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Work done by an external force on a system equal and opposite to a conservative force is stored as potential energy within the system.

We choose an arbitrary location x and define the potential energy at a point y to be the potential energy required to move an object from x to y.

Am I missing something? or is this the fundamental principle of potential energy?

when an object moves with zero force it can have any speed right? my textbook says it moves with an infinitesimal velocity which seems arbitrary, the velocity shouldn't matter, all that matter is that the force is zero.

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    $\begingroup$ Potential energy is always stored in a system. Not in a single object. $\endgroup$
    – M. Enns
    Commented May 28, 2020 at 13:46
  • $\begingroup$ Check out this answer to better understand what M. Enns means: physics.stackexchange.com/q/553741 $\endgroup$
    – Gert
    Commented May 28, 2020 at 13:54
  • $\begingroup$ An object that is not accelerating (zero net force acting on it) has a constant velocity. That velocity can be any value at all, as it will be entirely dependent on the frame of reference - while standing still, you might see a train go by at 100mph, but someone on the train sees themself at rest and you go by at 100mph. But you can both agree that neither of you are accelerating and that no net force is being applied. $\endgroup$
    – Nuclear Hoagie
    Commented May 28, 2020 at 14:14
  • $\begingroup$ @NuclearWang is the rest correct? $\endgroup$
    – MinigameZ more
    Commented May 28, 2020 at 14:44
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    $\begingroup$ It's a little better to define PE in terms of the work done by internal conservative forces. $PE=-W_\mathrm{internal, conservative}$. This makes it clear that the PE is a property of a system, not an object, identifies where the PE is "stored", and avoids potential problems if multiple, or non-conservative forces are in play. Your first sentence almost says this, but since PE is a property of a system, I favor defining it in terms of system forces, not external ones. $\endgroup$
    – garyp
    Commented May 28, 2020 at 14:53

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Work done by an external force on a body equal and opposite to a conservative force is stored as potential energy within the body.

Although one often speaks of a an object having potential energy, as already pointed out by @M.Enns and @Gert, potential energy is technically a system property, not the property of individual objects, particles, etc.. The answers provided by @Gert and @Philip Wood in the link provided by @Gert already do an excellent job of explaining why. So I will focus on the following questions at the end of your post.

when an object moves with zero force it can have any speed right?

If an object is subjected to zero net force, yes it can have any speed. I emphasize zero net force because an object can be acted upon by several forces, as long as the net force is zero.

my textbook says it moves with an infinitesimal velocity which seems arbitrary, the velocity shouldn't matter, all that matter is that the force is zero.

Again, all that matters for the velocity to be constant is that the net force on the object is zero.

The fact that you are bringing this up in connection with potential energy suggests to me that there is a connection between this statement in your book and potential energy, such as gravitational potential energy.

For example, regardless of how an object of mass $m$ gets to say a height $h$ in the earth's gravitational field, the earth-object system will have gained gravitational potential energy of $mgh$. It doesn't matter if its velocity was infinitesimal, finite, constant or not constant, or whether it started and ended at rest. The gravitational potential energy of the earth-object system is independent of the manner in which the object got to the height $h$. This is because gravity is a conservative force.

What does matter is whether or not the object will, in addition, undergo a change kinetic energy between its starting point and the height $h$. If it does, it means there was a net force applied to the object and therefore net work done on the object. Per the work-energy theorem, the net work done on an object equals it change in kinetic energy.

Hope this helps.

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