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)

Given: Radius of first particle $\left(r_1\right)=r$ and radius of second particle $\left(r_2\right)=2r.$
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 $v_1:v_2=1:\sqrt{2}$.