The gravitational potential at the center of a square due to four particles located at the vertices can be calculated by considering the gravitational potential energy of each particle with respect to the center.

Calculating the gravitational potential energy of each particle:
1. Consider one of the particles located at a vertex of the square.
2. The distance between this particle and the center of the square is half the length of the diagonal, which can be calculated using the Pythagorean theorem:
Diagonal length = √(L^2 + L^2) = L√2
Distance between particle and center = L√2/2 = L/√2
3. The gravitational potential energy (U) of this particle with respect to the center can be calculated using the formula:
U = -GMm/r
where G is the gravitational constant, M is the mass of the particle, m is the mass at the center (assumed to be negligible), and r is the distance between the particle and the center.
4. Substituting the values, we get:
U = -GMm/(L/√2)

Calculating the total gravitational potential at the center:
1. Since the gravitational potential is a scalar quantity, we can sum up the potential energies of each particle to find the total potential at the center.
2. Since all four particles are identical and equidistant from the center, the potential due to each particle will be the same.
3. Therefore, the total gravitational potential at the center is given by:
U_total = 4 * U
= 4 * (-GMm/(L/√2))
= -4GMm/(L/√2)

Final Answer:
The gravitational potential due to four particles each of mass M located at the vertices of a square with side L at the center of the square is -4GMm/(L/√2).

Explanation:
Gravitational potential is a measure of the potential energy of an object due to gravitational forces. In this case, we calculate the gravitational potential at the center of the square by considering the potential energy of each particle with respect to the center. Since the particles are equidistant from the center, we can sum up the potential energies of each particle to find the total potential at the center. By substituting the appropriate values and using the formula for gravitational potential energy, we arrive at the expression -4GMm/(L/√2) as the gravitational potential at the center of the square.