Two particles of equal masses m go round a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is `v=sqrt((Gm)/(nR))`. Find the value of n.

Answer

To solve the problem, we need to analyze the forces acting on each particle as they revolve around the center of the circle due to their mutual gravitational attraction.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two particles, each of mass \( m \), revolving in a circular path of radius \( R \).
- The distance between the two particles is \( 2R \) since they are on opposite sides of the circle.

2. **Gravitational Force**:
- The gravitational force \( F_g \) between the two particles can be calculated using Newton's law of gravitation:
\[ F_g = \frac{G m^2}{(2R)^2} = \frac{G m^2}{4R^2} \]

3. **Centrifugal Force**:
- The centrifugal force \( F_c \) acting on one of the particles due to its circular motion can be expressed as:
\[ F_c = \frac{m v^2}{R} \]

4. **Setting Forces Equal**:
- For the particles to be in circular motion, the gravitational force must equal the centrifugal force:
\[ F_g = F_c \]
Substituting the expressions we derived:
\[ \frac{G m^2}{4R^2} = \frac{m v^2}{R} \]

5. **Canceling Mass**:
- We can cancel one \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{G m}{4R^2} = \frac{v^2}{R} \]

6. **Rearranging the Equation**:
- Multiplying both sides by \( R \):
\[ \frac{G m}{4R} = v^2 \]

7. **Expressing \( v \)**:
- Thus, we can express \( v \) as:
\[ v = \sqrt{\frac{G m}{4R}} \]

8. **Comparing with Given Expression**:
- We are given that \( v = \sqrt{\frac{G m}{nR}} \).
- Setting the two expressions for \( v \) equal to each other:
\[ \sqrt{\frac{G m}{4R}} = \sqrt{\frac{G m}{nR}} \]

9. **Squaring Both Sides**:
- Squaring both sides to eliminate the square root:
\[ \frac{G m}{4R} = \frac{G m}{nR} \]

10. **Canceling Common Terms**:
- We can cancel \( Gm \) and \( R \) (assuming they are non-zero):
\[ \frac{1}{4} = \frac{1}{n} \]

11. **Finding \( n \)**:
- Taking the reciprocal gives:
\[ n = 4 \]

### Final Answer:
The value of \( n \) is \( 4 \).