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Logarithmic decrement

  1. May 2, 2010 #1
    1. The problem statement, all variables and given/known data

    Basically, I have LQ''(t) + RQ'(t) + (1/C)Q(t)=0, and I'm supposed to

    "Show that the ration of the charge Q between two successive maxima is given by exp(RTd/2L), where Td is the time between two successive maxima. The natural logarithm of this ration is called the logarithmic decrement.

    2. Relevant equations


    3. The attempt at a solution

    So I got a solution Q(t)=e(-Rt)/(2L) [ C1cos( (√(R2-4L/C) )/(2L)t) + C2sin( (√(R2-4L/C) )/(2L)t).

    But I can't figure out how to find Td. I mean, I could always find t when dQ/dt=0; but then I'd have to plug two values of t back into Q(t) and find the difference, and ............ So what's the right way to do this?
  2. jcsd
  3. May 2, 2010 #2


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    First notice the sin & cos terms have the same argument & the choice of c1 & c2 will just choose an overall phase. So for this argument set c2 = 0.

    Then the maxima will just be where cos is maximum and successive maxima will occur where the argument of cos has changed by 2pi
  4. May 3, 2010 #3


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    Not exactly. The max's don't agree with the max's of the cosine, but the right idea. To the OP, just look at e-btcos(at+c).
  5. May 3, 2010 #4


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    good pickup thanks - They will be pretty close when the natural frequency is much larger that the decay constant, but you do need to take the exponential into account
    Last edited: May 3, 2010
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