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Simple harmonic oscillator 
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#1
Oct308, 03:18 PM

P: 15

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
Oct308, 05:20 PM

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P: 12,068

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. 


#3
Oct408, 03:09 PM

P: 15

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 


#4
Oct408, 04:34 PM

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P: 12,068

Simple harmonic oscillator



#5
Oct508, 02:13 AM

P: 15

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 


#6
Oct508, 06:38 AM

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P: 12,068

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


#7
Oct508, 03:10 PM

P: 15

thanks,
if I want to estimate the variance of the elongation var = 1/N sum (x_{i}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*gk*xm*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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