Two vibrating strings of the same material but lengths L and $2$ L have radii $2$ r and r respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes, the one of length L with frequency $n_{1}$ and the other with frequency $n_{2}$ . The ratio $n_{1} / n_{2}$ is given by.

2

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1

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Solution

The correct option is D $1$
Fundamental frequency $n = \frac{1}{2 l} \sqrt{\frac{T}{π^{2} ρ}}$
where $m =$ Mass per unit length of wire

$\Rightarrow n \propto= \frac{1}{l r}$
$\Rightarrow \frac{n_{1}}{n_{2}} = \frac{r_{2}}{r_{1}} \times \frac{l_{2}}{l_{1}} = \frac{r}{2 r} \times \frac{2 L}{L} = \frac{1}{1}$

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