In the above problem, find total mechanical energy of the system .

Explanation

To find the total mechanical energy of the system consisting of three particles each of mass \( m \) at the vertices of an equilateral triangle of side \( a \), we will calculate both the potential energy and kinetic energy of the system.

### Step 1: Understanding the System
We have three particles, each of mass \( m \), positioned at the vertices of an equilateral triangle with side length \( a \). The particles are moving with a speed \( v = \sqrt{\frac{GM}{a}} \).

### Step 2: Calculate the Potential Energy
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 our system, we need to calculate the potential energy for each pair of particles.

1. There are three pairs of particles: (1,2), (2,3), and (3,1).
2. The distance between each pair is \( a \).

Thus, the potential energy for each pair is:
\[ U_{12} = U_{23} = U_{31} = -\frac{G m^2}{a} \]

The total potential energy \( U_{\text{total}} \) of the system is:
\[ U_{\text{total}} = U_{12} + U_{23} + U_{31} = 3 \left(-\frac{G m^2}{a}\right) = -\frac{3G m^2}{a} \]

### Step 3: Calculate the Kinetic Energy
The kinetic energy \( K \) of a single particle moving with speed \( v \) is given by:
\[ K = \frac{1}{2} m v^2 \]
Substituting \( v = \sqrt{\frac{G m}{a}} \):
\[ K = \frac{1}{2} m \left(\frac{G m}{a}\right) = \frac{G m^2}{2a} \]

Since there are three particles, the total kinetic energy \( K_{\text{total}} \) is:
\[ K_{\text{total}} = 3 \left(\frac{G m^2}{2a}\right) = \frac{3G m^2}{2a} \]

### Step 4: Calculate Total Mechanical Energy
The total mechanical energy \( E \) of the system is the sum of the total kinetic energy and the total potential energy:
\[ E = K_{\text{total}} + U_{\text{total}} \]
Substituting the values we calculated:
\[ E = \frac{3G m^2}{2a} - \frac{3G m^2}{a} \]
Combining the terms:
\[ E = \frac{3G m^2}{2a} - \frac{6G m^2}{2a} = -\frac{3G m^2}{2a} \]

### Final Answer
The total mechanical energy of the system is:
\[ E = -\frac{3G m^2}{2a} \]