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Question
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 °C = 343 m s–1.
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Solution 1
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 = m/l = 2.10/12 = 0.175 kg m^(-1)`
For tension T, velocity of the transverse wave can be obtained using the relation:
`v = sqrt(T/mu)`
`:. T = v^2mu`
= (343)2 × 0.175 = 20588.575 ≈ 2.06 × 104 N
Solution 2
Here l = 12.0 m, M = 2.10 kg
v = `343 "ms"^(-1)`
Mass per unit length = `M/l = 2.10/12.0 = 0.175 "kg m"^(-1)`
As `v = sqrt(T/m)`
`:. T = v^2. m = (343)^2 xx 0.175 = 2.06 xx 10^(4) N`
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