *GM^2)/d^2, where G is the gravitational constant. Then I simply just integrated this with respect to d to find the potential energy . I am guessing this gives me the potential energy for each particle, so to find the total energy I multiplied by 4. This did gave me total potential energy= - GM^2*(2+4sqrt 2)/d which is wrong. Where did I go wrong?

 

[Oldblood]

I appreciate any help. The goal is to calculate the potential energy in a 4 particle system making up a square

 

[Oldblood]

anyone?