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Potential energy of a displaced mass on a spring

  1. Jan 24, 2017 #1
    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. jcsd
  3. Jan 24, 2017 #2

    BvU

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

    [edit] hint: what about the discrepancy between 1st and 2nd relevant equation ?
     
  4. Jan 24, 2017 #3
    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
     
  5. Jan 24, 2017 #4

    haruspex

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    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.
     
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