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X(t) of a diatomic molecule with given v_0

  1. Jul 4, 2012 #1
    1. The problem statement, all variables and given/known data
    Two identical carts (of mass m) are free to move on a frictionless, straight horizontal track. The masses are connected by a spring of constant k and un-stretched length [itex]l_{0}[/itex]. Initially the masses are a distance [itex]l_{0}[/itex] apart with the mass on the left having a speed [itex]v_{0}[/itex] to the right and the mass on the right at rest. Find the position of mass on the left as a function of time.

    2. Relevant equations
    [itex]q=x_{2}-x_{1}-l_{0}[/itex]

    [itex]\dot{q}=\dot{x_{2}}-\dot{x_{1}}[/itex]

    [itex]\ddot{q}=\ddot{x_{2}}-\ddot{x_{1}}[/itex]

    3. The attempt at a solution
    [itex]m\ddot{x_{1}}=-k(x_{2}-x_{1}-l_{0})[/itex]

    [itex]m\ddot{x_{2}}=k(x_{2}-x_{1}-l_{0})[/itex]

    remembering [itex]\ddot{q}=\ddot{x_{2}}-\ddot{x_{1}}[/itex]

    [itex]\ddot{q}=\frac{2k}{m}q=\omega^{2}q[/itex]

    [itex]q(t)=c_{1}e^{\omega t}+c_{2}e^{-\omega t}[/itex]

    Just wondering if I'm on the right track? If so I'll do the initial conditions and then solve for [itex]x_{1}(t)[/itex]

    Thanks for the help!
     
  2. jcsd
  3. Jul 4, 2012 #2
    Hi AbigailM! :smile:

    It would help if you state what x1 and x2 are... :tongue2:
     
  4. Jul 4, 2012 #3
    :eek: oops, x1 is the position of cart 1 and x2 is the position of cart 2.
     
  5. Jul 4, 2012 #4
    What have you chosen as the origin? I believe q is the elongation/compression of the spring? Is cart 1 the left cart, and cart 2 the right?


    PS : Its good to state all assumptions before solving the problem :wink:
     
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