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A sitar wire is replaced by another wire of same length and material but of three times the earlier radius. If the tension in the wire remains the same, by what factor will the frequency change ?
Solution
We have, $\mu=\frac{\mathrm{M}}{\mathrm{L}}=\frac{\mathrm{V} \rho}{\mathrm{L}}=\frac{\mathrm{AL} \rho}{\mathrm{L}}=\mathrm{A} \rho$
$\therefore m=\left(\pi r^{2}\right) \rho$
$\ldots(1)$
Now, we have, $f=\frac{1}{2 \mathrm{~L}} \sqrt{\frac{\mathrm{T}}{\mu}}$
$\therefore f=\frac{1}{2 \mathrm{~L}} \sqrt{\frac{\mathrm{T}}{\pi r^{2} \rho}} \Rightarrow f \propto \frac{1}{r}$
( $\because$ other factors are constants)
$\therefore \frac{f_{1}}{f_{2}}=\frac{r_{2}}{r_{1}}=\frac{3}{1} \quad\left(\because r_{2}=3 r_{1}\right)$
$\therefore f_{2}=\left(\frac{1}{3}\right) f_{1}$
