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Math Help - Equilibria and Oscillations question

  1. #1
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    Equilibria and Oscillations question

    stuck again...

    A particle of mass m = 1 moves along the x-axis under a force whose potential is:

    V(x) = (1 - x^2)*e^(-(x^2)/4)

    Sketch the function V against x, find equilibrium positions and stability. Determine the periods of small oscillations about the stable equilibrium point(s). If the particle is started at x = sqrt(5) with speed U, determine the ranges of U for which the particle (i) oscillates, (ii) escapes to
    +infinity and (iii) escapes to −infinity.


    Many thanks
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by callumh167 View Post
    stuck again...

    A particle of mass m = 1 moves along the x-axis under a force whose potential is:

    V(x) = (1 - x^2)*e^(-(x^2)/4)

    Sketch the function V against x, find equilibrium positions and stability. Determine the periods of small oscillations about the stable equilibrium point(s). If the particle is started at x = sqrt(5) with speed U, determine the ranges of U for which the particle (i) oscillates, (ii) escapes to
    +infinity and (iii) escapes to −infinity.


    Many thanks
    The force on the particle is:

    F(x)=\left. \frac{dV}{dx}\right|_x

    The possible equilibrium positions are where there is no force, and so correspond to the extrema of V(x) .

    Stable equilibria are minima of V(x).

    RonL
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