MHT CET · Physics · Waves and Sound
' \(n\) ' 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
- A \(2 \mathrm{n}\)
- B \(\frac{\mathrm{n}}{2}\)
- C \(\frac{\mathrm{n}}{\sqrt{2}}\)
- D \(\sqrt{2} n\)
Answer & Solution
Correct Answer
(B) \(\frac{\mathrm{n}}{2}\)
Step-by-step Solution
Detailed explanation
Frequency of vibration of the string is given by
\(
\begin{aligned}
& \mathrm{n}=\frac{1}{2 \ell} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}=\frac{1}{2 \ell} \sqrt{\frac{\mathrm{T}}{\pi \mathrm{r}^2 \rho}}=\frac{1}{2 \ell \mathrm{r}} \sqrt{\frac{\mathrm{T}}{\pi \rho}} \\
& \therefore \frac{\mathrm{n}^{\prime}}{\mathrm{n}}=\frac{\mathrm{r}}{\mathrm{r}^{\prime}}=\frac{1}{2} \\
& \therefore \mathrm{n}^{\prime}=\frac{\mathrm{n}}{2}
\end{aligned}
\)
\(
\begin{aligned}
& \mathrm{n}=\frac{1}{2 \ell} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}=\frac{1}{2 \ell} \sqrt{\frac{\mathrm{T}}{\pi \mathrm{r}^2 \rho}}=\frac{1}{2 \ell \mathrm{r}} \sqrt{\frac{\mathrm{T}}{\pi \rho}} \\
& \therefore \frac{\mathrm{n}^{\prime}}{\mathrm{n}}=\frac{\mathrm{r}}{\mathrm{r}^{\prime}}=\frac{1}{2} \\
& \therefore \mathrm{n}^{\prime}=\frac{\mathrm{n}}{2}
\end{aligned}
\)
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