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Classical Mechanics 3 Masses

  1. Mar 2, 2008 #1
    Three equal masses arranged in a equilateral triangle are connected by 'springs' with force constants 'k'

    the coordinates of the masses are:

    mass 1 at [0, [tex]\sqrt{3}[/tex]/2*L]
    mass 2 at [L/2, 0]
    mass 3 at [-L/2,0]

    find the normal mode frequencies.

    The only part i am having trouble with is setting up the potential energy i know what to do after I have the potential energy.

    so far i have

    U = 1/2 * k (x_2 - x_3)^2

    i know there are more terms in the potential but i am having trouble projecting the deviations from equilibrium onto the diagonals.
  2. jcsd
  3. Mar 2, 2008 #2


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    well, there's 6 degrees of freedom, so there are 6 normal modes. 2 are translations as a whole and 1 is rotation as a whole. there are 3 vibrational mode, one of which is the "breathing" mode, and then there are 2 others.
  4. Mar 2, 2008 #3
    neglecting all rotation what will the other terms of the potential be?
  5. Mar 2, 2008 #4


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    U=\frac{k}{2}((\vec x^{(1)}-\vec x^{(2)})^2+(\vec x^{(1)}-\vec x ^{(3)})^2+(\vec x^{(2)}-\vec x^{(3)})^2)
  6. Mar 2, 2008 #5
    in your notation vector x 1 is the location of the first mass [x1,y1] , x 2 second mass [x2,y2],....

  7. Mar 2, 2008 #6


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