Frequency of a wave at resonance

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Homework Statement

A 1.2 m tube is closed at one end. A stretched wire is placed near the open end. The wire is
0.330 m long and has a mass of 9.6 g. It is fixed at both ends and oscillates in its fundamental
mode. By resonance, it sets the air column in the tube into oscillation at that column’s
fundamental frequency. Find (a) that frequency and (b) the tension in the wire.

I only have a problem finding the frequency since the wave speed is not given. Because both the wire and the air are oscillating at their fundamental frequency there must be a way to use this to find the frequency without the wave speed.

Homework Equations

kdx/dt -w = 0 => dx/dt = w/k = f λ => v = f λ => f = v/λ

3. The Attempt at a Solution
λ_string = 2L_string
λ_air = 4L_tube
f_string = f_air
v_string/λ_string = v_air/λ_air => v_string/2L_string = v_air/4L_tube => v_air = 7.42v_string

 

[TSny]

Are you sure that you are not given the speed of sound in air?

 

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Yes. Only the information given is available. Really makes me think that this was actually a mistake and that it cannot be solved. After thinking about it there is no way to cancel out the velocities with the known information making it impossible to find the frequency. Since this is only a year 1 physics question about waves it shouldn't be difficult anyways.

I've tried emailing my prof about this but apparently he ignores emails...

 

[TSny]

If this is a question from a textbook, you might check to see if the there is a statement at the beginning of the exercises/problem section that givens the velocity of sound to use.

 

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This question actually came from my prof. He claims to have used this on a previous exam. I got a .doc copy of his last exam. this is the 2nd question. Also though I did notice that in the .doc file the last 2 question had some numbers over lapping each other making it impossible to read. So it's entirely possible that some of the information is simply missing or in the wrong format. I'll try reading through my textbook again to see if he took the question from it.

It would make sense for the wave speed of the air to be given since you still have to do everything I already tried to get both the frequency and tension.

 

[TSny]

OK. I think this is a problem form the Halliday Resnick Walker text.

Your work looks correct to me.

 

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I found the question: Halliday Resnick 10th edition chapter 17 page 509 Q#50. It's the exact question from the book. No changes to the question have been made so apparently it should be possible to solve. However the book does not have a solution for it. The fact that my Prof used this on an exam makes believe there's a solution were missing.

 

[TSny]

At the beginning of the problem section the text says to use a specific velocity for the speed of sound.

 

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Well there's my problem right there. Guess i'll post the solution for anyone interested.
v_air = 343m/s
λ_string = 2L_string
λ_air = 4L_tube
f_string = f_air = 71.45833333Hz

v_string/λ_string = v_air/λ_air => v_string/2L_string = v_air/4L_tube =>v_string = 47.1625m/s
u = m/L_string = 0.29090909kg
v_string = √t/u => t = u√v_string = 0.066N

Thanks for the help :)

 

[TSny]

v_string = √t/u => t = u√v_string = 0.2N

Check this.

 

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Check this.

should be 0.066N. 0.0096kg⋅√47.01625 = 0.066N

 

[TSny]

You didn't solve for t correctly here:

v_string = √t/u => t = u√v_string

 

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Oh, I took the tension out of the equation: f = 1/λ⋅√t/μ which makes μλ²f² = t = 647.07N

Weird how I think 0.2N makes sense for a tension force on a string but 600N seems too high.

 

[TSny]

Oh, I took the tension out of the equation: f = 1/λ⋅√t/μ which makes μλ²f² = t = 647.07N

Decimal point is not in the correct place. I think the mistake goes back to your value for u.

Weird how I think 0.2N makes sense for a tension force on a string but 600N seems too high.

Good observation.

 

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Decimal point is not in the correct place. I think the mistake goes back to your value for u.

Good observation.

I should observe my calculator better as well. I may notice that I'm reading 64.707 and not 647.07