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

Options

A. `r_(1)//r_(2)`
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 find the ratio of the centripetal forces acting on two particles of equal masses revolving in circular paths of different radii but with the same period.

### Step-by-Step Solution:

1. **Understanding Centripetal Force**:
The centripetal force \( F \) acting on an object moving in a circular path is given by the formula:
\[ F = m \cdot a_c \]
where \( a_c \) is the centripetal acceleration. The centripetal acceleration can be expressed in terms of angular velocity \( \omega \) as:
\[ a_c = \omega^2 \cdot r \]
Thus, the centripetal force can be rewritten as:
\[ F = m \cdot r \cdot \omega^2 \]

2. **Identifying Variables**:
Let:
- Mass of both particles = \( m \)
- Radius of the first particle's path = \( r_1 \)
- Radius of the second particle's path = \( r_2 \)
- Angular velocity of both particles = \( \omega \) (since they have the same period)

3. **Calculating Centripetal Forces**:
For the first particle:
\[ F_1 = m \cdot r_1 \cdot \omega^2 \]
For the second particle:
\[ F_2 = m \cdot r_2 \cdot \omega^2 \]

4. **Finding the Ratio of Centripetal Forces**:
Now, we can find the ratio of the centripetal forces:
\[ \frac{F_1}{F_2} = \frac{m \cdot r_1 \cdot \omega^2}{m \cdot r_2 \cdot \omega^2} \]
The mass \( m \) and \( \omega^2 \) cancel out:
\[ \frac{F_1}{F_2} = \frac{r_1}{r_2} \]

5. **Conclusion**:
Therefore, the ratio of the centripetal forces acting on the two particles is:
\[ \frac{F_1}{F_2} = \frac{r_1}{r_2} \]

### Final Answer:
The ratio of their centripetal forces is \( \frac{r_1}{r_2} \).

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