Four masses, each equal to M are placed at the four corners of a square of side a. Calculate the force of attraction on another mass m, kept at the centre of the square.

Answer

To solve the problem of calculating the gravitational force of attraction on a mass \( m \) placed at the center of a square formed by four masses \( M \) at its corners, we can follow these steps:

### Step 1: Understand the Configuration
We have four masses \( M \) located at the corners of a square with side length \( a \). The center of the square is where the mass \( m \) is placed.

### Step 2: Determine the Distance from the Center to the Corners
The distance from the center of the square to any corner can be calculated using the Pythagorean theorem. The center of the square is at the coordinates \( \left(\frac{a}{2}, \frac{a}{2}\right) \) from any corner at \( (0, 0) \). The distance \( r \) from the center to a corner is given by:
\[ r = \sqrt{\left(\frac{a}{2}\right)^2 + \left(\frac{a}{2}\right)^2} = \sqrt{\frac{a^2}{4} + \frac{a^2}{4}} = \sqrt{\frac{a^2}{2}} = \frac{a}{\sqrt{2}} \]

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

### Step 4: Determine the Direction of the Forces
Each of the four masses \( M \) will exert a gravitational force on \( m \) directed towards itself. Due to symmetry, the forces from the masses at opposite corners will cancel out in the horizontal and vertical directions.

### Step 5: Calculate the Net Force
Since the forces from the masses at opposite corners cancel each other out, the net gravitational force on mass \( m \) at the center due to the four masses \( M \) is zero. Therefore, the total force of attraction on mass \( m \) is:
\[ F_{\text{net}} = 0 \]

### Final Answer
The force of attraction on the mass \( m \) kept at the center of the square is:
\[ F_{\text{net}} = 0 \]