Two strings of same material are stretched to the same tension. If their radii are in the ratio $1 : 2$ , then respective wave velocities in them will be in ratio

4 : 1

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2 : 1

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1 : 2

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1 : 4

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Solution

The correct option is B $2 : 1$
We know that the velocity of wave in a stretched string is given by:
$v = \sqrt{\frac{T L}{m}}$
Where $T =$ tension $=$ same for both
$L =$ length $=$ same for both
$m =$ mass
hence
$v \propto \frac{1}{\sqrt{m}}$
we know that
mass $=$ volume $\times$ density
$= π r^{2} L ρ$
Since L and $ρ$ are equal for both the strings, hence $m \propto r^{2}$
$\Rightarrow v \propto \frac{1}{\sqrt{m}} \propto \frac{1}{r}$
$\Rightarrow \frac{v_{1}}{v_{2}} = \frac{r_{2}}{r_{1}} = 2 : 1$ .

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