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

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Solution

The correct option is D 1
$n = \frac{v}{2 l} = \frac{\sqrt{\frac{T}{μ}}}{2 l}$ , where $μ$ = mass per unit length $= ρ s$
or $n = \frac{1}{2 l} \sqrt{\frac{T}{ρ s}}$ , but $s = π r^{2}$
$∴ n \propto \frac{1}{r l}$
or $\frac{n_{1}}{n_{2}} = \frac{r_{2} l_{2}}{r_{1} l_{1}} = \frac{r \times 2 l}{2 r \times l} = 1$

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