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Dynamics: impulse momentum

  1. Dec 1, 2011 #1
    I am having a really hard time with dynamics of rigid bodies, but i think doing this problem will clear up some confusion i have.

    1. The problem statement, all variables and given/known data
    A massless rope hanging over a frictionless pulley of mass M supports two monkeys (one of mass M, the other of mass 2M). The system is released at rest at t = 0. During the following 2 sec, monkey B travels down 15 ft of rope to obtain a massless peanut at P. Monkey A holds tightly to the rope during these two sec. Find the displacement of A during the time interval. Treat the pulley as a uniform cylinder of radius R.
    Relevant picture: diagram

    2. Relevant equations
    As mentioned in the topic title, the problem has to do with impulse momentum:
    ∫ƩFy = m(y'f - y'i) [for monkey A]
    ∫ƩFy = m(y'f - y'i) [for monkey B]
    ∫ƩM = (I[itex]^{c}_{zz}[/itex]w)[itex]_{f}[/itex] -(I[itex]^{c}_{zz}[/itex]w)[itex]_{i}[/itex] [for the pulley of mass M]

    3. The attempt at a solution
    This is what I tried to do
    (all initial velocities and angular velocities are zero)

    For monkey A:
    ∫[itex]^{2}_{0}[/itex](T[itex]_{1}[/itex] - Mg) dt = M*y'[itex]_{A}[/itex][itex]_{f}[/itex]
    -> T[itex]_{1}[/itex]*t - Mgt = M*y'[itex]_{A}[/itex][itex]_{f}[/itex]
    -> 2T[itex]_{1}[/itex] -2Mg = M*y'[itex]_{A}[/itex][itex]_{f}[/itex]

    For monkey B:
    ∫[itex]^{2}_{0}[/itex](T[itex]_{2}[/itex] - 2Mg) dt = 2M*y'[itex]_{B}[/itex][itex]_{f}[/itex]
    -> T[itex]_{2}[/itex]*t - 2Mgt = 2M*y'[itex]_{B}[/itex][itex]_{f}[/itex]
    -> 2T[itex]_{2}[/itex] - 4Mg = 2M*y'[itex]_{B}[/itex][itex]_{f}[/itex]

    For the pulley:
    ∫[itex]^{2}_{0}[/itex](MgR - 2MgR) dt = (I[itex]^{c}_{zz}[/itex]w)[itex]_{f}[/itex]
    -> -2MgR = (1/2)MR[itex]^{2}[/itex]w[itex]_{f}[/itex]

    From the conclusion of all three sets of equations, I have 5 unknowns, with only three equations
    I figure from here I need to some kinematics:
    By circular motion, the y'[itex]_{A}[/itex] = R*w[itex]_{f}[/itex], which helps eliminate one unknown
    In addition, there is the length of rope connecting the three bodies, but considering the pulley has mass, I am not sure how to relate the velocities of point A and point B
    I believe I get stuck a lot on the kinematic constraints, and here I am not sure how to solve the system of equations I developed

    Any help is greatly appreciated
     
  2. jcsd
  3. Dec 4, 2011 #2

    rude man

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    Homework Helper
    Gold Member

    Seems to me this problem is solvable by elementary energy conservation principles.
    Were you explicitly asked to analyze in terms of impulse momentum considerations?
     
  4. Dec 4, 2011 #3
    Yes, the problem is an exercise in using impulse momentum. I know there are other ways to solve it, but I was hoping to clear up some confusion I have with using impulse momentum.
     
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