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You want to manufacture a guitar such that the instrument will be in tune when each of the strings are tightened to the same tension. The middle (D) string on the guitar should have fundamental frequency 146.83 Hz. The highest (E) string should have fundamental frequency 329.63 Hz. If the D string has linear mass density 0.00256kg/m, what should be the mass density of the E string

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Answer:

0.000507 kg/m

Explanation:

L = Length of string

T = Tension

[tex]\mu[/tex] = Mass density of string

E denotes the E string

D denotes the D String

Frequency is given by

[tex]f=\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}[/tex]

So

[tex]f\propto \sqrt{\dfrac{1}{\mu}}[/tex]

[tex]\dfrac{f_D}{f_E}=\sqrt{\dfrac{\mu_E}{\mu_D}}\\\Rightarrow \mu_E=\dfrac{f_D^2}{f_E^2}\mu_D\\\Rightarrow \mu_E=\dfrac{146.83^2}{329.63^2}\times 0.00256\\\Rightarrow \mu_E=0.000507\ kg/m[/tex]

The mass density of the E string is 0.000507 kg/m

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