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About formula f=-dU/dx

  1. Feb 27, 2016 #1
    Is it true that we can't use the formula ##f##= ##\frac{-dU}{dx}## when external source of energy or some external force is present?

    I think it should rather be "we can't use ##f##= ##\frac{-dU}{dx}## when non conservative force is present" because potential energy is defined only for conservative force .
    (U in the above formula is potential energy , right?)
  2. jcsd
  3. Feb 27, 2016 #2
    If your f is a force then the force can be defined as negative gradient of potential energy - these systems are called conservative as the total energy is conserved
    however of dissipative forces are present and one can exactly account for the 'energy dissipated' or energy coming in -i do not think the total energy estimate can not be made in a closed system-what do you think?
  4. Feb 27, 2016 #3


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    I don't know. Where did you find this formula? In what context is it being used?
  5. Feb 27, 2016 #4
    I don't think it's that simple. The formula looks like the work-energy-theorem which is not limited to internal or conservative forces and potential energy. For example the kinetic energy of a braking car also follows this rule. In some cases this theorem doesn't hold (e.g. for open systems) but I don't know the general conditions for its validity.
  6. Feb 28, 2016 #5


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    I believe if the force is a function of position x, then F=-dU/dx applicable, where U is the position-dependent energy. In a spring, F=kx and hence, energy stored in the spring due to its displacement x is U=½kx2. Gravitational potential energy is also a position dependent energy. So, as per my understanding, F=-dU/dx is applicable to any position dependent force and energy system.
  7. Feb 28, 2016 #6


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    You did not say what f was. You always can define a function f(x) as the negative derivative of a differentiable function U(x) as f=-dU/dx. What is your problem???
    Try to ask a question which has sense.
    You asked the same question in an other thread recently, which has been closed .
  8. Feb 28, 2016 #7
    Sorry :frown:
  9. Feb 28, 2016 #8
    Okay, since you haven't specified what f and U is and what type of system you have in your mind I'm making the assumption that the body on which a force f acts is part of a system with potential energy U.
    If by "external force" you mean a force due to some influence outside of the system, then you're right, this relation won't hold. It doesn't matter whether the external force (external to the system) is conservative or not. This formula gives only the relation b/w the potential energy gradient and the force associated with THAT potential only.. Think of a spring- mass system in a gravitational field. Can you use this relation with f as the net force on the mass and U as either the gravitational potential energy? No. The gradient of gravitational potential energy gives the force on the mass due to gravity and the gradient of the spring potential energy gives the force due to the spring. You cannot mix up the two.
    This may not be the answer you were looking for, but hope I could make you see it clearly( somewhat).
  10. Feb 28, 2016 #9
    Don't be. Since this is Physics Forums and you are posting in the general physics sub, it should be abundantly clear what this formula is despite the fact that you didn't explicitly define the terms. You are asking for a clarification about the formula ##F=-\frac{dU}{dx}## where ##F## is force and ##U## is a scalar potential function (of position; I am not sure if it can be extended to have time dependance, but it doesn't in normal introductory courses). This is essentially the definition of a conservative force. If there is a ##U## such that a force can be written in this way, then ##F## is a conservative force.

    (This is a little off topic, but I think there are methods that treat friction as a force with a potential function even though it is not actually conservative. I could be totally wrong about this, but I think I recall one of my profs saying that friction terms are added to a effective potential function for certain situations)

    Aniruddha@94 gives a good answer. I assume your confusion is due the the term "external" which is probably not the best term, but it might be easier to grasp than "conservative force" for some people.
  11. Feb 28, 2016 #10
    Wow really?! I didn't know that. Could you please mention some source, it sounds interesting.
  12. Feb 28, 2016 #11
    I'll try to find one, but as I said, I only recall a professor telling me this. I may have just misunderstood what he was saying.
  13. Feb 28, 2016 #12

    Is F external to the system? How to know FORCE is external to the system or not?
  14. Feb 28, 2016 #13


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    When you analyze a problem, you are free to pick the boundaries of the system in any way you like. Having chosen those boundaries, a force is "external" if it acts on an entity inside the system and originates from an entity outside the system.
  15. Feb 29, 2016 #14


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    Instead of shoving a video, can you explain with your own words what the problem is? What is the set-up? F is a force, acting on what? U is the potential energy of what? What is x?
  16. Feb 29, 2016 #15
    Okay, I'll try to explain it. Are your concepts of potential energy, work and forces clear? If not then you should look into them. Whenever I have to understand something related to potential energy, I try to relate the system to a mass-earth system and think of gravitational potential energy ( because gravitation is very intuitive to us), so I'll use that to explain the idea behind the formula.
    In the case of capacitors, the parallel plates and the dielectric form the system. There are no frictional forces and no external forces ( we want the force b/w the plates and the dielectric slab). The total potential energy associated with this system is U=(1/2)*CV^2.
    Now think about a mass-earth system. The mass is at some height H and the potential energy of the SYSTEM is mgH ( we know mgh is only an approximation, but this will do for this discussion). mgH is the potential energy of the whole system, NOT of a single body. When you change the configuration of the system (I.e, change the height of the mass) then the potential energy of the system also changes, let's say it becomes mgH'. Now, the gradient of this potential energy (gradient is how U changes with distance x) gives us the force exerted ON a body.
    The forces in both the bodies are equal in magnitude (Newton's 3rd law), this fits perfectly. Can you see that both the forces ( on mass due to the earth and vice versa ) come from dU/dx .
    Coming back to our capacitor system, the same thing happens. The gradient of U gives the force on the dielectric. (Of course the dielectric is also pulling the plates with an equal force)
  17. Feb 29, 2016 #16
    The gradient of Potential energy gives force on just one body and not on entire system. Potential energy gradient gives force on one of the bodies constituting that system. I understand now. Thanks @Aniruddha@94
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