Waves

JEE Main level, single correct, medium

72 of 142

Question 72 of 142

A wire of length 2L,2L, is made by joining two wires AA and BB of same length but different radii rr and 2r2r and made of the same material. It is vibrating at a frequency such that the joint of the two wires forms a node. If the number of antinodes in wire AA is pp and that in BB is qq then ratio p:qp:q is:

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Answer: (D) 1:21:2

For rod A, f1=n12lTρA1{f}_{1}=\dfrac{{n}_{1}}{2l}\sqrt{\dfrac{T}{\rho {A}_{1}}}
For rod B, f2=n22lTρA2{f}_{2}=\dfrac{{n}_{2}}{2l}\sqrt{\dfrac{T}{\rho {A}_{2}}}
Given f1=f2{f}_{1}={f}_{2}
n12lTρA1=n22lTρA2\dfrac{{n}_{1}}{2l}\sqrt{\dfrac{T}{\rho {A}_{1}}}=\dfrac{{n}_{2}}{2l}\sqrt{\dfrac{T}{\rho {A}_{2}}}
n12lTρA1=n22lTρ(4A1)[∵rB=2rAA2=4A1]\dfrac{{n}_{1}}{2l}\sqrt{\dfrac{T}{\rho {A}_{1}}}=\dfrac{{n}_{2}}{2l}\sqrt{\dfrac{T}{\rho (4{A}_{1})}}\left[∵\begin{matrix}{r}_{B}=2{r}_{A} \\ {A}_{2}=4{A}_{1}\end{matrix}\right]
n1=n22{n}_{1}=\dfrac{{n}_{2}}{2}
n1n2=12\dfrac{{n}_{1}}{{n}_{2}}=\dfrac{1}{2}

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