A steel wire has a length of $12.0 \;m$ and a mass of $2.10 \;kg .$ What should be the tension in the wire so that speed of a transverse wave on the wire equals the speed of sound in dry air at $20\,^{\circ} C =343\; m s ^{-1}$
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Length of the steel wire, $l=12 \,m$
Mass of the steel wire, $m=2.10\, kg$
Velocity of the transverse wave, $v=343\, m / s$
Mass per unit length, $\mu=\frac{m}{l}=\frac{2.10}{12}=0.175\, kg\, m ^{-1}$
For tension $T$, velocity of the transverse wave can be obtained using the relation:
$v=\sqrt{\frac{T}{\mu}}$
$\therefore T=v^{2} \mu$
$=(343)^{2} \times 0.175=20588.575 \approx 2.06 \times 10^{4} \;N$
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