How much tension should be in the string if it is to produce resonance?

[Tomanadi]

Homework Statement

A 73.4 cm long guitar string of mass 2.21 g is placed near a tube open at one end, and also 73.4cm long. How much tension should be in the string if it is to produce resonance (in its fundamental mode) with the third harmonic in the tube?

Homework Equations

v=sqrt(T/(m/L))
f=(nv)/4L

The Attempt at a Solution

The wavelength of the note produced in the string = 2*.734 m
The velocity of the waves in the string = sqrt ( T / ( 0.00223 / 0.734 ) )
Frequency = sqrt ( T / ( 0.00221 / 0.734 ) ) / (2*0.734)
Frequency of the wave of the tube = 343 / (4*0.734 / 5)
So
343 / (4*0.734 / 5) = sqrt ( T / ( 0.00221 / 0.734 ) ) / (2*0.734)

The answer doesn't work for me, though.

 

[anathema]

Hello, thank you for your question. Based on the given information, we can use the equation v=sqrt(T/(m/L)) to calculate the velocity of the waves in the string. We know the length of the string (L) and its mass (m), so we can plug those values in and solve for T, the tension in the string.

v=sqrt(T/(m/L))
v=sqrt(T/(0.00221/0.734))
v=sqrt(T/0.003008)

Next, we can use the equation f=(nv)/4L to calculate the frequency of the third harmonic in the tube. We know the length of the tube (L) and the speed of sound (n), so we can plug those values in and solve for f.

f=(nv)/4L
f=(343/4*0.734)/3
f=117.1 Hz

Now, we can set the frequency of the string equal to the frequency of the tube, since they will be in resonance.

117.1 Hz = sqrt(T/0.003008)

Solving for T, we get a tension of 3.94 N in the string. This tension should allow the string to produce resonance with the third harmonic in the tube. I hope this helps!