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Simple harmonic oscillator

  1. Oct 3, 2008 #1
    Hi all,

    I have to determine the potential energy of a hanging spring with a mass m in the end and spring constant k. I try to write down the force in the system

    F = m*g + k*x

    and integrate the force in order to get the potential energy

    E_p = m*g*x+0.5*k*x*x

    Does this look correct and is it possible to derive the mean displacement from the potential energy if one could neglect the kinetic energy.

    Thanks in advance

    Best regards

  2. jcsd
  3. Oct 3, 2008 #2


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    You're expression for E_p is correct. I'm assuming you're taking upward as the positive direction. Your force expression has +/- sign issues, by the way.

    The mean displacement is where E_p has a minimum value. So yes, it's possible to derive mean displacement from your E_p expression.
  4. Oct 4, 2008 #3
    thanks for the answer. So the mean is when

    m*g = k*x

    solving for x

    x = m*g/k

    which results in the mean elongation of the spring is

    <dis> = 0.5*m*g/k

    Is this correct?

    Thanks in advance

    all the best
  5. Oct 4, 2008 #4


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    Not quite. It contradicts your previous statement.
  6. Oct 5, 2008 #5

    I am a little bit puzzled where my mistake is... I differentiate my expression for the potential energy in order to find a stationary point

    d(E_p) = m*g - k*x

    setting this equal to zero and solving for x

    x = m*g/k

    than I set this into the equation for the potential energy as I presume this is the minimum

    E_p = m*g*(m*g/k)-0.5*k*(m*g/k)^2
    = 0.5 * (m*g)^2/k

    this I would presume is the expression for the mean elongation? Where does I misunderstand thanks in advance

    All the best
  7. Oct 5, 2008 #6


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    When you get
    x = mg/k
    you can stop, because that is the mean elongation.
  8. Oct 5, 2008 #7

    if I want to estimate the variance of the elongation

    var = 1/N sum (xi-x_mean)2

    I know the mean is x_mean = m*g/k which I insert into the expression and integrate from minus to plus infinity

    var = [tex]\int(m*g-k*x-m*g/k)^2 dx[/tex]

    Could anyone give a hint if this is on the right track?

    Thanks in advance all the best

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