Ncert Exercises & Solutions

A sitar wire is replaced by another wire of the same length and material but three times the earlier radius. If the tension remains the same, by what factor will the frequency change?

Hint: Use the concept of standing waves.

Step 1: Find the frequency of the wave in the wire.

Frequency of vibrations produced by a stretched wire,

ν = \frac{n}{2 l} \sqrt{\frac{T}{μ}}

Step 2 : Find the relation between the radius of the wire and the frequency of the wave .

Mass per unit length, μ = \frac{mass}{length} = \frac{{πr}^{2} lρ}{l} = {πr}^{2} ρ [∵ M = Vρ = Alρ = {πr}^{2} lρ]

ν = \frac{n}{21} \sqrt{\frac{T}{{πr}^{2} ρ}} \Rightarrow ν \propto \sqrt{\frac{1}{r^{2}}}

ν \propto \frac{1}{r}

Hence, when the radius is tripled, the frequency will be

\frac{1}{3}

rd of the previous value.

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