What is the mass of the string

Question

A tight uniform string with a length of 2.50 m is tied down at both ends and placed under a tension of 100 N. When it vibrates in its second overtone mode the sound given off has a frequency of 75 Hz. What is the mass of the string?

Answer 1

Do your own homework. The formula that you seek, however, it in the source...

Answer 2

the speed of a wave on a string is:

v = sqrt (T / mu),

where T is the tension and mu is the linear mass density.

The speed of any wave is the wavelength times the frequency:

v = lambda * nu

Set those equal and plug in m/L for the mass density:

sqrt (T L / m) = lambda nu

And solve for your mass:

m = TL / (lambda nu)^2

One more thing you have to do.  The fundamental wavelength is twice the length.  The first overtone is the length.  The second overtone's wavelength is 2/3 the length.  If you were to keep going, the nth harmonic wavelength is 2L/n, and the nth overtone is 2L /(n+1).  If all that isn't obvious, draw pictures of all the possible standing waves that have nodes at either end of the string.

So lambda = 2/3 L.  Plug that in, simplify, and you get:

m = 9T / (4 L nu^2)

They give you the tension, length, and frequency.  Plugnchug.