What is the equation for relating tension to frequency?

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SUMMARY

The equation relating tension to frequency for a vibrating string is given by f = (n/2L) * √(T/μ), where f is the frequency, n is the mode number, L is the length of the string, T is the tension, and μ is the linear mass density. In the case of the middle C string of a piano, which vibrates at 261.6 Hz, the tuner needs to increase the tension from 900N to correct the frequency, which is currently flat by 15 Hz. This adjustment requires calculating the necessary tension increase using the established formula.

PREREQUISITES
  • Understanding of wave mechanics and harmonic frequencies
  • Familiarity with the properties of vibrating strings
  • Knowledge of linear mass density (μ) and its calculation
  • Basic algebra for manipulating equations
NEXT STEPS
  • Calculate the required tension increase using the formula f = (n/2L) * √(T/μ)
  • Explore the effects of changing tension on the frequency of different stringed instruments
  • Study the relationship between string length and frequency in musical acoustics
  • Investigate the concept of fundamental frequency and overtones in string vibrations
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Musicians, piano tuners, acoustics engineers, and physics students interested in the principles of sound and vibration in stringed instruments.

gtp405
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Ok this problem seems blatantly easy but for some reason I just cannot find the equation.

Homework Statement



The middle C string of a piano is supposed to vibrate at 261.6 Hz when excited in its fundamental mode. A piano tuner finds that in a piano that has a tension of 900N on this string, the frequency of the vibration is too low (flat) by 15 hz. How much must he increase the tension of the string to achieve the correct frequency?

Homework Equations



This is what I can't find.

The Attempt at a Solution



At first glance to me this problem seems extremely easy, just use a certain formula that relates tension to frequency in some way, but I've looked high and low in the book and can't seem to find it. All I really need is an equation to relate the two.

Thanks!
 
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[tex]f=\frac{n}{2L}{\sqrt{\frac{T}{\mu}}[/tex]

gives the resonant frequencies for a string under tension
 

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