Calculating Tension for Steel Wire in Piano

AI Thread Summary
To calculate the tension required for a steel wire in a piano to produce the fundamental frequency of middle C (261.6 Hz), the wave speed must first be determined using the formula v = 2Lf1, resulting in a speed of 183.12 m/s. The tension in the wire can then be calculated using the equation v = √(T/μ), where μ is the mass per unit length of the wire. Given the wire's mass of 4.300 x 10^-3 kg and length of 0.7000 m, μ is calculated as 6.14 x 10^-3 kg/m. By substituting the values into the tension formula, the required tension can be derived. Understanding the difference between the velocities of sound in air and transverse waves in the wire is crucial for accurate calculations.
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Homework Statement



Question: Standing Waves:

A steel wire in a piano has a length of 0.7000m and a mass of 4.300 x 10^-3 kg. To what tension must this wire be stretched in order that the fundamental vibration correspond to middle C (fc=261.6 Hz on the chromatic musical scale)?


Homework Equations



f1=v/wave1 = v/2L


The Attempt at a Solution



f1=v/2L
v=2Lf1= (.7000m)(261.6Hz)= 183.12 m/s
 
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Can you differentiate between the two velocities?

(1) Velocity of sound wave in air which is produced by vibrating wire.

(2) Velocity of transverse wave in a stretched string.
 
You know the speed required right? And the speed of sound in a string is given by v=\sqrt{\frac{T}{\mu}} where \mu is the mass per unit length.
 

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