# Potential energy of a displaced mass on a spring

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1. Jan 24, 2017

1. The problem statement, all variables and given/known data

A spring of negligible mass exerts a restoring force on a point mass M given by F(x)= (-k1x)+(k2x^2) where k1 and k2 >0. Calculate the potential energy U(x) stored in the spring for a displacement x. Take U=0 at x=0.

2. Relevant equations

ΔU=∫F(x)dx
U=½kx^2

3. The attempt at a solution
Using the equation above I tried to find the potential energy of the spring after it has been displaced by some distance x. I integrated F(x) from x=0 to some displacement x=x0.

ΔU=∫(-k1x)+(k2x^2)dx

ΔU=-k1½x0^2+k2⅓x0^3

Is this the right way in finding the potential energy?

2. Jan 24, 2017

### BvU

Almost. Basically you are going to refine your second relevant equation. So if $k_2 = 0$ you should find it back. Do you ?

 hint: what about the discrepancy between 1st and 2nd relevant equation ?

3. Jan 24, 2017

So then I can use the fact that ΔU= Uf-Ui= ½kxf^2-½kxi^2. Where f is final and i is initial?
If this is correct, then xi=0 and that term drops out. Then I would be left with ½kxf^2 = -k1½x0^2+k2⅓x0^3

4. Jan 24, 2017

### haruspex

No. As BvU was hinting, that equation is only valid if F(x)=-kx.
No, for the reason given above.
Yes, except that you have a Δ on the left, implying a difference between two different states. On the right, you have assumed that one of those states is x=0. Try to write it consistently.