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

Question

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:

Smart Achievers · Accepted Answer

(3)

$f = \frac{n}{2 l} \sqrt{\frac{T}{μ}}, where n = nth harmonic and is equal to number of antinodes.$

$f_{A} = \frac{1}{2 l} \sqrt{\frac{T}{ρ A_{0}}} \Rightarrow f_{B} = \frac{1}{2 l} \sqrt{\frac{T}{4 ρ A_{0}}}$

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

Solution

1 3 : 5

2 4 : 9

3 1 : 2

4 1 : 4

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