' $n$ ' waves are produced on a string in 1 second. When the radius of the string is doubled, keeping…
' $n$ ' waves are produced on a string in 1 second. When the radius of the string is doubled, keeping tension same, the number of waves produced in 1 second for the same harmonic will be
$2 \mathrm{n}$
$\frac{\mathrm{n}}{2}$
$\frac{\mathrm{n}}{\sqrt{2}}$
$\sqrt{2} n$
Solution
Frequency of vibration of the string is given by
$\begin{aligned}
& \mathrm{n}=\frac{1}{2 \ell} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}=\frac{1}{2 \ell} \sqrt{\frac{\mathrm{T}}{\pi \mathrm{r}^2 \rho}}=\frac{1}{2 \ell \mathrm{r}} \sqrt{\frac{\mathrm{T}}{\pi \rho}} \\
& \therefore \frac{\mathrm{n}^{\prime}}{\mathrm{n}}=\frac{\mathrm{r}}{\mathrm{r}^{\prime}}=\frac{1}{2} \\
& \therefore \mathrm{n}^{\prime}=\frac{\mathrm{n}}{2}
\end{aligned}$