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Confirm the differential equation for a falling rope using newtons law

  1. Oct 23, 2007 #1
    1. The problem statement, all variables and given/known data


    A uniform chain of length L, measured in feet, is held vertically so that the lower end just touches the floor. The chain weighs 2lb/ft. The upper end that is held is released from rest at t = 0 and the chain falls straight down. Ignore air resistance, assume that the + direction is downward, and let x(t) denote the length of the chain on the floor at time t. Use the fact that the net force F in (18) is 2L to show that a differential equation for x(t) is:

    (L - x) (d^2x / dt^2) - (dt/dx)^2 = Lg


    2. Relevant equations


    the problem mentions equation 18 in our book which is

    F = d/dt (mv)

    3. The attempt at a solution

    I rewrote the original equation as

    (L - x) (d^2x / dt^2) - Lg = (dt/dx)^2

    by analyzing the system I'm guessing the weight = 2(L-x)

    mass = 2(L-x)/g

    since F = ma

    2(L-x)/g * a = 2L

    a = Lg/(L-x)

    a = (d^2x / dt^2) or the second derivative of t with respect to x

    then

    (L-x)(Lg/(L-x) - Lg = (dx/dt)^2

    0 = (dx/dt)


    When I tried to reproduce the same d.e. for this system i get

    (L - x) (d^2x / dt^2) + (dt/dx)^2 = Lg

    *the + instead of the - before the (dt/dx)^2 is what throws me off.

    thanks in advance for all your help
     
  2. jcsd
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