If the radius of curvature of the path of two particles of same mass are in the ratio 3:2,then in order to have constant centripetal force,their velocities will be in the ratio of:

Options

A. `1:sqrt3`
B. `sqrt3:sqrt2`
C. `2:sqrt3`
D. `sqrt3:1`

Answer

To solve the problem, we need to find the ratio of the velocities of two particles moving along paths with different radii of curvature, given that the ratio of their radii of curvature is 3:2.

### Step-by-Step Solution:

1. **Understanding Centripetal Acceleration**:
The centripetal acceleration \( a_c \) for an object moving in a circular path is given by the formula:
\[ a_c = \frac{v^2}{r} \]
where \( v \) is the velocity of the object and \( r \) is the radius of curvature.

2. **Setting Up the Ratios**:
Let the radius of curvature for the first particle be \( r_1 \) and for the second particle be \( r_2 \). According to the problem, the ratio of the radii is:
\[ \frac{r_1}{r_2} = \frac{3}{2} \]
This implies:
\[ r_1 = 3k \quad \text{and} \quad r_2 = 2k \]
for some constant \( k \).

3. **Applying the Centripetal Acceleration Formula**:
Since we want the centripetal force (and hence acceleration) to be constant for both particles, we can equate their centripetal accelerations:
\[ a_{c1} = a_{c2} \]
This gives us:
\[ \frac{v_1^2}{r_1} = \frac{v_2^2}{r_2} \]

4. **Substituting the Radii**:
Substituting \( r_1 \) and \( r_2 \) into the equation:
\[ \frac{v_1^2}{3k} = \frac{v_2^2}{2k} \]
We can cancel \( k \) from both sides:
\[ \frac{v_1^2}{3} = \frac{v_2^2}{2} \]

5. **Cross Multiplying**:
Cross multiplying gives:
\[ 2v_1^2 = 3v_2^2 \]

6. **Finding the Ratio of Velocities**:
Rearranging this gives:
\[ \frac{v_1^2}{v_2^2} = \frac{3}{2} \]
Taking the square root of both sides, we find:
\[ \frac{v_1}{v_2} = \sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}} \]

7. **Final Answer**:
Thus, the ratio of the velocities \( v_1 : v_2 \) is:
\[ v_1 : v_2 = \sqrt{3} : \sqrt{2} \]