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General Atwood system with moment of inertia

  1. Feb 3, 2012 #1
    1. The problem statement, all variables and given/known data


    I have taken this image from http://farside.ph.utexas.edu/teaching/336k/lectures/node79.html

    Hello, I am currently doing a project on Lagrangian mechanics. As part of my project, it was advised that I should redo each of the problems done using the Lagrangian method in the Newtonian format.

    So essentially I am requesting help on determining the acceleration of the system

    2. Relevant equations




    3. The attempt at a solution

    I am getting increasingly confused especially when the signs don't match :(
    (a = radius of the pulley, not acceleration)
    (acceleration is denoted as [itex]\ddot{x}[/itex]



    Tangential acceleration of the pulley is the same as the acceleration of the mass, assuming that there is absolute grip of the rope to the pulley.


    However we can see that






    Adding the first 2 equations


    Replacing the Tensions


    Finally rearranging it



    The signs are everywhere and completely different to the derivation obtained through the Lagrangian method. :I Can anyone explain what my issue is?

    EDIT: Included the solution obtained via Lagrangian method

  2. jcsd
  3. Feb 3, 2012 #2


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    Gold Member

    The minus sign difference in the numerator is not a problem but the minus sign in the detonator of your solution is a problem, the inertia of the pulley should have a plus sign, its inertia tends to reduce the acceleration of the system.
  4. Feb 6, 2012 #3
    Thanks for your reply Spinnor,

    I agree the problem is within the negative moment of Inertia in the final acceleration equation.

    However, I can't seem to find a reason to justify it. (Doing so would somehow need to make the initial F=T-mg equation negative or somehow justify T_2 - T_1 instead of T_1 - T_2 for the angular acc. equation)
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