Question

One string of a certain musical instrument is 74.0cm long and has a mass of 8.71g . It is being played in a room where the speed of sound is 344 m/s.

Part A

To what tension must you adjust the string so that, when vibrating in its second overtone, it produces sound of wavelength 0.762m ? (Assume that the breaking stress of the wire is very large and isn

Answer 1

The second overtone is 3 times the frequency of the fundamental mode, so the fundamental will be 1/3f.
f = v/?
= 344/0.762 = 451.44 Hz
fundamental = f/3 = 150.48 Hz
Use the fundamental to find the tension necessary:
f = (1/2L)sqrt(T/?) where L = length of the string, T = tension, and ? = linedar mass of the string
? = 0.00871/0.74 = 0.0117 kg/m
solve for T:
sqrt(T/?) = f*2L
T/? = (f*2L)^2
T = ?*(f*2L)^2 = 0.0117*(451.44*2*.74)^2 = 179014 N
An impossibly high amount of tension, which is why higher strings are usually much shorter than 74cm.