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Three hanging masses and two pulleys, why does m3 accelerate?

  1. Feb 17, 2014 #1
    1. The problem statement, all variables and given/known data

    Figure CP7.57 shows three hanging masses connected by massless strings over two massless, frictionless pulleys.

    a) Suppose: m1 = 2.5 kg, m2 = 1.5 kg, and m3 = 4.0 kg. Find the acceleration of each.

    b) The 4.0 kg mass would appear to be in equilibrium. Explain why it accelerates.

    2. Relevant equations

    Tension of Rope A = 4*m1*m2*m3*g / (4*m1*m2 + m2*m3 + m1*m3)

    3. The attempt at a solution

    I managed to solve a) using the above equation which I derived thru some lengthy algebra (5 equations involving 5 unknowns -- a1, a2, a3, Tension of Rope A, and Tension of Rope B), and got the following answers: a1 = -2.21m/s^2, a2 = 2.85m/s^2, and a3 = -0.316m/s^2. These answers match up with the answer key.

    But I'm kind of stuck at part (b) -- even though I worked out mathematically in part (a) that box 3 does indeed accelerate, I'm unsure how to explain why it accelerates in a way that makes intuitive sense. There was no answer key for part (b), so I'd appreciate any helpful guidance. I've attached a picture illustration of the problem as well.
     

    Attached Files:

  2. jcsd
  3. Feb 17, 2014 #2
    Were m1 and m2 equal, the system should stay still, since m1 is heavier it pulls on the cord, adding momentum to the smaller pulley system and breaking the equilibrium state between m1, m2 and m3. It has been a long day, though, I haven't run any numbers, just an intuitive guess.
     
  4. Feb 17, 2014 #3
    Oh...thanks for posting lendav_rott. I think I might have gotten this. Just now, I imagined an extreme case in which m1 was massless, and m2 = m3. Well m2 would accelerate right down without any resistance, and due to this m3 would also accelerate right down. So, going back to this problem, m3 accelerates because m1 and m2 are not in equilibrium, so even though m1 + m2 = m3, the force pulling upward on m3 is less than m1 + m2. Does that sound right?
     
  5. Feb 17, 2014 #4

    PhanthomJay

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    Newtons laws don't work very well in non inertial frames. As a thought experiment, consider a block m1 resting on another block m2. There is no friction between blocks m1 and m2 or between the floor and m2. Now apply a horizontal pull force T to the lower block, and an equal but opposite force T to the top block. While there is no acceleration of the center of mass of the system, the blocks each accelerate separately. They do not remain at rest.
     
  6. Feb 17, 2014 #5

    BvU

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    Let me drop a coin in the hat as well: Hold B fixed for a moment.
    The center of gravity of m1 and m2 is accelerating downwards. You can calculate that a, and it must be the resultant from g(m1+m2) - tension in AB . So tension in AB < gm3. Now let go of B: acceleration again. m1+m2 accelerated upwards so a new balance equation is needed - the a has to be adapted slightly. How much follows from the tension, which is now (g - a')m3. Sort of, if I didn't miss something.

    Funny they ask a first, then b...
     
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