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Wine Glass Resonance Frequency

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  1. Mar 29, 2016 #1
    1. The problem statement, all variables and given/known data
    Question 1: When analysing AP French's Formula for Frequency of Wine glasses, what is the direct relationship between the Frequency heard (at different levels - fd) and the level of water from the top of the glass (d). I know that as the level of water from the glass decreases (meaning the water level is higher), the lower the frequency. However, is this relationship linear or quadratic, etc and why.

    Question 2: If you graph (f0/fd)2 on the x axis and (1-(d/H*)4 on the y axis, is this relationship going to be linear or something else.

    2. Relevant equations
    The equation is in the picture below and above (Link is http://imgur.com/a0pQVdN)
    upload_2016-3-29_9-35-14.png
    3. The attempt at a solution
    I believe that the answer to question 1 is quadratic and the answer to question 2 is linear. Please help
     
  2. jcsd
  3. Mar 29, 2016 #2
    (f0/fd)^2
     
  4. Mar 29, 2016 #3

    gneill

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    Hi BobDylan22, Welcome to Physics Forums.

    It's better to upload and insert in-thread any images that are critical to the problem. I've inserted a copy into your original post. I also fixed the equations in your text to make them easier to parse. Note that you can use the x2 and x2 icons in the editing panel to insert subscripts and superscripts for equations (or use LaTeX notation).

    You'll need to provide more explanation for your attempt at a solution. Describe your thinking behind your conclusions in more detail.
     
  5. Mar 29, 2016 #4

    rude man

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    If you let x = (1-(d/H*)4 and
    y = (f0/fd)2
    then if the other coefficients are constants, what do you get for your equation? Should be apparent if the relationship between y and x is linear or not.

    Also, if you let y = (f0/fd) and
    x = √{(1-(d/H*)4}, is that quadratic or not?

    BTW it may interest you that a portion of the vibration has an inertial component, i.e. if you set up the vibration nodes at 0 and 180 degrees, then turn the glass say 90 degrees, the vibrations will follow the glass only about 2/3 of the way, i.e. ~60 deg. The so-called "Hemispherical Resonator Gyroscope (HRG)" is based on this phenomenon.

    Cf. https://en.wikipedia.org/wiki/Hemispherical_resonator_gyroscope#Advantages
     
  6. Mar 30, 2016 #5
    I know that if you graph x = (1-(d/H*)4 and y = (f0/fd)2, you get a linear line,
    but what if you graph fd and d?
     
  7. Mar 30, 2016 #6

    rude man

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    OK, we've done question 2.

    Question 1: let y = fd and x = d.
    Then [f0/y]2 = 1 + a(1 - x/H)4
    where a is the given constant.
    In order for the relationship between y and x to be a quadratic, you have to be able to come up with constants α, β and γ such that
    y = α + βx + γx2.
    Considering just the binomial expansion of (1 - x/H)4, does that seem likely ?
     
  8. Mar 30, 2016 #7
    Ok, but what I don't get is that as the distance from the top of glass to the top of water level decreases (becoming fuller), the frequency decreases. But this formula does not show that relationship. It shows the opposite.
     
  9. Mar 30, 2016 #8

    gneill

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    Did you try plotting it just to get an idea of the shape of the curve? Assign some arbitrary values to the constants and plot fd versus d. Note that the way the given function is defined the fd is in the denominator on the left hand side. So if you rearrange to isolate it you'll get something of the form:

    $$f(d) = \frac{f_o}{\sqrt{1 + a\left(1 - \frac{d}{h} \right)^4 }}$$
    where fo, a, and h are chosen suitably.
     
  10. Mar 30, 2016 #9
  11. Mar 30, 2016 #10

    gneill

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    I can't see how your spreadsheet does its calculations, but your results are certainly different from what I see plotting a function of the form I showed in my previous post. Here's a snip from a Mathcad worksheet:
    upload_2016-3-30_22-5-25.png


    The constants fo, a, and h were chosen arbitrarily just to get a feel for the form of the function.
     
  12. Mar 30, 2016 #11
    I was graphing fd and d
     
  13. Mar 31, 2016 #12

    rude man

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    No way! As d goes down, f0/fd goes up but notice that fd is in the denominator on the left-hand side, so they both go down & up together.
     
  14. Mar 31, 2016 #13
    Fo will always be the same. As d decreases, that means that the level of water in the glass increases, meaning lower frequency.
     
  15. Mar 31, 2016 #14

    rude man

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    Right, contradicting your post #7!
     
  16. Mar 31, 2016 #15
    No it doesn't. As the distance between the top of glass decreases (that means d decreases), which means the glass is more full, hence the frequency decreases.
     
  17. Apr 2, 2016 #16

    No it doesn't. As the distance between the top of glass decreases (that means d decreases), which means the glass is more full, hence the frequency decreases.
     
  18. Apr 2, 2016 #17

    rude man

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    My last post on this thread: look again at your post 7. here is what you said: "Ok, but what I don't get is that as the distance from the top of glass to the top of water level decreases (becoming fuller), the frequency decreases. But this formula does not show that relationship. It shows the opposite."

    The formula does NOT " ... show the opposite"!
     
  19. Apr 2, 2016 #18
    So you are saying that the formula shows the correct relation? Can you please show me
     
  20. Apr 2, 2016 #19

    rude man

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    See my post 12. Read it carefully.
     
  21. Apr 18, 2016 #20
    Thankyou, but is the relationship quadratic?
     
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