How Does the Integral of Force Relate to Potential Energy?

[Hertz]

In my physics class, my teacher has been referring to Potential Energy in a 1D force field as the negative integral of the force with respect to position.

AKA:
$U(x)=-\int{F(x)dx}$
where U is the potential energy at x and F is the force at x.

Can someone please explain this to me in conceptual terms?

 

[Simon Bridge]

Sure - look at the relationship between gravitational potential energy and gravitational force.
Note: the rule only applies to conservative forces.

Objects like to go from high potential energy to low potential energy ...
In Newtonian mechanics, we describe this by saying the object experiences a force.
By thinking about how the change in potential relates to the instantanious acceleration you shoud be able to spot the connection.

You can also think of it in terms of the work energy relation ... where the force changes with position.

 

[UltrafastPED]

The "potential energy" is the amount of kinetic energy that would be gained if the object returned to where it started (without friction or other losses).

So think of kinetic energy is the energy of motion, and potential energy is the energy of "where things are".

 

[dauto]

That's just a definition. The potential energy is defined as (minus one) times the work done by the conservative force when moving the particle from some reference level. Combining that with the work-energy theorem gives you mechanical energy conservation.

 

[UltrafastPED]

It's an interesting "definition" if it describes real physics! Certainly it is sufficient for classical mechanics.

Definitions typically follow axioms or theorems, and give names to those results. So in that since potential energy is the "definition" of the quantity described by the work energy theorem.