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Wave velocity in a free-hanging rope

  1. Sep 23, 2010 #1
    1. The problem statement, all variables and given/known data

    A uniform rope of length L hangs freely from the ceiling. Show that the time for a transverse wave to travel the length of the rope is t0 = [tex] 2\sqrt{L/g}[/tex].

    2. Relevant equations

    v = [tex] \sqrt{\tau/\mu}[/tex]. (Where [tex]\tau[/tex] is the tension and [tex]\mu[/tex] the linear density of the rope.)

    3. The attempt at a solution

    Set up axes so that the rope is parallel to the x-axis, with the bottom of the rope at the origin.

    Let m(x) represent the mass of the rope below x. Then [tex] m(x) = \mu x [/tex]
    giving [tex]\tau (x) = m(x)g = \mu g x[/tex]
    so [tex] v (x) = \sqrt{\mu g x/\mu} = \sqrt{gx}[/tex]

    Also [tex] L = \int^{t_0}_{0} vdt[/tex]

    I can see that velocity is a function of time and that integrating will give me something at least similar to the required equation, but I can't figure out how to get v in terms of t. Or maybe I should be getting x in terms of t. I couldn't find that either, though.
  2. jcsd
  3. Sep 23, 2010 #2


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    Homework Helper

    v = dx/dt = sqrt(gx)

    dx/sqrt(gx) = dt.

    Now find the integration and put the limits x = 0 to x = L
  4. Sep 23, 2010 #3
    Ohhh, I get it now. Thanks!
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