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What is a simple harmonic oscillator

  1. Jul 24, 2014 #1
    Definition/Summary

    An object (typically a "mass on a spring") which has a position (or the appropriate generalization of position) which varies sinusoidally in time.

    Equations

    [tex]
    x(t)=A\sin(\omega t)+B\cos(\omega t)
    [/tex]

    [tex]
    \omega^2 =\frac{k}{m}
    [/tex]

    Extended explanation

    According to Hooke's law and Newton's 2nd Law, a point mass of mass [itex]m[/itex] attached to a spring of spring constant [itex]k[/itex] obeys the equation
    [tex]
    m\frac{d^2 x}{dt^2}=-kx\;,\qquad(1)
    [/tex]
    where [itex]x[/itex] is the position of the point mass.

    The solution of equation (1) is given by
    [tex]
    x(t)=A\sin(\omega t)+B\cos(\omega t)\;,\qquad(2)
    [/tex]
    where A and B are constants that may be chosen so that x(t) satisfies the appropriate initial conditions, and
    where
    [tex]
    \omega=\sqrt{\frac{k}{m}}\;.
    [/tex]

    For example, in terms of the initial position [itex]x_0[/itex] and initial velocity [itex]v_0[/itex], equation (2) can be written as
    [tex]
    x(t)=\frac{v_0}{\omega}\sin(\omega t)+x_0\cos(\omega t)\;.
    [/tex]

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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