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 solve the problem of finding 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 1: Understand the Configuration
We have a ring of mass \( M \) and radius \( R \). The point \( P \) is located on the axis of the ring at a distance \( R \) from the center of the ring.

### Step 2: Identify the Gravitational Field Due to the Ring
The gravitational field \( E \) at a point along the axis of a ring of mass \( M \) and radius \( R \) at a distance \( x \) from the center of the ring is given by the formula:

\[ E = \frac{GMx}{(R^2 + x^2)^{3/2}} \]

Here, \( G \) is the gravitational constant.

### Step 3: Substitute the Values
In our case, since the distance \( x \) from the center of the ring to point \( P \) is equal to the radius \( R \) of the ring, we substitute \( x = R \):

\[ E = \frac{GM \cdot R}{(R^2 + R^2)^{3/2}} = \frac{GM \cdot R}{(2R^2)^{3/2}} \]

### Step 4: Simplify the Expression
Now, simplify the denominator:

\[ (2R^2)^{3/2} = 2^{3/2} \cdot R^3 = 2\sqrt{2} \cdot R^3 \]

So, we can rewrite the gravitational field \( E \):

\[ E = \frac{GM \cdot R}{2\sqrt{2} \cdot R^3} = \frac{GM}{2\sqrt{2} \cdot R^2} \]

### Step 5: Calculate the Gravitational Force on Mass \( m \)
The gravitational force \( F \) acting on the mass \( m \) due to the gravitational field \( E \) is given by:

\[ F = m \cdot E = m \cdot \frac{GM}{2\sqrt{2} \cdot R^2} \]

Thus, the final expression for the gravitational force on mass \( m \) is:

\[ F = \frac{GmM}{2\sqrt{2} \cdot R^2} \]

### Conclusion
The gravitational force on mass \( m \) placed at point \( P \) is:

\[ F = \frac{GmM}{2\sqrt{2} \cdot R^2} \]

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