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Well, as far as I have been taught and know :$$\vec F = -dU/dx $$ But then my teacher also told me that when the derivative of potential energy is zero the $"net"$ Forces are zero.

But this contradicts both my book and my professor, as he had also said that the above statement comes from the fact that :

The work done by any force is integral of the dot product Force $\vec F$ and an infinitely small displacement $d\vec r$; where $\vec F$ represents a particular force and not the net force.

And I know that :

Work done by a conservative force is equal to negative of the change in potential energy.

So am I missing something here?

I viewed the similar questions but those didn't answer my query.

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    $\begingroup$ In your professor's scenario were all forces conservative so that the net force is also conservative? $\endgroup$
    – BioPhysicist
    Commented Apr 23, 2020 at 17:30
  • $\begingroup$ Not always, in one scenario that he took to explain this, there was friction acting only along horizontal part of the surface $\endgroup$
    – Koustubh Jain
    Commented Apr 23, 2020 at 17:39

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The concept of potential energy is only valid for a conservative force, and it is only used for a specific force. For example we can have gravitational potential energy, associated with gravitational force, and we can have electrostatic potential energy associated with Coulomb force. There is nothing to say that both of these forms of potential energy could not appear in the same question, and they would be treated as distinct. There may be other forces in the question, which have no bearing on the force associated with potential energy. I won't say "believe your book and your professor", because you are not missing something, and I think you can probably understand this from your own knowledge.

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If the net forces are not zero, then there would be some acceleration from Newton's law. The acceleration would change the energy of the object.

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  • $\begingroup$ But if a specific Potential energy is constant, then wouldn't it mean that the force providing that one PE would be zero and not others ? $\endgroup$
    – Koustubh Jain
    Commented Apr 23, 2020 at 17:10
  • $\begingroup$ in the previous one i meant, for example if the Elastic Potential energy by a spring is constant then wouldn't it mean that only the spring force is zero and not the other forces $\endgroup$
    – Koustubh Jain
    Commented Apr 23, 2020 at 17:19
  • $\begingroup$ I don't think the elastic potential energy of a spring would be constant. Otherwise it wouldn't be a spring. The potential energy of a spring changes when you stretch and compress it. $\endgroup$
    – Dapianoman
    Commented Apr 30, 2020 at 20:31
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You are correct. The equation $\vec{F}=- \vec{\nabla} U$ is only valid for a conservative force.

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  • $\begingroup$ i want to know that why is the net of all the forces is zero when the potential energy is constant $\endgroup$
    – Koustubh Jain
    Commented Apr 23, 2020 at 17:32
  • $\begingroup$ The net force is not necessarily zero. $\endgroup$
    – Arsaces
    Commented Apr 23, 2020 at 17:34

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