Two particles of equal masses are revolving in circular paths of radii `r_(1)` and `r_(2)` respectively with the same speed. The ratio of their centripetal force is

Options

A. `(r_(2))/(r_(1))`
B. `sqrt((r_(2))/(r_(1)))`
C. `((r_(1))/(r_(2)))^(2)`
D. `((r_(2))/(r_(1)))^(2)`

Explanation

To solve the problem, we need to understand the relationship between centripetal force, mass, speed, and radius in circular motion.

### Step-by-Step Solution:

1. **Identify the Formula for Centripetal Force**:
The centripetal force \( F \) acting on an object moving in a circular path is given by the formula:
\[ F = \frac{mv^2}{r} \]
where \( m \) is the mass of the object, \( v \) is its speed, and \( r \) is the radius of the circular path.

2. **Set Up the Forces for Both Particles**:
Let:
- For particle 1 (with radius \( r_1 \)):
\[ F_1 = \frac{m v^2}{r_1} \]
- For particle 2 (with radius \( r_2 \)):
\[ F_2 = \frac{m v^2}{r_2} \]

3. **Calculate the Ratio of the Centripetal Forces**:
To find the ratio of the centripetal forces \( F_1 \) and \( F_2 \), we can write:
\[ \frac{F_1}{F_2} = \frac{\frac{m v^2}{r_1}}{\frac{m v^2}{r_2}} \]
The mass \( m \) and speed \( v \) are the same for both particles, so they cancel out:
\[ \frac{F_1}{F_2} = \frac{r_2}{r_1} \]

4. **Final Result**:
Therefore, the ratio of the centripetal forces is:
\[ \frac{F_1}{F_2} = \frac{r_2}{r_1} \]

### Conclusion:
The ratio of the centripetal forces acting on the two particles is equal to the ratio of their respective radii.