Potential Energy of a System of Four Particles

To calculate the potential energy of a system of four particles placed at the vertices of a square of side length l, we need to consider the gravitational potential energy between each pair of particles.

Gravitational Potential Energy
The gravitational potential energy between two particles of masses m1 and m2 separated by a distance r is given by the formula:

E = -G * (m1 * m2) / r

Where:
- E is the gravitational potential energy
- G is the gravitational constant (approximately 6.674 × 10^-11 N m^2/kg^2)
- m1 and m2 are the masses of the two particles
- r is the distance between the two particles

Calculating Potential Energy for the System
Let's label the four particles as A, B, C, and D, with A being the top-left vertex of the square and moving clockwise. We will calculate the potential energy for each pair of particles and sum them up to obtain the total potential energy of the system.

1. Particles A and B:
- Distance between A and B: l
- Mass of each particle: m
- Potential energy between A and B: E1 = -G * (m * m) / l

2. Particles B and C:
- Distance between B and C: l
- Mass of each particle: m
- Potential energy between B and C: E2 = -G * (m * m) / l

3. Particles C and D:
- Distance between C and D: l
- Mass of each particle: m
- Potential energy between C and D: E3 = -G * (m * m) / l

4. Particles D and A:
- Distance between D and A: l
- Mass of each particle: m
- Potential energy between D and A: E4 = -G * (m * m) / l

Total Potential Energy of the System
To obtain the total potential energy of the system, we sum up the individual potential energies:

Total Potential Energy = E1 + E2 + E3 + E4

Potential at the Center of the Square
To calculate the potential at the center of the square, we consider the center particle as the fifth particle and calculate the gravitational potential energy between the center particle and each of the four corner particles.

5. Particle at the center and particle A:
- Distance between the center and A: l/√2 (diagonal of the square)
- Mass of each particle: m
- Potential energy between the center and A: E5 = -G * (m * m) / (l/√2)

6. Particle at the center and particle B:
- Distance between the center and B: l/√2 (diagonal of the square)
- Mass of each particle: m
- Potential energy between the center and B: E6 = -G * (m * m) / (l/√2)

7. Particle at the center and particle C:
- Distance between the center and C: l/√2 (diagonal of the square)
- Mass of each particle: m

potential energy of system= 4mghand on center is mgh