Answer:
The tension is 75.22 Newtons
Explanation:
The velocity of a wave on a rope is:
[tex] v=\sqrt{\frac{TL}{M}}[/tex] (1)
With T the tension, L the length of the string and M its mass.
Another more general expression for the velocity of a wave is the product of the wavelength (λ) and the frequency (f) of the wave:
[tex]v= \lambda f[/tex] (2)
We can equate expression (1) and (2):
[tex]\sqrt{\frac{TL}{M}}[/tex]=[tex] \lambda f [/tex]
Solving for T
[tex] T= \frac{M(\lambda f)^2}{L}[/tex] (3)
For this expression we already know M, f, and L. And indirectly we already know λ too. On a string fixed at its extremes we have standing waves ant the equation of the wavelength in function the number of the harmonic [tex]N_{harmonic} [/tex] is:
[tex] \lambda_{harmonic}=\frac{2l}{N_{harmonic}}[/tex]
It's is important to note that in our case L the length of the string is different from l the distance between the pin and fret to produce a Concert A, so for the first harmonic:
[tex] \lambda_{1}=\frac{2(0.425m)}{1}=0.85 m[/tex]
We can now find T on (3) using all the values we have:
[tex] T= \frac{2.42\times10^{-3}(0.85* 440)^2}{0.45}[/tex]
[tex] T=75.22 N[/tex]
