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U(x) = -du/dx

  1. Mar 29, 2014 #1
    I couldn't manage to find substantial contents on the web explaining why U(x) = -du/dx.
    But in thinking about why U(x) = -du/dx, I constructed an x-y axis where the x represents the displacement and y represents the potential energy U(x).

    Suppose I started with an object moving down slope where m = -ve at an Θ where Θ< 90°. Then as time progress the object continues to move in the +ve x-direction while potential energy decreases. The converse holds true if the object were to move up wards where m = +ve at the same angle. Potential energy U(x) would increase as it moves in the +ve x-direction.

    From this, it can then be seen mathematically that U(x) = -du/dx for an object in the first case. In the second case, the negative sign in U(x) would be replaced with a positive.

    But what happens when the object is fell at an angle Θ = 90° to the x-axis? It is fairly obvious the the answer would be a real number/ 0 but would this scenario make any mathematical sense?
  2. jcsd
  3. Mar 29, 2014 #2


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    Staff: Mentor

    OK, U(x) is potential energy. What is u?
  4. Mar 29, 2014 #3
    u is the force?

    Edit: thinking further it does make sense. Force is the derivative of PE.
    U(x) =du/dx
    If this is true, all thats left is to understand the existence of the -ve sign
    Last edited: Mar 29, 2014
  5. Mar 30, 2014 #4


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    Staff: Mentor

    Then it should be F = -dU/dx.

    U is the (negative) integral of F: ##U(x) = - \int {F(x) dx}##

    The minus sign is needed in order to make conservation of energy work. As the force accelerates the object (increasing its kinetic energy), the potential energy must decrease, and vice versa.
    Last edited: Mar 30, 2014
  6. Mar 30, 2014 #5
    Well, it may be a good a idea to figure out what "u" is.
    If it's a force then the relationship is not true so it does not make sense to try to understand it.

    You may be mixed up.
    There is a relationship between force and potential energy. The force is the negative gradient of the potential energy. Is this what you had in mind?
  7. Mar 30, 2014 #6
    Yes, what you suggested was one of the mental bits I had in mind. As the gradient of the PE is negative, force is positive-and vice versa. Why does this relationship holds? Could this be expressed in mathematical language ?
  8. Mar 30, 2014 #7
    And then it makes sense. As an object travels down slope, towards +ve x-direction, PE decreases as KE increases.
    And should I want to express the change in energy in terms of KE instead of PE.
    It would then be F = dK/dx which suggests that as the object tends towards the +ve x-direction as it slopes down, KE increases.

    On a tanget, the difference between conservative forces and non-conservative forces is that the former path independent. In other words, the way I think about it, it is dependent on displacement rather than distance as non-conservative force depends on.
    Last edited: Mar 30, 2014
  9. Mar 30, 2014 #8


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    Staff Emeritus
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    In the future, please never simply write down a symbol without giving proper explanation of what the symbol represents. This is not only a bad practice here in this forum, but a bad practice when you are trying to learn something.

  10. Mar 30, 2014 #9
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