MHT CET · Physics · Waves and Sound
Two wires of same material of radius ' \(r\) ' and ' \(2 r\) ' respectively are welded together end to end. The combination is then used as a sonometer wire under tension ' \(\mathrm{T}\) '. The joint is kept midway between the two bridges. The ratio of the number of loops in the wires with that the joint is a node is
- A \(1: 5\)
- B \(1: 2\)
- C \(1: 4\)
- D \(1: 3\)
Answer & Solution
Correct Answer
(B) \(1: 2\)
Step-by-step Solution
Detailed explanation
Frequency of vibration will be same for both the segments. If \(\mathrm{p}_1\) and \(\mathrm{p}_2\) are the number of loops for the wires of radius \(r\) and \(2 r\) then we have
\(
\begin{aligned}
& \mathrm{n}=\frac{\mathrm{p}_1}{2 \ell} \sqrt{\frac{\mathrm{T}}{\pi \mathrm{r}^2 \rho}}=\frac{\mathrm{p}_2}{4 \ell} \sqrt{\frac{\mathrm{T}}{\pi(2 \mathrm{r})^2 \rho}} \\
& \therefore \frac{\mathrm{p}_1}{\mathrm{p}_2}=\frac{1}{2}
\end{aligned}
\)
\(
\begin{aligned}
& \mathrm{n}=\frac{\mathrm{p}_1}{2 \ell} \sqrt{\frac{\mathrm{T}}{\pi \mathrm{r}^2 \rho}}=\frac{\mathrm{p}_2}{4 \ell} \sqrt{\frac{\mathrm{T}}{\pi(2 \mathrm{r})^2 \rho}} \\
& \therefore \frac{\mathrm{p}_1}{\mathrm{p}_2}=\frac{1}{2}
\end{aligned}
\)
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