Three equal masses of 1 kg each are placed at the vertices of an equilateral triangle and other particle of mass 2kg is placed on centroid of triangle which is at a distance of `sqrt 2` m from each of the vertices of the triangle .The force in newton , acting on the particle of mass 2 kg is

Options

A. 2
B. 1
C. 1
D. Zero

Explanation

To solve the problem step by step, we need to analyze the forces acting on the mass of 2 kg placed at the centroid of the equilateral triangle formed by the three 1 kg masses.

### Step 1: Identify the Setup
We have three equal masses (1 kg each) at the vertices of an equilateral triangle and a 2 kg mass at the centroid. The distance from the centroid to each vertex is given as \( \sqrt{2} \) m.

### Step 2: Calculate the Gravitational Force Between Each Mass
The gravitational force \( F \) between two masses can be calculated using Newton's law of gravitation:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
Where:
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses,
- \( r \) is the distance between the masses.

For our case:
- \( m_1 = 1 \) kg (mass at the vertex),
- \( m_2 = 2 \) kg (mass at the centroid),
- \( r = \sqrt{2} \) m.

Thus, the force exerted by one of the 1 kg masses on the 2 kg mass is:
\[ F = \frac{G \cdot 1 \cdot 2}{(\sqrt{2})^2} = \frac{2G}{2} = G \]
This means each of the three 1 kg masses exerts a force \( G \) on the 2 kg mass.

### Step 3: Determine the Direction of Forces
Since the triangle is equilateral, the angles between the lines connecting the centroid to the vertices are 120 degrees. The forces exerted by the three masses will have components in both the x and y directions.

### Step 4: Resolve Forces into Components
Let’s denote the force from each mass as \( F \). The components of the forces can be resolved as follows:
- The x-components of the forces from the three vertices will cancel each other out due to symmetry.
- The y-components will also cancel each other out.

### Step 5: Calculate the Net Force
Since the forces from the three masses are equal in magnitude and symmetrically arranged, the net force acting on the 2 kg mass can be calculated as follows:
- The upward forces (y-components) will balance the downward forces (y-components).
- The leftward forces (x-components) will balance the rightward forces (x-components).

Thus, the net force acting on the 2 kg mass is:
\[ F_{\text{net}} = 0 \]

### Conclusion
The net gravitational force acting on the particle of mass 2 kg at the centroid is **0 N**.