To solve this problem it is necessary to apply the concepts related to string vibration. A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch,
The speed of propagation of the wave in a string is proportional to the square root of the tension and inversely proportional to the root of the linear density. Mathematically this can be expressed as
[tex]v = \sqrt{\frac{T}{\rho}} \Rightarrow v^2 = \frac{T}{\rho}[/tex]
At the same time we have that the frequency is subject to the wavelength and the speed. That is to say
[tex]f = \frac{v}{\lambda} \rightarrow v = 343m/s[/tex] Speed of sound at 20°C.
Therefore the frequency would be
[tex]f = \frac{v}{\lambda}[/tex]
[tex]f = \frac{343}{0.6}[/tex]
[tex]f= 571.7Hz[/tex]
While that is the auditory frequency, the frequency caused in the string would be
[tex]\lambda = 2L[/tex]
[tex]\lambda = 2*0.4[/tex]
[tex]\lambda = 0.8m[/tex]
So reusing the frequency formula we can find that now the velocity in the string is of
[tex]f = \frac{v}{\lambda}[/tex]
[tex]v = \lambda f[/tex]
[tex]v = 0.8*571.7[/tex]
[tex]v = 457.3m/s[/tex]
Finally using the vibration in the string we can find the tension, so:
[tex]v^2 = \frac{T}{\rho}[/tex]
[tex]T = v^2 \rho[/tex]
[tex]T = 457.3 * 0.0013[/tex]
[tex]T = 209N[/tex]
