Simple harmonic oscillator

  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

    M
     
  2. jcsd
  3. Redbelly98

    Redbelly98 12,049
    Staff Emeritus
    Science Advisor
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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. Hi,
    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. Redbelly98

    Redbelly98 12,049
    Staff Emeritus
    Science Advisor
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    Correct.

    Not quite. It contradicts your previous statement.
     
  6. hi,

    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. Redbelly98

    Redbelly98 12,049
    Staff Emeritus
    Science Advisor
    Homework Helper

    When you get
    x = mg/k
    you can stop, because that is the mean elongation.
     
  8. thanks,

    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

    M
     
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