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Falling Rope on a scale

  1. Jun 5, 2008 #1
    1. The problem statement, all variables and given/known data
    A uniform flexible rope is suspended above a scale, with the bottom of the rope just touching the scale (gravity points downward). The rope has a length L and a total mass of M. The mass is uniformly distributed along it's length.

    The rope is released. After a length x<L has fallen onto the scale, what does the scale read? Assume the scale can measure the force applied to it instantaneously. (Hint: the force exerted by the rope on the scale has two components).

    2. Relevant equations

    3. The attempt at a solution
    so p(t)=(M/L)*(v(t)^2)
    thus dp/dt=(M/L)*(g^2)*t^3

    also, x(t)=g(t^2)/2--> t=sqrt(2xg). plug that into the eqn for dp/dt, and get

    However, that solution doesn't treat the problem with a force of two components, nor do the units seem to work out. Any ideas?
  2. jcsd
  3. Jun 5, 2008 #2
    Let M=mass, L=length of the rope, v=velocity

    You have to express in component form,
    [tex]F_{net} + \frac{dm}{dt}v=m\frac{dv}{dt} [/tex]
    [tex]= F_{n} - mg+\frac{dm}{dt}v=0[/tex] (Eq 1), then

    [tex]\frac{dm}{dt}=\frac{-M}{L}v[/tex] (*), where dm=mass of the section of the rope that falls on the scale.

    Substitute (*) back into (Eq 1) and find v using [tex]v^2=v_{0}^2+2a\Delta y[/tex]
    Then find [tex]F_n[/tex]
  4. Jun 5, 2008 #3
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