# A wire of length $2L,$ is made by joining two wires A and B of same length but different radii r and 2r 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 A is p and that in B is q then the ratio p:q is : Option 1) $1:2$ Option 2) $4:9$ Option 3) $1:4$   Option 4) $3:5$

$\begin{matrix} &A &B \\ Length&L & L\\ radii&r & 2r\\ density& \rho & \rho \\ no. of antinodes &p &q \end{matrix}$

So $f_{A}=\frac{p}{2l_{A}}\sqrt{\frac{T}{\rho \: A_{A}}}=\frac{p }{2l_{A}}\sqrt{\frac{T}{\rho (\pi r^{2})}}$

$f_{B}=\frac{q}{2l_{B}}\sqrt{\frac{T}{\rho \: A_{B}}}=\frac{ q }{2l_{B}}\sqrt{\frac{T}{\rho (\pi r^{2})\times 4}}$

$\frac{f_{A}}{f_{B}}=\frac{p}{q}\times 2=1$

$So\; \frac{p}{q}=\frac{1}{2}$

$p:q\; ::\; 1 : 2$

Option 1)

$1:2$

Option 2)

$4:9$

Option 3)

$1:4$

Option 4)

$3:5$

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