# Where is the potential energy?

I read this question What is potential energy truly? and I find the answers not really satisfying.

When I move an object upward in a gravitational field, I have to work against that gravity. There is not really any energy mysteriously stored inside the object. Conversely, if the object is pushed over the edge it will fall because of that gravitational field and on impact it will exert the energy it obtained because of the acceleration.

If I move such a system into a zero gravity zone in space, then the object would still be "above" (even though "above" may not make much sense in zero gravity), but suddenly the potential energy is gone?

So isn't potential energy just an effect of gravitation and not something "hidden" inside an object?

• [...] and not something "hidden" inside an object? Who claims that it is? Something that has energy is capable of performing work. Is an object suspended on an edge capable of that?
– Gert
May 1, 2021 at 15:56
• @Gert, Who claims that it is? That's how I understand all the explanations as far as I understand them. May 1, 2021 at 15:59
• This may help. Potential energy is caused by what force? May 1, 2021 at 18:57

If I move such a system into a zero gravity zone in space, then the object would still be "above" but suddenly the potential energy is gone?

It is not suddenly gone. You had to get the object there.

In your first example of lifting the object, it is presumed that the force of gravity is constant because the distance moved is small compared to the radius of the earth. But each time you lift it an additional height the work you need to lift it the same distance is theoretically less because the force of gravity gets smaller (being proportional to the inverse square of the radius r of the earth).

Eventually, if you get far enough away from the earth (and other gravitating bodies), an infinitesimal amount of work is required to move it any farther away from the earth. For this reason the gravitational potential energy is assigned a value of zero at r equal to infinity, making all values of gravitational potential energy negative.

So isn't potential energy just an effect of gravitation and not something "hidden" inside an object?

Potential energy is not something "hidden" inside an object. First of all, objects themselves don't posses gravitational potential energy. Potential energy is a system property. In the case of the object lifted by an external (to the Earth/object system) force the increase in gravitational potential energy is that of the Earth/object system, not the object alone or the Earth alone.

If the object then falls, the potential energy of the Earth/object system is converted to kinetic energy of the object due to the work done by the gravitational field.

In any case, we are normally only interested in the changes in potential energy, not the absolute values of potential energy.

Hope this helps.

The "zero" potential energy point has to be arbitrarily assigned to some position. In a zero gravity zone in space, there is no up or down, and if you don't select your zero potential energy point, there is nothing to measure against. This means that the potential energy has not gone anywhere, as the problem statement is incomplete, eliminating the possibility of calculating a potential energy for the object that is in the zero gravity zone in space.

Also note that there is no zero gravity zone in space. There are points where the gravitational force from one object balances the gravitational force from another object such that there is zero net gravitational force from those objects at that point, but the gravitational force from any given object does not quit after a given distance.

• It's probably useful to add that we're never interested in PE's absolute value but rather in changes of the PE of an object or system.
– Gert
May 1, 2021 at 16:06

So, I mean, let's start with, “yes, you’re right.” Potential energy is “stored” in the abstract coördinates of a system. You can sometimes connect that to a particular object in the system, for example the position of a small mass in a large mass’s gravitational well, assuming that the large mass is immobile. Then it’s just stored in the small mass’s radial position and you can pretend that the energy is stored in there. Or, if you have two masses attached by a spring, the coördinate is the distance between them and, handily enough, you have a spring stretched for that length so it is handy to say that energy is “stored in the spring,” even though if we’re super-pedantic we might say it’s stored in “the strain coördinate of the spring” or so.

Then you run into more strange phenomena. The field components of the electromagnetic field are components that “store energy.” It also stores momentum and angular momentum! If you want those to be stored in concrete objects you have to go all the way to an advanced but of physics called “quantum field theory” to argue that these are stored “in the photons” or so.

What's really happening is that energy is a different way to look at a system, compared with forces. It is just a different mental accounting. And just like real accounting, where a ton of details have to do with “ledger entries” (value transfers) but you can often get a good understanding from summing them up into overall “account balances”, in the energy perspective we sum up a lot of momentum flows and directions to only focus on the speeds of our particles. We lose some information, but we gain an easier way to reason about the system. And in fact if we are lucky, we can sometimes recover the exact dynamics (the transfers underneath, the forces) from a picture of the energy possibilities, often in the form “If I started here and I ended there, what was the set of transfers that had to happen to get me there?”[1] For this in the energy picture we usually need to ignore energy dissipation (friction etc) so that we can say “you had to get this energy from somewhere and it came from here, and such-and-so was the force that needed to happen to perform that work on you.”

The energies that can be unambiguously associated with the particles are kinetic energies, but it is also a mistake to think of those as a stuff because they are not reference-frame invariant. You see a grapefruit standing still on a table and say there is no kinetic energy, I am juggling on a train and I pass by your house and I think that’s a 70 km/hr grapefruit, I am very glad I do not have to catch that right now! The proof is if you tried to gently toss it to me, and I tried to add it to the balls I am juggling: I would lose a hand! Or at least cause a citrus explosion.

Potential energy is just what forces look like in this energy language. Not all forces can easily be brought into this language, but the ones that can are described by a potential energy function that says “given where everything is, here is a number representing the potential energy, and if that number decreases then by the work-energy theorem the kinetic energies have commensurately increased because I have precomputed that such-and-so work was performed on that system by these forces.” So, if forces can be associated with the interaction between two particles then so can energies, if you can take one of those particles for granted (like the Earth) then you can associate the force/energy purely with the other particle.

That we can almost always precompute this value is the surprising thing, that is a manifestation of the “unreasonable effectiveness of mathematics” in describing the world.

1. This is kind of like a bus map if you aren’t sick of analogies. :) “You were going North from 3rd and Hamilton on route 7, now you are going East at 12th and Main on route 11, the only way you could have done that without looping all the way around on one of these routes is if you got off at 15th street and took the route 3 bus to State St, then got off there... yeah, from there you could get on Route 7.” For us it is more abstract, the energies are assembled into a “Lagrangian” and a branch of mathematics called “calculus of variations” handles the “if you start here and end there how did you travel between them?” questions.

Where is the potential energy ?

Potential energy does not have a location. It is an attribute of the system - specifically, it is a function of the configuration of the system - but it does not have a location any more than the total mass of the system or the distance between the two objects has a location.

First, use the word gravitational potential energy when talking in the context of gravitation.

When I move an object upward in a gravitational field, I have to work against that gravity. There is not really any energy mysteriously stored inside the object.

Well Yes! When you move the object at some height, you have used your energy to do work against gravitational force, this energy stored inside the object. If you want, you can test it by a change in mass of the object ($$E=mc^2$$).

if the object is pushed over the edge it will fall because of that gravitational field and on impact, it will exert the energy it obtained because of the acceleration.

Yes! that's true.

isn't potential energy just an effect of gravitation and not something "hidden" inside an object?

I don't understand what do you mean by something hidden inside an object. But you can use $$E=mc^2$$ to see potential energy. $$m_\text{extra}=\frac{E_\text{potential}}{c^2}$$

• If you want, you can test it by a change in mass of the object ($E=mc^2$) Correct but unless $E$ is absolutely ginormous that effect is almost imperceptible (and thus hardly testable)
– Gert
May 1, 2021 at 16:11

"If I move such a system into a zero gravity zone in space, then the object would still be "above" (even though "above" may not make much sense in zero gravity),but suddenly the potential energy is gone?"

Nope. The potential energy is now stored in the object.

If the object is in gravity it will " fall back" and the potential energy is converted to the kinetic energy while "falling back".

But if there is no gravity then the energy is still there in the object and will be released whenever finds any chance.