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Energy in damped harmonic motion

  1. May 29, 2014 #1
    Hey PF,
    my book either got sloppy in a derivation or I am not connecting two very obvious dots.
    It gives the energy of the damped harmonic oscillator as
    E = (1/2)mv^2 + (1/2)kx^2
    then takes the derivative with respect to time to get dE/dt.

    then it gives the differential equation of motion as
    ma + kx = -cv

    okay cool i'm following so far...
    then it says with this equation of motion we know that
    dE/dt = -cv^2

    what am I missing here?
  2. jcsd
  3. May 29, 2014 #2
    Ff= -cv is the damping (friction) force.
    dE/dt is the rate of dissipation of the mechanical energy.
    Or the power dissipated due to the action of the friction force.
  4. May 29, 2014 #3

    D H

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    Staff Emeritus
    Science Advisor

    Which your text should have as ##\dot E = v(ma + kx)##.

    That's just a rearrangement of Newton's second law, with the external forces being the spring, linearly directed against displacement, and drag, linearly directed against motion. Mathematically, ##F=ma = -kx - cv##. Adding ##kx## to both sides yields ##ma+kx=-cv##. Substituting that result back into the expression for the time derivative of energy yields ##\dot E = -cv^2##.
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