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Rotation: mass transfer and angular momentum conservation

  1. Jul 5, 2013 #1
    1. The problem statement, all variables and given/known data

    a drum of mass M[itex]_{a}[/itex] and radius a rotates freely with initial angular velocity ω[itex]_{a}[/itex](0). A second drum with mass M[itex]_{b}[/itex] and radius b greater than b is mounted on the same axis and is at rest, although it is free to rotate. a thin layer of sand with mass M[itex]_{s}[/itex] is distributed on the inner surface of the smaller drum. At t=0 small perforations in the inner drum are opened. the sand starts to fly out at a constant rate λ and sticks to the outer drum. Find the subsequent angular velocities of the two drums ω[itex]_{a}[/itex] and ω[itex]_{b}[/itex]. Ignore the transit time of the sand.



    3. The attempt at a solution

    torque on drum A = [itex]\frac{1}{2}[/itex](M[itex]_{a}[/itex] + M[itex]_{s}[/itex]- λt)a[itex]^{2}[/itex]dω[itex]_{a}[/itex]/dt + [itex]\frac{1}{2}[/itex]λa[itex]^{2}[/itex]ω[itex]_{a}[/itex](t)

    torque on drum B = [itex]\frac{1}{2}[/itex](M[itex]_{b}[/itex]- λt)b[itex]^{2}[/itex]dω[itex]_{b}[/itex]/dt - [itex]\frac{1}{2}[/itex]λb[itex]^{2}[/itex]ω[itex]_{b}[/itex](t)

    now applying angular momentum conservation on the system I got a relation between ω[itex]_{a}[/itex] and ω[itex]_{b}[/itex].
     
    Last edited: Jul 5, 2013
  2. jcsd
  3. Jul 5, 2013 #2
    I don't think that the Torque equations are needed. all you need to do is to conserve the angular momentum.
     
  4. Jul 5, 2013 #3
    but i can not find any more relations between ω[itex]_{a}[/itex] and ω[itex]_{b}[/itex].
     
  5. Jul 5, 2013 #4

    haruspex

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    That can't be enough. For any given distribution of sand between the drums there will be a continuum of solutions for the two angular velocities that give the same overall angular momentum.

    Avi Nandi, I'm unconvinced by your expression for torque on the inner drum. If a cart is travelling along and some of the load on the cart falls off, what force does that exert on the cart?
     
  6. Jul 6, 2013 #5
    thank you haruspex and darkxponent.
     
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