For a mass of m potential energy of a test mass at distance r is given as (G⋅mr)

Total potential energy of the system is Gm2l×4+2⋅Gm2√2l or,4Gm2l+√2Gm2l ( Due to 4 mass at the vertices and 2 diagonally present,length of diagonal is √2l)

Distance of each mass from the centre is =length of diagonal/2 i.e l√2

So,potential energy at the centre for 4 such mass is ⎛⎜⎝Gml√2⋅4⎞⎟⎠ i.e 4√2Gml

Gravitational Potential at the Center of a Square

The gravitational potential at the center of a square of side length a can be calculated using the formula for gravitational potential energy.

Gravitational Potential Energy Formula
- The gravitational potential energy at a point in a gravitational field is given by the formula: U = -G * (m1 * m2) / r, where U is the gravitational potential energy, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between the two objects.

Calculating Gravitational Potential at the Center of a Square
- In the case of a square of side length a, we can consider the square to be made up of four point masses at its corners.
- The gravitational potential at the center of the square would be the sum of the gravitational potentials due to each of these point masses.
- Since the distances from the center of the square to each of the point masses are equal, we can use the formula for gravitational potential energy to calculate the total gravitational potential at the center of the square.

Final Result
- The gravitational potential at the center of a square of side length a would be the sum of the gravitational potentials due to each of the four point masses at the corners of the square.
- By plugging in the relevant values into the formula, we can calculate the gravitational potential at the center of the square.