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The double Atwood machine has frictionless, massless pulleys

  1. Sep 13, 2015 #1
    1. The problem statement, all variables and given/known data
    The double Atwood machine has frictionless, massless pulleys and cords. Determine (a) the acceleration of masses ma, mb, and mc, and (b) the tensions Fta and Ftc in the cords.

    2. Relevant equations
    F=ma

    3. The attempt at a solution

    So I drew free body diagrams for the mass A, mass B, mass C, and the lower hanging pulley.

    I then computed these equations from F=ma:


    Fta - ma = maaa
    Fta - mb = mbab
    2Fta = Ftc
    Ftc - mc = mcac

    This is all assuming that there is currently no relationship between the accelerations. And this is where I am stuck. I do not know how to represent ac as that of aa and ab. Or maybe that isn't the correct way to solve the system either. The only thing that I assume is that the tensions for ma and mb are the same. Any guidance would be appreciated. Thanks!
     

    Attached Files:

  2. jcsd
  3. Sep 13, 2015 #2

    Nathanael

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    Can you relate the acceleration of the lower pulley to the acceleration of mass C?

    There is a simple way to relate the acceleration of mass A and mass B to the acceleration of the lower pulley. I don't want to spoil it, yet I'm not sure how to hint at it o0)

    Have another go at it, just remember that the length of each string is constant.
     
  4. Sep 13, 2015 #3
    So I believe that the acceleration of mass B is the negative of mass A; or aa = -ab.
    But I have looked into how the system would react to m3 accelerating downwards with a constant string length, and I just cannot understand it. Like, I want to believe that as m3 moves down a distance L, that the length of the lower hanging pulley also moves at that same L as do the masses ma and mb.
     
  5. Sep 13, 2015 #4

    Nathanael

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    This would be true for a single atwood-machine with two blocks A and B, but it is not quite right in our situation.
    (For example, if ma=mb, then they will have the same acceleration in the same direction and it will not necessarily be zero.)

    m3 is mass C?

    Anyway, consider the acceleration of the center of the lower pulley. What is it's relationship to ac?

    When you think about the acceleration of A and B, try to imagine it as being from two parts: partly from the acceleration of the lower pulley and partly from the movement of the string.
     
  6. Sep 13, 2015 #5
    Wait, why exactly would the accelerations not be opposite for these masses? If the pulley system is moved up, wouldn't they be affected in opposite directions?

    I would think that the lower pulley's acceleration would be the opposite of the acceleration of mass c.
     
  7. Sep 13, 2015 #6

    Nathanael

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    A special case to imagine, pretend A and B have the same mass: if the lower-pulley moves up then they will both move up with it, so the accelerations are not opposite.

    Right, because the length of string can't change (if C moves up an amount, then the center of the pulley must move down by the same amount).

    Consider the reference frame of the lower-pulley (just ignore the upper pulley and mass C for now). In this reference frame it is a normal atwood-machine, and so A and B have equal and opposite accelerations. Let's call the accelerations of A and B in this reference frame +aD and -aD. Now when we switch back to the original frame, we must add the acceleration of the lower-pulley, which I'll call ap. This means the accelerations of A and B are ap+aD and ap-aD.
     
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