1. For circular motion, the centripetal force is

$F_c = \frac{mv^2}{r}$

2. The two particles have the same mass and must have the same constant centripetal force.

So,

$\frac{mv_1^2}{r_1} = \frac{mv_2^2}{r_2}$

Since masses are equal, they cancel:

$\frac{v_1^2}{r_1} = \frac{v_2^2}{r_2}$

3. Given the radii of curvature are in the ratio

$r_1 : r_2 = 3 : 4$

Thus,

$\frac{v_1^2}{v_2^2} = \frac{r_1}{r_2} = \frac{3}{4}$

Taking square root:

$\frac{v_1}{v_2} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$

Hence,

$v_1 : v_2 = \sqrt{3} : 2$

4. Checking options:

• A: $1 : \sqrt{3}$
• B: $2 : \sqrt{3}$
• C: $\sqrt{3} : 2$ ✅
• D: $\sqrt{3} : 1$

Therefore, the correct option is C.