Two vibrating strings of same material but lengths $L$ and $2 L$ have radii $2 r$ and $r$ respectively. They are stretched under the same tension and vibrate in their fundamental modes with frequencies $f_{1}$ and $f_{2}$ respectively. The ratio of $\frac{f_{1}}{f_{2}}$ is given by (strings are fixed at both ends)

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Solution

The correct option is D 1
Fundamental frequency for vibration is given by,
$f = \frac{v}{2 L} = \frac{1}{2 L} \sqrt{\frac{T}{μ}}$
$\Rightarrow f = \frac{1}{2 L} \sqrt{\frac{T}{ρ s}}$ , where $μ = ρ s$
$s \to$ cross - sectional area of string
$∵ s = π r^{2})$
$\Rightarrow \frac{f_{1}}{f_{2}} = \frac{r_{2} L_{2}}{r_{1} L_{1}} = \frac{r \times 2 L}{2 r \times L}$
$∴ \frac{f_{1}}{f_{2}} = 1$

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