Four point masses each mass m kept at the vertices of a square. A point mass m is kept at the point of intersection of the diagonal of a square. What be the force experienced by central mass m ?

Explanation

To solve the problem of finding the force experienced by the central mass \( m \) placed at the intersection of the diagonals of a square formed by four point masses \( m \) at the vertices, we can follow these steps:

### Step 1: Understand the Configuration
We have a square with four point masses \( m \) located at its vertices. The central mass \( m \) is placed at the intersection of the diagonals of the square.

### Step 2: Determine the Distances
Let the side length of the square be \( a \). The distance from each vertex to the center (intersection of the diagonals) can be calculated using the Pythagorean theorem. The distance \( r \) from a vertex to the center is given by:
\[ r = \frac{a}{\sqrt{2}} \]

### Step 3: Calculate the Gravitational Force from Each Vertex
The gravitational force \( F \) exerted by each mass \( m \) at a distance \( r \) on the central mass \( m \) is given by Newton's law of gravitation:
\[ F = \frac{G m^2}{r^2} \]
Substituting \( r = \frac{a}{\sqrt{2}} \):
\[ F = \frac{G m^2}{\left(\frac{a}{\sqrt{2}}\right)^2} = \frac{G m^2}{\frac{a^2}{2}} = \frac{2G m^2}{a^2} \]

### Step 4: Analyze the Direction of Forces
Each of the four masses exerts a force on the central mass. The forces from the masses at opposite corners will have equal magnitudes but will act in opposite directions.

### Step 5: Calculate the Net Force
Since the forces from the masses at opposite corners cancel each other out, we can analyze the forces in pairs:
1. The force from mass at (0,0) and (a,a) will cancel.
2. The force from mass at (0,a) and (a,0) will also cancel.

Thus, the net force \( F_{net} \) on the central mass \( m \) is:
\[ F_{net} = 0 \]

### Conclusion
The total force experienced by the central mass \( m \) is zero due to the symmetry of the configuration.

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