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2D isotropic oscillator

  1. Feb 4, 2006 #1
    I'm given some initial conditions for a 2-d isotropic oscillator:
    At t=0: x=A, y=4A, dx/dt = 0, dy/dt = 3wA

    Solving the differential equations of motion and using those conditions, I get the following:
    [tex]let\ \gamma = tan^{-1}(-3/4)[/tex]
    [tex]x(t) = A cos(\omega t)[/tex]
    [tex]y(t) = 5A cos(\omega t + \gamma)[/tex]

    The problem then asks to show that the motion is confined to a box of dimensions 2A and 10A. To me this seems inherent just by looking at the amplitudes of x and y, but maybe I'm missing something?

    The book (Fowles & Cassiday, 7th ed) goes into this big long spiel to show the confinement of motion. It rewrites y in terms of x, skips a million trig substitutions, and ends up with an equation of the form:
    [tex]ax^2 + bxy + cy^2 + dx +ey = f[/tex]

    And it says this can tell you, based on the discriminant, whether it's an ellipse, a parabola, or a hyperbola, and what it's bounds are.

    So I took my x and y (listed above), put y in terms of x, did some trig substitutions, rearranged, squared both sides, and ended up with:
    [tex]x^2 - 8xy + y^2 = 9[/tex]

    Now, how does this help me describe the motion any more than my original equations for x and y? And how does this help me to show that the motion is confined to a box of dimensions 2A and 10A any more than the amplitudes of the original equations do?

    Or should I ignore that whole part of the book? (probably not, but you never know)

    My position equations seem ok since they agree with the IC's, and this graph I made of them seems sane, and is clearly between -A,A and -5A,5A (which is what the problem text suggested).
     
  2. jcsd
  3. Feb 4, 2006 #2
    Here's the graph, so you don't have to follow the link:
     

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