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Initial acceleration of the charge

  1. Nov 13, 2015 #1
    1. The problem statement, all variables and given/known data
    Three identical charged balls of mass m and charge q bound in a triangle thread length l. One of the strands break. Calculate the acceleration of the middle ball at the initial moment.

    2. Relevant equation

    a. Newton's laws

    ##m\vec a_1 = \vec F_{12} +\vec F_{13} + \vec T_{13}##
    ##m\vec a_2 = \vec F_{21} +\vec F_{23} + \vec T_{23}##
    ##m\vec a_3 = \vec F_{31} +\vec F_{32} + \vec T_{31} + \vec T_{32}##


    b. Constraint Equations

    ##(\vec r_1 -\vec r_2)^2 = (\vec r_1 -\vec r_3)^2 = const##

    c. May be relations between ##\vec r##-vectors in CM-system
    ##\vec r_1 +\vec r_2+\vec r_3=0 ##

    12-02.gif
    3. The attempt at a solution
    The unknown T-forces should to be exluded using Constraint Equations, but what to do with it, I have no idea.
     
  2. jcsd
  3. Nov 13, 2015 #2

    haruspex

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    How could you obtain another equation concerning accelerations from you constraint equations (b)?
    I believe your equation (c) should be deducible from the other equations.
     
  4. Nov 13, 2015 #3
    Yes (c) is the cosequence of the Newton's laws.
    I don't know answers. Now I have no idea, how to start to solve this problem.
     
  5. Nov 14, 2015 #4
    From the (b), I can obtain
    First differentiating:
    ##(\vec r_1 - \vec r_3)(\vec v_1 - \vec v_3)=0##
    From the second differentiating:
    ##(\vec v_1-\vec v_3)^2 + (\vec r_1 - \vec r_3)(\vec a_1 - \vec a_3)=0##

    And for the second constraint
    ##(\vec v_2-\vec v_3)^2 + (\vec r_2- \vec r_3)(\vec a_2 - \vec a_3)=0##
     
    Last edited: Nov 14, 2015
  6. Nov 14, 2015 #5

    haruspex

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    Right. But we are only interested in the initial acceleration. What simplification does that provide?
     
  7. Nov 14, 2015 #6
    For the initial conditions:
    ##\vec v_1= \vec v_2 = \vec v_3=0##.
    Then
    ##(\vec r_1 - \vec r_3)(\vec a_1 - \vec a_3)=0##

    And
    ##(\vec r_2 - \vec r_3)(\vec a_2 - \vec a_3)=0##
     
    Last edited: Nov 14, 2015
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