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A String is Connected to a Vibrating Arm and Passes Over a Light Pulley

  1. Nov 29, 2012 #1
    Here is a tricky one (for me) that uses linear mass density and two masses. I don't really know which to do so I did both!

    1. The problem statement, all variables and given/known data
    An object can be hung from a string (with linear mass density μ=0.00200kg/m) that passes over a light pulley. The string is connected to a vibrating arm (of constant frequency f), and the length of the string between the vibrating arm and the pulley is L=2.10m. When the mass m of the object is either 25.0kg or 36.0kg, standing waves are observed; no standing waves are observed with any mass between these values, however.
    What is the frequency of the vibrating arm?
    What is the largest object mass for which standing waves could be observed?

    In the picture, there is a vibrating sinusoidal transverse wave with 3 periods.

    2. Relevant equations
    I used μ=T/v2, v=fλ, and T=m*9.81m/s2.


    3. The attempt at a solution
    First I found the wavelength, λ=2.1m/3=0.7m.

    Next, I found the tensionS... Using 25kg and 36kg.
    T=25*9.81=245.25N OR T=36*9.81=353.16N.

    With these tensions, I found the velocitieS
    μ=245.25/v2=350.2m/s OR μ=353.16/v2=420.2.

    Always fun getting two different solutions, right?

    Finally, found the frequencies using these velocities and my wavelength.
    f=350.2/.7=500.3Hz OR f=420.2/0.7=600.3Hz.

    I don't know which one to pick :(

    Also, I had no idea where to begin for the second part of the question, so if you could give me a hint for that, it would be greatly appreciated.
     
    Last edited by a moderator: Nov 29, 2012
  2. jcsd
  3. Nov 29, 2012 #2

    haruspex

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    I suspect that's only for illustration. The harmonics involved should be inferred from the information regarding the two weights (and lack of a standing wave for anything in between).
     
  4. Nov 29, 2012 #3

    collinsmark

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    Edit: Deleted my comment.

    (I think I figured out my misinterpretation of the problem now: The vibrating arm oscillates in a sinusoidal fashion, such that it doesn't produce any harmonics itself [i.e. there are no significant Fourier series harmonic components]. So the fundamental resonant frequency of the string is always less than or equal to the frequency of the vibrating arm. -- Previously, I mistook the contraption to operate the other way around such that the string was picking up on the Fourier decomposition harmonics of the arm.)
     
    Last edited: Nov 29, 2012
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