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Hard limit problem.

  1. Sep 12, 2009 #1
    1. The problem statement, all variables and given/known data

    "Demonstrate that in the limit (nu)---> 2w0 the solution to the damped harmonic oscillator equation becomes x(t) = (x0 + [v0 +(nu/2)x0]t)e^(-(nu)t/2)


    2. Relevant equations

    Solution to damped harmonic oscillator equation; x(t) = a*e^(-(nu)t/2)cos(w1t - (theta))

    Where a = [x0^2 + (v0 + (nu)x0/2)^2/w1^2]1/2 and

    (theta) = tan^-1((v0 + (nu)x0/2)/w1x0)

    and w1^2 = w0^2 - (nu)^2/4


    3. The attempt at a solution

    So I plugged (nu) ---> 2w0 into my equations and got ...

    x0^2 + [(v0 + w0x0)^2)/(w0^2 - w0^2)]^1/2 * e^(-w0t) * cos (-tan^-1 (( v0 + w0x0)/((w0^2-w0^2)^1/2*x0))

    and I note that the divide by 0 in my tan inverse goes to infinity which equals pi/2 and the cos of pi/2 is 0

    but then before my e term I have (w0^2 - w0^2) on the denominator which means that's going to infinity... so I have one part going to 0 and one part going to infinity...

    and I don't know how to deal with this.
     
    Last edited: Sep 12, 2009
  2. jcsd
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