Four particles each of mass `m` are kept at the four vertices of a square of side 'a' . Find gravitational potential energy of this system.

Explanation

To find the gravitational potential energy of a system of four particles each of mass \( m \) located at the vertices of a square of side \( a \), we can follow these steps:

### Step 1: Identify the pairs of particles and their distances
In a square configuration with four particles, we have:
- Four pairs of particles that are separated by a distance \( a \) (the sides of the square).
- Two pairs of particles that are separated by a distance \( \sqrt{2}a \) (the diagonals of the square).

### Step 2: Calculate the potential energy for pairs separated by distance \( a \)
The gravitational potential energy \( U \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[ U = -\frac{G m_1 m_2}{r} \]
For the pairs separated by distance \( a \):
- There are 4 such pairs.
- The potential energy for one pair is:
\[ U_a = -\frac{G m m}{a} = -\frac{G m^2}{a} \]
Thus, the total potential energy for the four pairs is:
\[ U_{total_a} = 4 \cdot U_a = 4 \left(-\frac{G m^2}{a}\right) = -\frac{4G m^2}{a} \]

### Step 3: Calculate the potential energy for pairs separated by distance \( \sqrt{2}a \)
For the pairs separated by distance \( \sqrt{2}a \):
- There are 2 such pairs.
- The potential energy for one pair is:
\[ U_{\sqrt{2}} = -\frac{G m m}{\sqrt{2}a} = -\frac{G m^2}{\sqrt{2}a} \]
Thus, the total potential energy for these two pairs is:
\[ U_{total_{\sqrt{2}}} = 2 \cdot U_{\sqrt{2}} = 2 \left(-\frac{G m^2}{\sqrt{2}a}\right) = -\frac{2G m^2}{\sqrt{2}a} \]

### Step 4: Combine the total potential energies
Now we can combine the total potential energies from both sets of pairs:
\[ U_{total} = U_{total_a} + U_{total_{\sqrt{2}}} \]
Substituting the values we found:
\[ U_{total} = -\frac{4G m^2}{a} - \frac{2G m^2}{\sqrt{2}a} \]

### Step 5: Simplify the expression
To simplify, we can factor out \( -\frac{G m^2}{a} \):
\[ U_{total} = -\frac{G m^2}{a} \left( 4 + \frac{2}{\sqrt{2}} \right) \]
Since \( \frac{2}{\sqrt{2}} = \sqrt{2} \), we can rewrite it as:
\[ U_{total} = -\frac{G m^2}{a} \left( 4 + \sqrt{2} \right) \]

### Final Answer
Thus, the gravitational potential energy of the system of four particles is:
\[ U_{total} = -\frac{G m^2}{a} (4 + \sqrt{2}) \]