If the radius of the circular path of two particles of same masses are in the ratio of $1 : 2$ and have equal centripetal force their velocities should be in the ratio of

1 : \sqrt{2}

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\sqrt{2} : 1

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4 : 1

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1 : 4

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Solution

The correct option is D $1 : \sqrt{2}$
Given : $r_{1} : r_{2} = 1 : 2$ $m_{1} = m_{2} = m$
Centripetal force $F = \frac{m v^{2}}{r}$
$∴$ $\frac{m v_{1}^{2}}{r_{1}} = \frac{m v_{2}^{2}}{r_{2}}$
$⟹$ $\frac{v_{1}}{v_{2}} = \sqrt{\frac{r_{1}}{r_{2}}} = \sqrt{\frac{1}{2}}$
$⟹$ $v_{1} : v_{2} = 1 : \sqrt{2}$

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