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Show that potential energy is conserved

  1. Apr 19, 2017 #1
    1. The problem statement, all variables and given/known data

    potential energy function of :

    $$
    U(x) = 4x^2 + 3
    $$

    And have to

    i) Work out the equation of motion

    ii) Prove explicitly that the total energy is conserved


    2. Relevant equations


    $$
    F = \frac{dU}{dt}
    $$

    3. The attempt at a solution

    I'm not too sure how to go about this.

    I would say that I have the force of

    $$
    F = 8x
    $$

    By differentiating the given potential energy function. I need to work out the
    equation of motion, what I have an object with mass ##m##.

    So this means that I have

    $$
    F = 8x = ma
    $$

    Then I have that

    $$
    a = \frac{8x}{m}
    $$

    Is this an equation of motion? I mean, it's acceleration, or should I find for
    ##v(t)## and ##x(t)## as well as this?

    In which case I would have

    $$
    v(t) = \int a(t) dt
    $$

    Which in this case is found as (having the mass in the equation seems unusual?)

    $$
    v(t) = v_0 + \frac{1}{2m}8x^2 = v_0 + \frac{4}{m} x^2
    $$


    So then from this I have that

    $$
    x(t) = x_0 + v_0t + \frac{4}{3m}x^3
    $$

    And this would be all of the equations of motion for this 1D case?

    Then I need to prove that energy is conserved here, and I've no idea how to go
    about that.

    I've not been given any frictional forces, so it seems like it's just a given
    that I'm going to have

    $$
    W + PE_0 + KE_0 =
    PE_f + KE_f + \text{Energy(Lost)}
    $$

    Here I can remove work ##W## and the energy lost for

    $$
    PE_0 + KE_0 =
    PE_f + KE_f
    $$

    And I need to do something with these?

    Potential energy - I have the potential energy function given as part of the
    problem which is

    $$
    U(x) =
    4x^2 + 3
    $$

    Then I can sub this into the energy expression as
    $$
    4x_0^2 + 3
    + KE_0 =
    4x_f^2 + 3
    + KE_f
    $$

    Getting rid of the constants seems pretty harmless

    $$
    4x_0^2
    + KE_0 =
    4x_f^2
    + KE_f
    $$

    Now I'm really not sure what I should do from here, sub in kinetic formulas of
    ##K = \frac{1}{2}mv^2##?

    $$
    4x_0^2
    +
    \frac{1}{2}mv_0^2
    =
    4x_f^2
    +
    \frac{1}{2}mv_f^2
    $$

    I'm not sure if I can arrange this to be 'nicer' in any way either, I'm purely
    thinking in algebra at the moment though not physics :S


    $$
    8(x_0^2 - x_f^2) =
    m(v_f^2 - v_0^2)
    $$

    I'm not sure if differentiation should do anything nice here, but I really have
    no idea what I'm doing with this.

    Thanks
     
  2. jcsd
  3. Apr 19, 2017 #2

    haruspex

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    Dividing energy by time gives power, not force.
     
  4. Apr 20, 2017 #3
    OK ##F = - \frac{dU}{dx}## sorry , i'm still unsure about the question
     
  5. Apr 20, 2017 #4

    haruspex

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    You got a=8x/m ok, but you cannot integrate that wrt t directly. The expression you got for v(t) was the integral wrt x (which just gets you back to U).

    There is a useful trick for solving equations like ##\ddot x=f(x)##. Multiply both sides by ##\dot x##, then integrate dt.
     
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