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Wave through uniform rope

  1. May 26, 2008 #1
    1. The problem statement, all variables and given/known data
    A uniform rope of length L and negligible stiffness hangs from a solid fixture in the ceiling
    The free lower end of the rope is struck sharply at time t=0. What is the time t it takes the resulting wave on the rope to travel to the ceiling, be reflected, and return to the lower end of the rope?

    2. Relevant equations

    Equation for wave motion - is it this:
    y(t ) = A cos (kt - w),
    for some constants k,w.

    3. The attempt at a solution
    v = sqrt (F/u) = sqrt (mg/ (m/L) ) = sqrt (gL)
    this implies that velocity is constant for the wave. I was wondering if this was correct, since it looks suspicious.
    From here, I'm not exactly sure where to go - which wave equation should I use to go further (to solve and find out what I want to find out?). Would plugging in to the equation I had before, y( t) = A cos(kt-w), be the way to go?
     
  2. jcsd
  3. May 26, 2008 #2
    F is not a constant in this case, each part of the rope is beeing stretched differently depending on how close it is to the ceiling
     
  4. May 26, 2008 #3
    In that case, how would I deal with a situation where F is not constant? I have only learnt so far how to deal with cases where it is constant.
     
  5. May 26, 2008 #4
    the equation for v is still valid you just have to write F in terms of x (where x is distance from the bottom part of the rope)
     
  6. May 26, 2008 #5

    Doc Al

    User Avatar

    Staff: Mentor

    You'll need to integrate. Find the speed of the wave as a function of position along the rope. (What's the tension as a function of position?)
     
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