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Two masses and two pulleys

  1. Sep 9, 2016 #1
    1. The problem statement, all variables and given/known data
    Masses M1 and M2 are connected to a system of strings and pulleys as shown. The strings are massless and inextensible, and the pulleys are massless and frictionless. Find the acceleration of M1.

    ATAeetS.jpg

    2. Relevant equations
    Newton's 2nd Law of motion

    3. The attempt at a solution

    So here is my line of reasoning. We have four objects with which we can use Newton's second law to derive relationships which will get us an explicit expression of the acceleration of block 1.

    Newton's law for block 1:
    ##T_1 - m_1 g = m_1 a_1##

    Newton's law for block 2:
    ##T_2 - m_2 g = m_2 a_2##

    Newton's law for pulley with block 2:
    ##T_1 - 2T_2 = 0##
    ##T_1 = 2T_2##

    Next, if we combine the equation for block 1 with the equation for block 2, and if we assume ##a_1 = a_2##, then we get,

    ##\displaystyle a_1 = \frac{g(2m_2 - m_1)}{m_1 - 2m_2}##

    However, this is not the correct expression, because we must have that if M1 = M2 then ##\displaystyle a_1 = \frac{g}{5}##

    Where am I going wrong with my reasoning?
     
  2. jcsd
  3. Sep 10, 2016 #2

    ehild

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    a1 is not equal to a2. Think: the moving pulley sinks by dx, how much does m2 move?
     
  4. Sep 10, 2016 #3
    But how does the second pulley move if it is massless?
     
  5. Sep 10, 2016 #4

    ehild

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    How does m2 move if the pulley does not move? The other end of the string is fixed to the ground.
     
  6. Sep 10, 2016 #5
    Okay... But my equation ##T_1 - 2T_2 = 0## is correct since the mass of that pulley is zero?
     
  7. Sep 10, 2016 #6

    ehild

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    Yes.
     
  8. Sep 10, 2016 #7
    Say for example that M1 is super heavy, and goes down. How does the second pulley accelerate upwards if it is massless?
     
  9. Sep 10, 2016 #8

    ehild

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    Why not? It pulls m2 upward.
     
  10. Sep 10, 2016 #9
    But it's massless, so the net force on it is zero, which means there would be no acceleration it seems
     
  11. Sep 10, 2016 #10

    ehild

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    A massless object can accelerate without any force. F=ma. What can be a if both F and m are zero?
     
  12. Sep 10, 2016 #11
    The tension on string 1 is twice the tension on string 2, so would it be fair to suppose that the acceleration of mass 2 would be twice that of mass 1?
     
  13. Sep 10, 2016 #12

    ehild

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    You get the relation between the accelerations from the constraints that the strings do not change their lengths.
     
  14. Sep 10, 2016 #13

    ehild

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  15. Sep 10, 2016 #14
    I'm just not seeing how to get the relation out of that... I know that I can say that the length of the string = so and so, and then differentiate twice... But I can't figure out the so and so.
     
  16. Sep 10, 2016 #15

    ehild

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    See the figure. If the pulley sinks by dx, the left piece of string becomes shorter by dx and the right one must become longer by dx. So the mass m2 moves downward by dx with respect to the pulley. But the pulley has moved downward by dx with respect to the ground, so the mass had to move by 2 dx downward with respect to the ground.
    upload_2016-9-10_8-23-47.png
     

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  17. Sep 10, 2016 #16
    How would you derive this from the fact that the length of the string is constant? If we say that ##l = x_L + x_R + \pi r##, where r is the radius of the pulley, and the two x's are the left and right sections of the string, then if we differentiate twice we just get that ##\ddot{x}_L = - \ddot{x}_R##, which doesn't show that the rate at which the mass moves is twice the rate at which the pulley moves.
     
  18. Sep 10, 2016 #17

    ehild

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    The acceleration is with respect to the ground. You need to differentiate h twice, instead of xr.
     
  19. Sep 10, 2016 #18
    So is it something like ##l = h + \pi r + (h - d_g)## where ##d_g## is the distance of the block from the ground. So if we differentiate twice we get that ##\ddot{d_g} = 2\ddot{h}##, which means that the rate at which the block accelerates is twice the rate at which the pulley accelerates...?
     
  20. Sep 10, 2016 #19

    ehild

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    Yes, and twice the acceleration of the other mass, only opposite direction.
     
  21. Sep 10, 2016 #20
    Do you get this from looking at the constant length of the other string for the other pulley?
     
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