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Damped Oscillator and time

  1. Aug 13, 2009 #1
    1. The problem statement, all variables and given/known data
    A mass M is suspended from a spring and oscillates with a period of 0.880 s. Each complete oscillation results in an amplitude reduction of a factor of 0.96 due to a small velocity dependent frictional effect. Calculate the time it takes for the total energy of the oscillator to decrease to 0.50 of its initial value.


    2. Relevant equations

    N oscillations=(initial amplitude)x(factor)^N
    E=Eo*e^(-t/Tau)
    Tau = m/b


    3. The attempt at a solution
    I really have no idea on how to approach this problem. I need to find tau, which is m/b, but idk what b is. if i have tau, the E on both sides cancel and i'm left with
    1/2 = e^-t/tau. t = tau ln (2)
    So basicly i need to find tau.

    Thanks!
     
  2. jcsd
  3. Aug 13, 2009 #2

    kuruman

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    Ignore m/b. What fraction of the initial energy is left after one oscillation?
    Ans. 0.962
    After two oscillations?
    Ans. 0.962*0.962
    After n oscillations?
    .....
     
  4. Aug 13, 2009 #3
    i got 16.97 as n. it works because .96^17 = ~ 0.499

    So if thats true, 0.88 which is the period * 16.97 which gives me 14.94 seconds.

    This makes sense except the answer's still wrong?
     
  5. Aug 13, 2009 #4
    OK i'm retarded. n = 7.47

    My friend here said not to do .96squared and woulnd't tell me why. So i blame her.

    Thanks again =D
     
  6. Aug 13, 2009 #5

    kuruman

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    Your friend is correct. I got 7.53 s (close enough). Initially, I assumed linearity where there was none.

    I will get you started. Assume that the rate of change of the amplitude is proportional to the amplitude. Call the proportionality constant C. Then

    [tex]\frac{dA}{dt} = - c A[/tex]

    Solve this equation for A(t), and use the fact that A(0.88) = 0.96 A0

    Once you have A(t) you can find E(t), etc. etc.

    This is a good problem. I learned something from it. Thanks.
     
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