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Momentum of a falling chain

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  1. Sep 22, 2015 #1
    1. The problem statement, all variables and given/known data
    1. (30 points) A very flexible uniform chain of mass M and length L consisting of very small links is suspended from one end so that it hangs vertically, the lower end just touching the surface of a table. The upper end is suddenly released so that the chain falls onto the table and coils up in a small heap, each link coming to rest the instant that it strikes the table.

    Find: (a) The force F exerted by the chain on the table at any instant, in terms of the weight of the chain already on the table at that moment.

    When the chain falls, we know that T=0, and net force is Mg
    2. Relevant equations
    F=dp/dt=Mv +Mv'

    Possibly CM?
    R=(m1r1+m2r2)/(m1+r1)

    3. The attempt at a solution
    I know that we need to solve for the actual mass of the chain hitting the table as a function of its acceleration g, and that we need to likely integrate to find mass per unit length, so my attempt at this equation was
    M/L=dL/dt
    Mdt=dL*L

    Then, once mass per unit length was found, we could put it in for M in the equation F=dp/dt=Mv +Mv', and solve for F. I'm just not quite sure I'm on the right track. My calculus teacher deliberately avoided applications in physics, so I'm really having a rough time setting up the differentials. *Note, we have to use Newtonian methods, not lagrange or hamiltonian. Thank you in advance!
     
  2. jcsd
  3. Sep 22, 2015 #2
    Momentum is a good place to start. What is the momentum ## dp ## of a small length of chain ## dl ## falling at velocity ## v ##?
     
  4. Sep 22, 2015 #3

    mfb

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    2016 Award

    Staff: Mentor

    The chain is uniform, mass per length is constant. The length on the table (and its rate of change) is not constant, but you can find it first from free fall.
     
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