Question

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

• A. $4 : 1$
• B. $1: \sqrt{2}$
• C. $1 : 4$
• D. $\sqrt{2} : 1$

Answer 1

Answer: (B)

$1: \sqrt{2}$

Answer 2

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 $v_{1}: v_{2}=1: \sqrt{2}$.