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Δt between consecutive frequency configurations

  1. Jul 18, 2017 #1
    1. The problem statement, all variables and given/known data

    At a long, vertical tube, water is poured in, with an inflow R = 1.00 L/min = 1.67 * 10-5 m3/s. The radius, r = 5.00 cm. At the open end, a tuning fork is oscillating with a frequency of f = 512 Hz. What's the time difference between two consecutive configurations, while the water slowly rises?

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    2. Relevant equations

    f = n*v/4L

    3. The attempt at a solution

    Alright, in this case, we have a constant f that doesn't change, an L that keeps getting shorter, and we're in a "tube with one end open, one end closed" environment.

    >Volume of the water: V = π*r2*h = 7.85 * 10-3m2*h
    >hstart = 0 & Lstart = max
    >h keeps going up, while L keeps going "down"
    >every 1s, a volume of V = 1.67 * 10-5 m3 is added. So, (using the V formula from the first >), we have h = 2.13 * 10-3 m. So the height of the water goes up by said h every 1s.

    And that's where I get stuck. Initially I figured I'd say that the given f is the harmonic, put n=1, v = 343 m/s, and find the initial/max length L. But as I kept upping the ante on n (3,5,7), the L kept growing, which wasn't realistic, since it should go down. So that's one out.

    Any help is appreciated!
     
  2. jcsd
  3. Jul 18, 2017 #2

    haruspex

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    Why do you assume the first resonance is at the fundamental frequency?
     
  4. Jul 18, 2017 #3
    Well, at first I did so because I thought that I'd use it to find the entire length of the tube. Obviously that was wrong. I'm out of ideas though.
     
  5. Jul 18, 2017 #4

    haruspex

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    But you do not need to find that.
    What L values do you get for n=1, 2, 3?
     
  6. Jul 18, 2017 #5
    It's an open/closed tube, so f = n*v/4L & n = 1,3,5,7..., right?

    >n = 1: L = 0.168 m

    >n = 3: L = 0.502 m

    >n = 5: L = 0.837 m
     
  7. Jul 18, 2017 #6

    haruspex

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    So how big is each step?
     
  8. Jul 18, 2017 #7
    From 1 to 3 it's 0.334 m & from 3 to 5 it's 0.335 m.
     
  9. Jul 18, 2017 #8

    haruspex

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    Within rounding error, they're the same, right? Can you generalise to the step from n to n+1?
    Can you relate those distance steps to the time intervals?
     
  10. Jul 19, 2017 #9
    My problem is that if I go higher (5, 7, 9), then the steps change a bit. From 5 to 7 it's 0.333. From 7 to 9 it's 0.331. I guess it's to be attributed to the rounding error, but there is a difference.

    Something like this?

    >fn = n*v/4Ln
    >fn+1 = (n+1)*v/4Ln+1
    >fn = fn+1
    ___________________________
    fn+1/fn = ((n+1)*v/4Ln+1)/(n*v/4Ln) <=> n*Ln+1 = (n+1)*Ln <=> Ln+1 = ((n+1)/n)*Ln


    I'm a bit stuck on that.
     
  11. Jul 19, 2017 #10

    haruspex

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    The frequency is fixed.
     
  12. Jul 19, 2017 #11
    Yeah, it's stable, 512 Hz, right? I factored that in (fn = fn+1) and ended up with Ln+1 = ((n+1)/n)*Ln
     
  13. Jul 19, 2017 #12

    haruspex

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    True, but not as useful as finding Ln+1-Ln.
     
  14. Jul 19, 2017 #13
    Well, in that case: Ln+1 - Ln = Ln/n
     
  15. Jul 19, 2017 #14

    haruspex

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    Still not that useful. Try to get an expression that has no L term on the right. It will have v.
     
  16. Jul 19, 2017 #15
    Yeah, I forgot about that: f = n*v/4*Ln <=> Ln = n*v/4*f || So we have: Ln+1 - Ln = (n*v/4*f)*(1/n) = v/4*f = 343 m/s / 4*512 Hz = 0.167 m
     
  17. Jul 19, 2017 #16

    haruspex

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    Right. How long will it take for the air column to change in length by that mch?
     
  18. Jul 19, 2017 #17
    Based on what I found previously (first post):

    In 1 s, the water rises by 2.13 * 10-3 m, so the Length decreases by so.
    How much time, x, do we need for the Length to change by 0.167 m ?

    By doing the Rule of Three I end up with 78.4 s, which is about half of the book's answer (158 s).
     
  19. Jul 19, 2017 #18

    haruspex

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    Sorry, I did not check this equation:
    That is not right for a half open tube.
     
  20. Jul 19, 2017 #19
    Isn't it a tube with one end closed, and one end open? The top is the open part, and the bottom is the closed one. That's what my book has, with n being able to take the values of 1,3,5,7...
     
  21. Jul 19, 2017 #20

    haruspex

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    Oh ok, if that is how n is being defined... But you took consecutive values as n, n+1 instead of n and n+2.
     
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