# Simple harmonic oscillator

by mhellstrom
Tags: harmonic, oscillator, simple
 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
 Mentor 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.
 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 = 0.5*m*g/k Is this correct? Thanks in advance all the best
Mentor
P: 12,068
Simple harmonic oscillator

 Quote by mhellstrom Hi, thanks for the answer. So the mean is when m*g = k*x solving for x x = m*g/k
Correct.

 which results in the mean elongation of the spring is = 0.5*m*g/k Is this correct?
Not quite. It contradicts your previous statement.
 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
 Mentor P: 12,068 When you get x = mg/k you can stop, because that is the mean elongation.
 P: 15 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 = $$\int(m*g-k*x-m*g/k)^2 dx$$ Could anyone give a hint if this is on the right track? Thanks in advance all the best M

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