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Resistance of an oscillating system

  1. May 10, 2017 #1
    1. The problem statement, all variables and given/known data
    93f07c45ade95211f6d3d72b23fe6ce5.png



    2. Relevant equations
    ##F = -kx = m\ddot{x} ##
    ## f = \frac{2\pi}{\omega}##
    ## \omega = \sqrt{\frac{k}{m}} ##
    ##\ddot{x} + \gamma \dot{x}+\omega_o^2x = 0 ##
    ##\gamma = \frac{b}{m}##

    3. The attempt at a solution
    I'm stuck on part c of this question. Using the above equations I got k = 90.5 and ##\ddot{x}## = 135.75 m/s##^2##. I believe I have to use the damped oscillator case for this question (equation 4) but I'm not sure how to find ##\dot{x}##
     
  2. jcsd
  3. May 10, 2017 #2

    DrClaude

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    Staff: Mentor

    It would help to have an analytical form for x(t), solution to the equation of motion for the damped oscillator.
     
  4. May 10, 2017 #3
    Sure
    ##x(t) = Acos(\omega t) + B sin(\omega t)##
    ##x(t) = Acos(\omega t + \phi)## where ##\phi## is the initial phase

    Is that what you're looking for?
     
  5. May 10, 2017 #4

    DrClaude

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    Staff: Mentor

    Nope. These are equations for an undamped oscillator. You have a damped oscillator, so the amplitude will not be constant.
     
  6. May 10, 2017 #5
    I've got
    ## x = Ae^{-\frac{1}{2} \gamma t} cos(\omega_d t +\phi)##
    where ##\omega_d = \omega_o \sqrt{1-(\gamma / 2\omega_o)^2}##
     
  7. May 10, 2017 #6

    DrClaude

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    You can now use that to figure out the constant γ based on the decrease in amplitude.
     
  8. May 10, 2017 #7
    Sorry, still confused. Don't I need to know ##\gamma## to solve for ##\omega_d##? I'm still not sure how to solve for ##\gamma## since I have two unknowns in my equation.
     
  9. May 10, 2017 #8

    DrClaude

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    You don't need to know ##\omega_d##. You have information on how the amplitude decreases, and this is enough to find ##\gamma##.
     
  10. May 10, 2017 #9
    Sorry, I don't fully understand why I don't need ##\omega_d##. The equation contains ##cos(\omega_d t + \phi)##.
     
  11. May 10, 2017 #10

    DrClaude

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    Yes, but you don't need the full motion. Someone has measured the amplitude of the oscillation at two points in time, and that is all that is needed. By how much the frequency was reduced is not relevant to what you are trying to find.
     
  12. May 10, 2017 #11
    Ah, okay. So the cosine term can be neglected can it?

    So I take A = 0.06, x = 0.3, t = 8.0s and solve ##x = Ae^{-\frac{1}{2} \gamma t}##?
     
  13. May 11, 2017 #12

    DrClaude

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    No, it can't be neglected. But you don't need to know the exact position of the mass at t = 8.0 s. Concentrate on the amplitude.

    Edit: It may help you to have a look at this picture http://hyperphysics.phy-astr.gsu.edu/hbase/images/oscda9.gif
     
  14. May 11, 2017 #13
    That diagram was quite helpful (if I think I know what I'm doing). ##A = \frac{x}{x_o}##, so I'll just equate ##e^{-\gamma t} = A## and solve for ##\gamma## which I can then use to solve for b?

    EDIT: Also, thanks for being so patient with me.
     
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