2 masses connected by a string wrapped around a massless pulley, F applied to pulley.

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

    m1 and m2 are connected by a massless string wrapped around a massless pulley. An external force F is applied to the pulley. m1 does not equal m2

    find the acceleration of each mass, the tension in the string, and the acceleration of the pulley.

    F external and m1 and m2 are known.

    there is no gravity or friction in the problem.
    2. Relevant equations

    F=ma



    3. The attempt at a solution

    What i've worked out so far is that we must consider two reference frames to determine the accelerations. If we can calculate the acceleration of the pulley and the acceleration of one of the masses, the acceleration of the other mass should be determined.

    If we consider the pulley's reference frame, one of the masses will be accelerating toward the pulley and the other mass will be accelerating with an equal and opposite acceleration.

    If we consider m1, we see the pulley accelerating toward m1 and m2 accelerating with an equal acceleration.

    how can we combine these two reference frames to determine the accelerations in the lab frame?

    is the acceleration of the center of mass of the system relevant?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. tiny-tim

    tiny-tim 26,041
    Science Advisor
    Homework Helper

    welcome to pf!

    hi tonicandgin! welcome to pf! :wink:
    no :confused:

    call the position of the pulley x, call the length of string between the pulley and m1 and m2, respectively, L1 and L2

    and use the fact that L1 + L2 must be constant …

    what do you get? :smile:
     
  4. Re: 2 masses connected by a string wrapped around a massless pulley, F applied to pul

    thanks for the response.

    to clarify: what do you mean by "call the position of the pulley x"? is that its position before the force is applied and everything starts moving? or does x change as the system moves?


    what i have so far is:

    call position of the pulley x. call the length between pulley and the 2 masses L1 and L2, respectively
    then position of m1 = x - L1 + 1/2 am1t2
    and position of m2 = x - L2 + 1/2 am2t2

    i've done a bunch of algebra trying to make use of L1 + L2 = L but nothing seems useful.

    I think what i should do is write position equations for each mass and then take two time derivatives to get acceleration, but the only position equations i can write include the accelerations I'm looking for in the first place.
     
  5. tiny-tim

    tiny-tim 26,041
    Science Advisor
    Homework Helper

    hi tonicandgin! :smile:

    (just got up :zzz: …)
    yes, that's the easiest way …

    then you get …

    position of m1 = x - L1
    position of m2 = x - L2

    then call the tension T, and do F = ma for each mass separately …

    what do you get? :smile:
     
  6. Re: 2 masses connected by a string wrapped around a massless pulley, F applied to pul

    thanks for the help. so far I have:

    a1 - ar = ap
    a2 + ar = ap

    where ar is the relative acceleration of each mass with the pulley.

    ( m1x1 + m2x2 ) / ( m1 + m2 ) is the center of mass of the system. if we take two time derivatives we get ( m1a1 + m2a2 ) / ( m1 + m2 ) which must equal the only external force, F.

    F must = 2T

    this is as much as i could get out of my professor today, he seemed to think it was solvable from here, but i still feel like there are too many unknowns and not enough equations.
     
  7. tiny-tim

    tiny-tim 26,041
    Science Advisor
    Homework Helper

    hi tonicandgin! :smile:

    (just got up :zzz: …)

    yes, there are two ways doing it

    your can either start with F = ma for the centre of mass, as your professor suggests,

    or you can just do F = ma for each of the three bodies separately
    whichever method you choose, write out all the equations you have, and we'll see :smile:
     
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