Close tube with string oscillation

[IIK*JII]

Homework Statement

In attached figure, a closed tube is placed near a string that is fixed at one end and has a weight attached to its other end. When bridges A and B are positioned at the points shown,plucking the string between A and B causes the tube to resonate at its fundamental frequency. Points a-e divide the length between A and B into 6 equal segments

Next, A is fixed in place, and B is gradually moved toward A while the closed tube is shifted so that it stays at the center of A and B. During this process, the string is repeatedly plucked between A and B. When B is at a certain point, the tube resonates at the next overtone above the fundamental frequency. Which of a-e represents that point? Here, the string's oscillation is only fundamental oscillation

Homework Equations

Close tube; L= (2n-1)\frac{λ}{4} ; n= 1,2,3...
String fixed end; λ=2L/n

The Attempt at a Solution

In 2nd period the tube oscillate at f₃

L = \frac{3}{4}λ
∴λ_tube=\frac{4}{3}L ...(1)
String ; λ_string=2L (oscillate at f₁) ...(2)

Divide (1)/(2) I got \frac{λ_tube}{λ_string} = 2/3

From that I guess the point is a point number 2 from first 3 points, so I don't know that I should choose point b or d as my answer. Also, I don't know that my method is correct or not...

Help is appreciated :)
Thanks

 

[Simon Bridge]

The question is not how much you have to reduce the entire wavelength on the string but how far to move the blocks ...

 

[IIK*JII]

Thank you very much Simon Bridge :)
Did you mean the block should move up?? and I should find height of the block when it moves up??

 

[Simon Bridge]

What I am saying is that you have calculated a ratio in whole wavelengths, but the distance you need to find (in order to know where to put the block) is that for a half-wavelength. You need to check to see what sort of difference, if any, that makes.

Presumably, the ratio of the fundamental to the first harmonic in the tube is the ratio of string wavelengths needed right? You already know the half-wavelength needed to make the string oscillate at the tube's fundamental frequency and you are keeping the tension, and so the wave-speed, fixed.

I hope I'm not confusing you - it is really hard to write about without actually telling you the answer. Basically the numbers you got look good - I'm trying to get you to work out if the numbers you got are the ones you need ... what you really need is a relationship along the lines of x_2=ax_1 where x₁ is the distance |AB| that got you the fundamental in the tube and x₂ is the distance between the blocks that gets you the second fundamental and a is the ratio between them.[1]

What you have is that \lambda_{tube} = \frac{2}{3}\lambda_{string}

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[1] actually you can finesse it by looking for the relation x_2=\frac{n}{6}x_1 since n will tell you which of the lettered points to move the block to :)

 

[IIK*JII]

Thank you Simon Bridge,,
your explanation is good help me imagine what this problem want

I think, for example, x2 from your meaning is length of string that I can find from wavelength right??

 

[Simon Bridge]

Bear in mind that I think you are very close and I have not actually done the problem myself. It's intreguing - I'll have to set it up as an experiment sometime.

 

[IIK*JII]

Thank you Simon Bridge
I got it now :)

 

[Simon Bridge]

Cool: well done :)