the density of the stretched string is changed by 2% without change in tension and radius. the change in transverse wave velocity is ? (ans : 1% increase or decrease)  

Dear Student,

Please find below the solution to the asked query:

Equation for the transverse wave velocity in a stretched string is,

$V=\sqrt{\frac{T}{\mu }} \Rightarrow V=\sqrt{\frac{T}{\rho A}}$
If tension and radius (area) are constant. Then the transverse wave velocity depends on the density as,

$V\propto \frac{1}{\sqrt{\rho }}$
Here, the change in density is 2%. Which is  < 5%. Therefore, the change in transverse wave velocity is,

$$\begin{gathered} \frac{∆V}{V}=\frac{1}{2}\left(\frac{∆\rho }{\rho }\right) \Rightarrow \frac{∆V}{V}\times 100=\frac{1}{2}\left(\frac{∆\rho }{\rho }\times 100\right) \\ \Rightarrow \frac{∆V}{V}\times 100=\frac{1}{2}\left(2\%\right) \\ \Rightarrow \frac{∆V}{V}\times 100=1\% \end{gathered}$$

Therefore, the percentage change in transverse wave velocity is 1%. This may be increase or decrease, depends on the nature of change of density.

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