15.3 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.
The velocity (v) of the transverse wave in the string is given by the relation:
, where T is the tension
T = = = 20588.575 N = 2.06 N
<p>Length of the steel wire, l = 12 m</p><p>Mass of the steel wire, m = 2.1 kg</p><p>Velocity of the transverse wave, v = 343 m/s</p><p>Mass per unit length, <span title="Click to copy mathml"><math><mi>μ</mi></math></span> = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>m</mi></mrow></mrow><mrow><mrow><mi>l</mi></mrow></mrow></mfrac></math></span> = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>2.1</mn></mrow></mrow><mrow><mrow><mn>12</mn></mrow></mrow></mfrac></math></span> = 0.175 kg/m</p><p>The velocity (v) of the transverse wave in the string is given by the relation:</p><p><span title="Click to copy mathml"><math><mi>v</mi><mo>=</mo><mi></mi><msqrt><mrow><mfrac><mrow><mrow><mi>T</mi></mrow></mrow><mrow><mrow><mi>μ</mi></mrow></mrow></mfrac></mrow></msqrt></math></span> , where T is the tension</p><p>T = <span title="Click to copy mathml"><math><msup><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>×</mo><mi>μ</mi></math></span> = <span title="Click to copy mathml"><math><msup><mrow><mrow><mn>343</mn></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>×</mo><mn>0.175</mn></math></span> = 20588.575 N = 2.06 <span title="Click to copy mathml"><math><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>4</mn></mrow></mrow></msup></math></span> N</p>
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