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Help Me With The Three Springs Problem

  1. Dec 21, 2011 #1
    1. The problem statement, all variables and given/known data

    Two walls are positioned a distance of 3L apart. Three identical springs of rest length L and spring constant k are connected in series between the walls. Two particles of mass m and αm are positioned at the junctures of the springs, respectively.

    Write down the Largrangian for the system.
    Use the Lagrangian to derive the equations of motion for the system.
    Find the normal frequencies ωi
    Assume α is very large. Expand ωi up to order O(1/α) and explain the result obtained.
    Assume α is very small. Expand ωi up to order O(1) and explain the result obtained.

    3. The attempt at a solution

    I have already found correct answers for the first three questions. The normal frequencies are √[(k ± 1/σ)/m]. The first corresponds to the rate of change of the distance between the two particles. The second to them moving back and forth while their distance remains the same.

    So if we start with start with basis
    (x = position of particle of mass m,0) and (0, y = position of particle of mass αm)
    And change from (x,0); (y,0) to (x,y);(x,-y) the Lagrangian simplifies and the frequencies above drop out.

    This gives the correct result for α = 1.

    When we assume α is large expand the frequencies up to order 1, nothing changes, but the quotient of α makes very little contribution, so the frequencies are almost identical.

    Likewise for when we assume α is small and expand to order zero, the α drop off and the frequencies are exactly zero. (I do not know why you would eliminate the higher order terms for small α)

    In either case the frequencies are almost equal.

    This is what I think will happen in those cases: the frequencies of the together//apart and back/forth together motions are the same. So the particles move together while moving to the right, and then apart while moving to the left, or vice versa. This corresponds to a "punching motion", where the particles take turns moving away from the centre while the other stays still.

    The motion is the same for either case, because in either case one of the masses becomes negligible.

    Am I right?
     
  2. jcsd
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