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Pulley question assigning conventions

  1. Feb 3, 2016 #1
    upload_2016-2-3_18-17-9.png

    In the diagram, the pulleys and the strings are massless. The strings do not stretch. The masses of the suspended blocks are m1 and m2, as shown. The magnitudes of the accelerations of the blocks are a1 and a2, respectively. I'm having trouble assigning the conventions for the variables. I assigned the motion downwards as positive.

    1. The problem statement, all variables and given/known data

    T1 = The string connecting m2, and the string connecting the two pulleys.
    T2 = Left string connecting the left pulley.
    a1 = acceleration of mass 1
    a2 = acceleration of mass 2
    m2
    m1
    2. Relevant equations
    Result of FBD:

    T1 + m2g = m2a2
    T2 + T1 - m1g = - m1a1

    3. The attempt at a solution

    So... I can't isolate a1 in terms of a2.
     
  2. jcsd
  3. Feb 3, 2016 #2

    gneill

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    Staff: Mentor

    If the pulleys and strings are massless, and the string is continuous, can there really be different tensions in different parts of the string?
    Make a sketch where you draw mass m1 in its initial position and then the situation where it has moved down by some distance d. How much more string, in total, is in the sections of string supporting the lowered pulley? Where must that string have come from? So by how much must mass m2 have risen?
     
  4. Feb 3, 2016 #3
    If m1 is lowered by d, then m2 rose by d?
     
  5. Feb 3, 2016 #4
    This is a constrained motion, wherein, the motion of the first pulley is linked to the second block. Try to understand that if the first pulley goes down by a distance d, then the second block will have to go up by a distance of 2d. If you are able to catch this, your question will be solved.
    Hint: Draw INDIVIDUAL free body diagrams. That will make you clear.
     
  6. Feb 3, 2016 #5

    gneill

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    Staff: Mentor

    Here's a diagram:

    upload_2016-2-3_22-44-51.png

    The pulley on the left descends a distance d. The added length to each of the string segments that support it are shown in red. The total added length has to come from somewhere...
     
  7. Feb 7, 2016 #6
    Thank you for showing me how we get 2d. Do we do a FBD for each mass, pulley, or both? I don't understand how figuring out the distance to be 2d would lead to the solution.
     
  8. Feb 7, 2016 #7

    gneill

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    Staff: Mentor

    In your first post you indicated that you were having a hard time relating the two accelerations. You now have a relationship for the two displacements. The same relationship will hold for velocities and accelerations since the motions are constrained by this fixed relationship.
     
  9. Feb 7, 2016 #8
    How does the mathematical formalism work?
     
  10. Feb 7, 2016 #9

    gneill

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    Staff: Mentor

    It's a matter of differentiating displacement to get the velocity relationship, and differentiating again to get the acceleration relationship. The constant 2 is carried along. So for example, if x1 and x2 are the two displacements related by x2 = 2x1, then

    ##\frac{dx_2}{dt} = 2\frac{dx_2}{dt}~~~## or, ##~~~v_2 = 2 v_1##
     
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