A mass m is placed at point P lies on the axis of a ring of mass M and radius R at a distance R from its centre. The gravitational force on mass m is

Options

A. `(GMm)/(sqrt(2)R^(2))`
B. `(GMm)/(2R^(2))`
C. `(GMm)/(2sqrt(2)R^(2)`
D. `(GMm)/(4R^(2))`

Explanation

To find the gravitational force on a mass \( m \) placed at point \( P \) on the axis of a ring of mass \( M \) and radius \( R \) at a distance \( R \) from its center, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Setup**:
- We have a ring of mass \( M \) and radius \( R \).
- A mass \( m \) is placed at point \( P \), which is on the axis of the ring, at a distance \( R \) from the center of the ring.

2. **Identify the Gravitational Field Due to the Ring**:
- The gravitational field \( E_g \) at a distance \( d \) along the axis of a ring of mass \( M \) and radius \( R \) is given by the formula:
\[ E_g = \frac{G M d}{(R^2 + d^2)^{3/2}} \]
- Here, \( G \) is the gravitational constant, \( M \) is the mass of the ring, \( R \) is the radius of the ring, and \( d \) is the distance from the center of the ring to the point where the mass \( m \) is located.

3. **Substitute the Values**:
- In our case, the distance \( d \) is equal to \( R \) (the distance from the center of the ring to point \( P \)).
- Substitute \( d = R \) into the formula:
\[ E_g = \frac{G M R}{(R^2 + R^2)^{3/2}} = \frac{G M R}{(2R^2)^{3/2}} \]

4. **Simplify the Expression**:
- Simplifying the denominator:
\[ (2R^2)^{3/2} = (2^{3/2})(R^3) = 2\sqrt{2} R^3 \]
- Thus, we have:
\[ E_g = \frac{G M R}{2\sqrt{2} R^3} = \frac{G M}{2\sqrt{2} R^2} \]

5. **Calculate the Gravitational Force on Mass \( m \)**:
- The gravitational force \( F \) on mass \( m \) is given by:
\[ F = m \cdot E_g \]
- Substituting \( E_g \):
\[ F = m \cdot \frac{G M}{2\sqrt{2} R^2} = \frac{G M m}{2\sqrt{2} R^2} \]

### Final Expression
The gravitational force on mass \( m \) is:
\[ F = \frac{G M m}{2\sqrt{2} R^2} \]