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Q.
If the radii of circular paths of two particles of same masses are in the ratio $1 : 2$, then to have a constant centripetal force, their velocities should be in a ratio of
Given: Radius of first particle $\left(r_{1}\right)=r$ and radius of second particle $\left(r_{2}\right)=2 r .$
We know that when a particle is moving in a circular path, then the centripetal force
$(F)=\frac{m v^{2}}{r}$ or $F \cdot r \propto v^{2}$
or $r \propto v^{2} .$ Therefore,
$\frac{r_{i}}{r_{2}}=\left(\frac{v_{1}}{v_{2}}\right)^{2}$
or $\frac{v_{1}}{v_{2}}=\sqrt{\frac{r_{1}}{r_{2}}}=\sqrt{\frac{1}{2}}$ or $v_{1}: v_{2}=1: \sqrt{2}$.