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)`

Answer

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 speed.

### Step-by-Step Solution:

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

2. **Set up the centripetal force for both particles**:
For the first particle (mass \( M_1 \), radius \( r_1 \)):
\[ F_{c1} = \frac{M_1 v^2}{r_1} \]
For the second particle (mass \( M_2 \), radius \( r_2 \)):
\[ F_{c2} = \frac{M_2 v^2}{r_2} \]

3. **Given conditions**:
- The masses of the two particles are equal: \( M_1 = M_2 = M \).
- The speed of both particles is the same: \( v_1 = v_2 = v \).

4. **Substituting the equal masses and speeds into the centripetal force equations**:
Substituting \( M_1 \) and \( M_2 \) with \( M \) and \( v_1 \) and \( v_2 \) with \( v \):
\[ F_{c1} = \frac{M v^2}{r_1} \]
\[ F_{c2} = \frac{M v^2}{r_2} \]

5. **Finding the ratio of the centripetal forces**:
To find the ratio \( \frac{F_{c1}}{F_{c2}} \):
\[ \frac{F_{c1}}{F_{c2}} = \frac{\frac{M v^2}{r_1}}{\frac{M v^2}{r_2}} = \frac{M v^2}{r_1} \cdot \frac{r_2}{M v^2} \]
The \( M \) and \( v^2 \) terms cancel out:
\[ \frac{F_{c1}}{F_{c2}} = \frac{r_2}{r_1} \]

6. **Final answer**:
Therefore, the ratio of the centripetal forces is:
\[ \frac{F_{c1}}{F_{c2}} = \frac{r_2}{r_1} \]