I believe the energy stored as mass business has to do with the relativistic understanding of the problem, which is all good and well, but, as Pasta noted, it will probably be useful for you to appreciate the matter from a high school-level Newtonian perspective as well (or even just instead), as I tried to explain in my previous post. In this framework, there truly is no mass change, as you note, yet there still is something to the notion of potential energy, arising from a non-trivial law of conservation of energy.
This is really playing with words more than anything else and the responses are unnecessarily complicated. Potential energy is not something intrinsic to a mass at all and you cannot try to pretend it is without incurring in contradictions. Potential energy of a mass is the energy which could conceivably be extracted if the mass were allowed to fall in a gravitational field. If the mass cannot do that for whatever reason then there is no potential energy. And the only way to know if the mass has potential energy is to find out some information external to the mass.
So the question is like asking how you can find out if a naked person is an American citizen without looking at anything but his naked body. The answer is “you can’t”. Or how can you know if a person is rich by just looking at his body and nothing else.
Weird hair splitting. The system gains mass. The boulder is part of the system. Ergo it doesn’t gain mass? Umm, yeah right.
I also see nothing in the OP that restricts the discussion to Newtonian Physics. And if it did it would be a big problem. One of the great things about Einstein’s work is that it fixed a lot of problems in terms of correctly balancing the books for energy.
The question I asked my high school physics teacher after he introduced the concept of potential energy went something like this:
Me: So if a boulder is lying on flat ground it has no potential energy?
Teacher: That’s right.
Me: Then, if we dig a hole right next to the boulder, even if we don’t do anything to the boulder itself, we’re giving it potential energy?
Teacher: That’s right.
Me: And the deeper we dig the hole, the more potential energy the boulder has?
Teacher: That’s right.
Me: Even if we’re not doing anything at all to the boulder?
Teacher: That’s right.
Me: Huh.
It could be my teacher was wrong. But if not, then the explanation of potential energy as being *part of the total energy within a system *makes some sense, while the claim that PE is nothing but a function of mass in relation to the center of the earth doesn’t quite seem to fit…
A counter thought experiment:
So. Do you believe in kinetic energy?
Potential energy, at least of the gravitational kind, is measured in relation to some reference. The numerical value of the potential energy isn’t important in and of itself, it’s the change in energy (as in a boulder rolling down a hill) that’s important, which is independent of a reference. So the choice of reference is somewhat arbitrary, and is usually chosen to be convenient and intuitive.
Usually we assume some convenient “lowest level” altitude as the reference zero point for gravitational potential energy calculation, such as “the ground” in your example, simply because it makes intuitive sense. However, there’s no real issue with setting the reference lower or higher than that.
Digging a hole beneath a boulder does not change the potential energy of the boulder, it only changes what altitude you’re assuming as a zero-point reference. And it’s not at all necessary to change that reference point, just convenient.
Quoth Pasta:
I’ll have to think about this some more. And that’s all I’ll say on the topic for the moment, lest I mis-speak.
Either your teacher was wrong, or you misunderstood him/her.
Potential energy is not changed because you dig hole next to, or even under a boulder.
Relative to a given reference elevation, the boulder has the same potential energy regardless of the presence, or absence, of a hole.
I think we could eliminate that solution more concretely by having one room at the top of a hill in Florida and the other at the bottom of a hill in Tibet. The one in Florida is closer to the center of the earth and thus has more mass, but if we consider the systems to be (boulder + room + hill), it also has more potential energy than the one in Tibet.
And if you start to move your measuring equipment towards the boulder, you now see that it has kinetic energy even though the boulder hasn’t changed. The way that I see it, energy is an abstraction that helps with bookkeeping but you have to get your frames of reference correct.
I’ve said for a long time that conservation of energy ideas don’t work to explain physics to a layman. In order to use COE, you first have to have been convinced that this abstract calculation really always is conserved, but if you haven’t done that part, as most laymen haven’t, you need to explain stuff in terms of masses and velocities. I think that helps you understand it better yourself, because the COE is just a shortcut.
** Indistinguishable’s **explanation in post #34 is spot on. I just want to stress that this conservation of energy applies only within a closed system, say within the OP’s houses.
I too struggled with this concept through high school and college physics and some teacher’s insistence that some magical “potential” energy is actually stored in objects didn’t help. If only objects would live up to their potential, we could solve the energy crisis!
If it makes it easier on your brain, consider that potential energy doesn’t just apply to rocks on hills. A bottle of nitroglycerin has oodles of potential *chemical *energy, for example. Nitro is an unstable compound producing a very expansive & exothermic reaction that can be triggered by shaking the liquid. The explosion doesn’t come out of thin air, does it ? Yet is there a Asplod-o-meter that can find the boom in liquid nitro ? Nope. By all measurements you can conceive, that nitro is just an inert liquid - until someone drops it.
But considerable energy was used in creating the nitro, is “stored” in it (in the form of chemical bonds), and is released all in one fell swoop. Preferably nowhere near me. Just as, in the real world, energy is expended to put a rock on top of a hill.
Potential energy is energy which can be extracted by dropping a mass in a gravitational field a certain distance and you cannot extract any if you cannot drop it or if there is no gravitational field and you cannot know if it exists without knowing if it can be dropped.
A rich man is just a poor man with money. A man is not rich or poor independently of his money. Similarly, a boulder does not have or lack potential energy independently of outside factors so asking how can it be determined without taking those outside factors into account is like asking about the notorious one hand clapping.
Potential energy is not an intrinsic property of the rock but rather it is the property of the system comprised by (1) mass, (2) gravitation field and (3) distance. Energy (or work, same thing) is force over distance. (1) and (2) determine the force, take that, multiply (or integrate as needed) by (3) and you have the energy in the system.
Mass by itself has no potential energy and it makes no sense to ask about it. Mass is just mass, a kiss is just a kiss, a sigh is just a sigh but the fundamental thruths still apply. Here and in Casablanca.
Ftg you need to think about this a little more, this is a pretty important concept. Mass and energy are not things, they’re properties of a system.
For instance, a system comprised of a rock and a photon has more mass than that of the rock even though the mass of the two items taken individually don’t change.
Or a system of photons, that has a zero momentum frame, has mass even though each individual photon has zero mass.
This all stems from Einstein’s relativistic equation that relates energy, mass and momentum:
E[sup]2[/sup] = m[sup]2[/sup]c[sup]4[/sup] + p[sup]2[/sup]c[sup]2[/sup]
Or setting c = 1
m[sup]2[/sup] = E[sup]2[/sup] - p[sup]2[/sup]
Ring: I think you are misreading my post. (Or perhaps intended to reply to someone else entirely.)
Reread the OP. No photons. Two boulders, one of which is at a different energy potential than the other. The OP wants to know how they are different.
Potential energy, in any medium (batteries, springs, whatever), changes the mass of the system and is a fact of Physics. Energy is always conserved, even if it is relabeled as mass.
Exactly. Potential energy just means it is energy which can be extracted. A raised weight, an electron in an electro-magnetic field, a compressed spring, a car battery, an explosive, all have potential energy.
A tense bow, ready to shoot an arrow, has potential energy. The energy is stored in the deformation of the bow when the archer pulls back and is extracted when the arrow is shot.
For some reason, it’s a lot easier for me to grasp potential energy in nitroglycerin’s chemical bonds or a bowstring’s tension than with my “two boulders” example.
I’d just like to say that I am soooooooooooooooooooo very glad that I am not the only person who harassed their teachers with questions like these. 
I think this is because it seems like there’s an obvious, non-arbitrary “zero point” for potential energy in those two cases. You think to yourself “Ah, yes, the potential energy is all the actual kinetic energy I could get flowing out of this, if I completely drained it; there’s some maximum value there”.
But this is a poor way to look at things; it makes it seem like there’s a well-defined, substantial notion of being at 0 potential energy, as though you can drain potential energy till you reach that, but go no further.
But, remember, the thing about potential energy is that it’s not directly defined; only change in potential energy is directly defined (as the opposite of change in kinetic energy). Potential energy itself is only defined from that, and thus only makes sense up to an additive constant. It’s like “height”. There’s no such thing as absolute height; just change in height, just height relative to some arbitrarily chosen zero level. It’s perfectly cromulent to view a situation as having negative potential energy; it just means you’ve arbitrarily chosen to set your zero point somewhere else.
Not to toot my own horn too much, but have you tried reading my post #24? I think it sets the matter out in a pretty straightforward way, directly addressing your original concern without bringing in red herrings.
It may help to consider the potential energy in a chemical bond, which comes from the donation or sharing of one or more electrons between two atoms (and combinations thereof). This energy is stored not in the electrons or nuclei (whose masses are invariant) but in the valence bonds between nuclei in the same way that gravitational potential energy is stored not in either of the attractive masses but in the gravitational bond between two masses. This energy is a property of the system, not its components in isolation.
Stranger
Quoth Pasta, again:
OK, I’ve thought about this some more, and the place where I’m getting stuck is the infinite sheet of mass. If General Relativity is in effect (as it is, if we’re talking about gravitational potential energy itself gravitating), then we can’t construct an infinite sheet of mass in the first place, not even in a limiting sense. That is to say, if you take a sequence of mass distributions which approaches an infinite sheet of mass, such that the surface gravity is the same for all of them, then you’ll eventually come to a mass distribution somewhere in the sequence which collapses into a black hole.