# Potential Energy Explained: Does It Contradict Conservation of Mass/Energy?

• michael879
In summary, the conversation discusses the concept of gravitational potential energy and how it relates to the conservation of mass/energy. The speaker questions whether this theory is made up, but it is explained that potential energy exists due to the presence of a gravitational field. The conversation also touches on the relationship between mass and energy and how potential energy comes from the big bang. There is also a discussion on the concept of kinetic energy and how it can be used to measure the difference in potential energy between two points. The conversation ends with a mention of entropy and the tendency of objects to be in a state of lowest potential energy.
michael879
I never really thought about this until the other day when someone asked me why something accelerating towards a planet doesn't make the planet lose mass. It makes sense according to conservation of mass/energy right? the object gains energy (speed) therefore the planet must lose energy or mass. I know that potential energy explains this energy gain, the object gains kinetic energy while losing potential. But doesn't this seem like some theory made up just to make conservation of energy work?

No, it exists. Gravitational potential energy is a property of an earth/ball system in which the ball is initially a distance r away from the centre of the Earth, for example. If r > R, the latter being the Earth's radius (which is a fancy way of saying, if the object is above the ground, lol), the ball will begin to fall. How much work was needed to get it there in the first place? You would have to do work on the ball against gravity to get it there. That is how much potential energy is stored in the system. The energy exists by virtue of the presence of the gravitational field.

We're dealing with gravitational potential energy, which I underlined. Conversion of mass to energy, on the other hand, occurs when nuclear forces binding nuclei together are overcome. That's a whole different ball game.

cepheid said:
No, it exists. Gravitational potential energy is a property of an earth/ball system in which the ball is initially a distance r away from the centre of the Earth, for example. If r > R, the latter being the Earth's radius (which is a fancy way of saying, if the object is above the ground, lol), the ball will begin to fall. How much work was needed to get it there in the first place? You would have to do work on the ball against gravity to get it there. That is how much potential energy is stored in the system. The energy exists by virtue of the presence of the gravitational field.

We're dealing with gravitational potential energy, which I underlined. Conversion of mass to energy, on the other hand, occurs when nuclear forces binding nuclei together are overcome. That's a whole different ball game.
well actually, I am pretty sure mass and energy are basically the same thing (E=mc^2), but any way I get what your saying. However, what if the ball was never put there. What if the ball has never been on earth, it started on the other side of the universe. What about when it starts to accelerate towards earth. why would it have potential energy in this case?

michael879 said:
I never really thought about this until the other day when someone asked me why something accelerating towards a planet doesn't make the planet lose mass. It makes sense according to conservation of mass/energy right? the object gains energy (speed) therefore the planet must lose energy or mass. I know that potential energy explains this energy gain, the object gains kinetic energy while losing potential. But doesn't this seem like some theory made up just to make conservation of energy work?

This is more of a philosophical question, I believe. Pretty much any quantity is "made up" into order to make a theory work.

juvenal said:
This is more of a philosophical question, I believe. Pretty much any quantity is "made up" into order to make a theory work.
not rly, kinetic energy exists, we can feel it whenever something hits it. But never mind I answered my own question: the potential energy comes from the big bang, when the ball and the Earth were in 1 place.

michael879 said:
well actually, I am pretty sure mass and energy are basically the same thing (E=mc^2), but any way I get what your saying. However, what if the ball was never put there. What if the ball has never been on earth, it started on the other side of the universe. What about when it starts to accelerate towards earth. why would it have potential energy in this case?

One cannot measure an absolute potential energy. It's only useful to talk about the difference in potential energy between two points.

michael879 said:
not rly, kinetic energy exists, we can feel it whenever something hits it. But never mind I answered my own question: the potential energy comes from the big bang, when the ball and the Earth were in 1 place.

You feel something. In principle, I think one could formulate a theory of mechanics that never used the concept of kinetic energy.

juvenal said:
One cannot measure an absolute potential energy. It's only useful to talk about the difference in potential energy between two points.
Im not talking about absolute potential. I was talking about two points very far apart that were "never" together. However they were right before the big bang.

juvenal said:
You feel something. In principle, I think one could formulate a theory of mechanics that never used the concept of kinetic energy.
I misphrased that, the reason a baseball thrown at you hurts is because of energy. Also, our bodies live off energy which definitely exists. And how can you deny heat?

michael879 said:
well actually, I am pretty sure mass and energy are basically the same thing (E=mc^2), but any way I get what your saying. However, what if the ball was never put there. What if the ball has never been on earth, it started on the other side of the universe. What about when it starts to accelerate towards earth. why would it have potential energy in this case?

because it takes work/energy to place the ball in its location...

the entire universe is conducting through entropy and ethalpy... disorder and lowest potential state..

if you think about it, it makes sense... a ball has a tendency to be on the ground... so to lift it up 5 meters you need to apply a force, thus giving it potential energy

michael879 said:
Im not talking about absolute potential. I was talking about two points very far apart that were "never" together. However they were right before the big bang.

I misphrased that, the reason a baseball thrown at you hurts is because of energy. Also, our bodies live off energy which definitely exists. And how can you deny heat?

I'm not denying anything. Kinetic energy and heat are defined by a theory used to explain physical phenomena, i.e. classical mechanics and thermodynamics. There is no external meaning.

Hessam said:
because it takes work/energy to place the ball in its location...

the entire universe is conducting through entropy and ethalpy... disorder and lowest potential state..

if you think about it, it makes sense... a ball has a tendency to be on the ground... so to lift it up 5 meters you need to apply a force, thus giving it potential energy

ok what I was trying to say is that this ball wasn't lifted by anyone, it just popped into existence far away from earth. However it would still have potential energy right? so the whole work theory doesn't rly apply here.

michael879 said:
well actually, I am pretty sure mass and energy are basically the same thing (E=mc^2), but any way I get what your saying.

*Sigh*...yes, of course. But how do we harness that energy? The point I was trying to make is that we can only convert mass directly to energy in nuclear reactors. You need to blow up nuclei, or fuse particles together to form them. For example, if I remember right, two hydrogen nuclei and two neutrons when considered separately have a total mass greater than that of a helium nucleus, despite the fact that a helium nucleus consists of just those very same four particles, bound together! So where did the extra mass go? It was converted to energy in the process of fusion. That's what happens in the sun (well, to simplify things).

What I was saying is that plain simple everyday MOTION in a gravitational field is does not involve the conversion of mass to energy (and I'm not sure where you got the idea that it ought to). It involves the interchange of energy between only two forms: kinetic and potential. We generally call this total "energy of motion" (when gravitational fields are involved) mechanical energy. In a conservative field, in the absence of other forces (non conservative forces such as friction), mechanical energy is conserved. We say that gravity is a conservative force. The energy is not "lost" (note the quotes), i.e. converted to other non - mechanical forms such as heat, light, etc.

michael879 said:
However, what if the ball was never put there. What if the ball has never been on earth, it started on the other side of the universe. What about when it starts to accelerate towards earth. why would it have potential energy in this case?

Consider an analogy: I want to assemble a system of electric charges. The system includes two unlike charges separated by a distance r. In order to separate them, I need to do work, because there is an attractive force trying to keep them together. So are you surprised that when I take away whatever is keeping them separated, they shoot back together? Do you ask, where did the energy come from? No, because you saw me put in a lot of work to painstakingly assemble this system. The mere existence of this system of two charges and their associated fields, separated by a distance, means that there is certain amount of energy stored in it, equal to that work. We usually associate this energy "with the electric field", because knowing the strength of the total electric field of this system of charges, we can calculate the electrostatic energy. What if I want to move my charges really far apart? Your first impulse would be to say that I'd have to do a hell of a lot of work to separate them this huge distance. But remember that the attractive force between them diminishes, the farther away they get. So if the second charge starts out* really far away in the first place, is there some potential at that far away point (essentially, "at infinity") due to the field of the first charge? Yes. Is it small enough to be considered essentially zero? Yes. Will the second charge accelerate towards the first if they are separated by this vast distance? No. But the potential due to the Earth's/source charge's field at intermediate points between the Earth and the ball is higher, so if it somehow ends up at those intermediate points (closer to the earth), its potential energy will increase.

*I know what you're going to ask about, that's why I starred it. How might the two charges "start out" that far apart in the first place? Well, consider what you are asking me. We were talking about assembling a system of masses (or charges). Then what did you do? You went and extended that "system" to include...

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...THE ENTIRE KNOWN UNIVERSE

And now you are asking me: Where did the energy come from to "assemble" this system? I.e., why does matter exist, and why are objects in their current locations/how did they get there/what is their entire history since the beginning of time? Sorry, not qualified to answer that. If you're satisfied with "big bang", then fine.

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Im just responding to the first part of ur post. I get that we can't harness the energy by converting mass to energy. But whenever something loses energy it loses mass since they are the same thing. This wasnt rly part of my question but I was saying that (and I know this is wrong) if a ball gained energy from a planet as it accelerated towards it, it would also gain mass while the planet loses mass. The mass would be negligible since Mlost = E/c^2 however, there would be a mass loss whenever energy is lost.

michael879 said:
Im just responding to the first part of ur post. I get that we can't harness the energy by converting mass to energy. But whenever something loses energy it loses mass since they are the same thing.

Not true. You have to be careful how you interpret E = mc^2. Mass and energy are not the same thing. This equation simply states that it is possible to get one from the other, i.e. that they are interchangeable. Possible under the circumstances I outlined.

exactly, so that when there is an energy loss, there is also a much smaller loss of mass.

You're not really listening. That's exactly what I said is NOT true.

o I thought you were saying something else. are you sure that isn't true? because I am pretty sure there is a way to get free energy if it isnt.

michael879 said:
ok what I was trying to say is that this ball wasn't lifted by anyone, it just popped into existence far away from earth. However it would still have potential energy right? so the whole work theory doesn't rly apply here.

well w/ hypothetical situations that are not possible, a lot of stuff can occur that can make your head spin

what I'm saying is plain and simple... gravity is a force... it pulls on the ball... so, for the ball to stay stationary it would require another force of equal magnitude

because this force is necessary to stay stationary there is some form of energy on the ball, which we label as "potential energy"

the force theory however does apply, because you do need a force to keep it stationary

so you ask yourself... why is this ball spontaneously falling towards a planet?

spontaniety of a process is caused eitehr by enthalpy or entropy

in this case enthalpy, reaching lower potential... the ball wishes to reach lower potential, thus its gravitational potential energy is converted into kinetic energy and is launched towards the earth

michael879 said:
potential energy = BS?

My copy of Halliday and Resnick does not have this as a boxed eqn, so my guess is the answer is no.

haha ok I give up, I got my answer thx

michael879 said:
o I thought you were saying something else. are you sure that isn't true? because I am pretty sure there is a way to get free energy if it isnt.

as far as i know (feel free to correct me)

we only use E=MC^2 to represent the binding energy that is holding certain atomic particles together

you're acting as if though we can use this equation whenever and w/ whatever we wish to.. which is of course incorrect

Hessam said:
as far as i know (feel free to correct me)

we only use E=MC^2 to represent the binding energy that is holding certain atomic particles together

you're acting as if though we can use this equation whenever and w/ whatever we wish to.. which is of course incorrect
I know the only time it is rly used is during nuclear reactions. However, I am pretty sure it applies everywhere, but the effect is so small its ignored. I am tired and can't think of an example right now but I am almost positive if you can take energy from something without taking any mass from it, you will have free energy.

It sounds like the usual mix-up between relative and rest masses to me. When an object loses energy, say kinetic energy, it loses relative mass, but all relative mass is, really, is the total energy of that object including mass energy. It does not lose rest mass, but then rest mass only describes the object when it is at rest, so is of little use when talking about losing or gaining energy outside the abstraction of classical mechanics. It seems favorable to mean 'rest mass' when you say mass, and 'relative mass' when you say energy, so no - an object does not lose mass as it loses energy other than in the case of binding energy. Having said that, on a fundamental level the idea of rest mass seems to become more absurd as the lack of reasonable frames of reference and the 0th law of TD are hard to get around. The rest mass of an electron, for instance, seems something of a non-concept. A photon even more. Down at this level, where only relative mass matters, yes - a particle loses mass as it loses energy, but we're really talking about something different when we say mass.

I dig the question about potential energy, but it seems to assume the energy is within the particle itself (i.e. part of the relative mass). Bit of a beginner to QED, and know nothing about QFT, but I'd kind of figured this was a redundant concept. If I'm wrong, can anyone tell me in layman's terms where potential energy resides in quantum theory? Being old-fashioned I'm still a subscriber of the notion that potential energy exists in the space between objects, not in them, but then no-one's ever given me any other physical description of potential energy to consider.

El Hombre:

You know more than I do, but I thought the idea of "relativistic mass" was not a particularly useful concept. In any case, I realize that as far as the relativity aspect went, I was abrupt with the OP and should have explained things better, but I'm shaky with the stuff myself. I'd best be keeping my mouth shut. In any case, I do know that we have rest energy ("mass energy" as you called it) given by:

$$E_0 = m_0 c^2$$

and as I understand it, rest energy + kinetic energy (for an object in motion) is given by:

$$E = \frac{m_0 c^2}{\sqrt{1 - \frac{v^2}{c^2}}}$$

What I'm not clear on are the nuances/interpretations of this result. For instance, I understand that this formula for "relativistic energy" clearly shows that there is an upper limit to the velocity of an object having mass, that it can never attain c. If nothing else, infinite energy would be required. But is it really correct to say that the object's "mass" in some sense, increases as it speeds up? As you pointed out, you have to clear on what you mean by the term "mass". Rest mass does not change, But I guess "relativistic mass" does, if you want to call it that. So, does that increase affect the inertial properties of the body? I.e. can we actually say that the object cannot be accelerated to "c" because it tends towards infinite mass? I think that is somehow misleading or maybe just plain incorrect. But I don't really know.

Why not? It's a reasonable physical reason for a speed limit - the more energy you absorb the more inertia you have, meaning more energy is required to accelerate by the same amount, meaning more inertia, so on and so forth. In the picture of the whole thing in your head, though, you have to make sure you're using the right frames of reference. A man accelerated near to c (and somehow surviving) observes no mass increase in himself as in his frame of reference he is at rest. Only in someone else's frame of reference in which he is traveling at such speeds would he appear to gain inertia, and gravity (more appropriate in the case of photons). And at that speed the interactions between body and observer and going to be slight.

To be honest, it's like telling a boy with no legs he's not allowed to run down the corridor. How would you speed something up to c if you could? Fire photons at it? In your frame of reference, as the particle gets faster and faster, the photons take longer and longer to catch up with it. In the particle's frame of reference, the Doppler effect would reduce the energy of the photons it absorbs down to practically nothing. Either way, regardless of inertia, that last push is not feasible.

"I'd best be keeping my mouth shut." No way, man! 'Twas a good question. Potential energy is something that seems to be described in different ways that don't stick together. Is it abstract or real? Is it within the body, as classical mechanics infers, within the space between, as field theories infer, or is it an illusion created by something else? I don't know, someone else probably would, but I wouldn't assume potential energy is internal and so a contributor to relative mass is all I'm saying.

Hessam said:
as far as i know (feel free to correct me)

we only use E=MC^2 to represent the binding energy that is holding certain atomic particles together

you're acting as if though we can use this equation whenever and w/ whatever we wish to.. which is of course incorrect

ok hessam, E=MC^2 is a universal equation, u can use it everywhere!, it applys in every situation, it says that mass equals energy, energy and mass are the same thing, and we don't use E=MC^2 to represent the binding energy that is holding atomic particles together, that is the strong nuclear force, we use E=MC^2 to represent that when we break that bond and lose mass we get a massive amount of energy. we get this energy from the lost mass because mass is energy,and a lot of it.

I think $$E=mc^{2}$$ is a dumb and uninformative version of a truly revolutionary equation:
$$E^{2}=c^{2}p^{2}+E_{0}^{2}$$

michael879 said:
I know the only time it is rly used is during nuclear reactions. However, I am pretty sure it applies everywhere, but the effect is so small its ignored. I am tired and can't think of an example right now but I am almost positive if you can take energy from something without taking any mass from it, you will have free energy.
Your question is a good one. I'll attempt an answer for you.

If you take a distant mass that has 0 gravitational potential energy and 0 kinetic energy with respect to the Earth or sun, its energy is its relativistic mass x c^2 relative to the Earth frame (at this point its potential energy is 0). So:

(1) $$E + U(r) = mc^2 + 0 = \sqrt{(m_0c^2)^2 + (cp)^2} = m_0c^2 = E_0$$ since p = 0

Since energy of the object-earth system cannot be created nor destroyed, the object loses potential energy (or gains negative potential energy) as it falls toward the Earth in an amount equal to its gain in relativistic energy:

(2) $$E + U(r) = mc^2 + GMm(-\frac{1}{r} - -\frac{1}{\infty})= \sqrt{(m_0c^2)^2 + (cp)^2} - \frac{GMm}{r}$$

But, since $E - m_0c^2 = KE = \frac{GMm}{r}$, (2) becomes

$$E + U(r) = \sqrt{(m_0c^2)^2 + (cp)^2} - \frac{GMm}{r} = m_0c^2 = E_0$$

So, energy is conserved.

Now you might conclude that the potential energy is just made up - as a sort of bookkeeping tool to keep the energy books balanced. Ok. But it is made up for a reason: If we abandoned the concept of potential energy, we would still have to keep track of it. We would have to keep track of how much energy is being created or destroyed because the amount being 'destroyed' (as the object moves further away from the earth) is exactly the amount that can be 'recreated' later as kinetic energy.

Is potential energy real? That question is equivalent to another question: does an object with rest mass m_0 with potential energy U = 0 have more mass (inertia) than the same object with potential energy U < 0?

This is very difficult to measure for gravity because gravity is so weak. But we know this is true for charged particles under the strong nuclear force. Two deuterons stuck together in a helium nucleus (and, therefore, with a very large negative binding potential) have less rest mass than two separated deuterons (He mass = 4.00151 u;
Deuteron mass = 2.013553 u). So, it appears that potential energy of unbound deuterons adds to their inertia. We believe that gravitational potential adds to an object's inertia as well.

AM

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So does potential energy add to the mass of a sole fundamental particle? Or only to the system in which those interactions occur? To put it another way, if you could weigh the moon where it is now, would its GPE be weighed or would you have to weigh the Earth and moon together to get it?

This is the kind of question that usually no-one on this forum ever seems to agree on (is it an electron absorb photons, or an atom, etc).

It only contributes to the rest mass of the whole system. That is, the planet and the mass it's interacting with. The planet and particle themselves do not get more massive individually.

Bang on! And this extends to EM, right? The binding energy in an atom could not be hypothetically weighed in the individual electrons and nucleons..?

Nope, I don't believe it could, because you'd have to separate the parts to weigh them, and then the binding energy would have been released.

I'm not talking practically here. I mean like a really little man in an atom with a magic weighing device to weigh things without moving them. And one of them Heisenberg Compensators from Star Trek too so he can pin the basts down. THEN would the mass of the separate parts include the EM potential?

I'm almost certain the answer is still no. Hopefully someone will correct me if not. The mass of the individual parts that make up the atom will not include the potential energy of the field that holds the nucleus together, or the coulomb potential that holds the electrons to the atom. However, if those individual parts are made up of still smaller parts, like the quarks that make up the nucleons, then the potential energy that binds those quarks together does figure into the mass of the nucleon.

Bang on again. It's always good when someone cleverer than myself tells me something that I tell myself I always knew really. So potential energy really is the energy stored in the field acting between the parts, or in QT the bosons knocking around at the time... energy the parts WILL get, just not yet..?

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