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

[Cyrus]

If 1 kg of ice at 0C melts into water at 0C... the water gains mass.

The molecules in the ice or the fields between the molecules (I'm not sure which) weigh more in the water state. But total rest mass increases and this increase is due to the increase in potential energy.

I don't think that's true learningphysics. E=K+mc^2. Melting the ice increases the internal kinetic energy of the water molecules. So K goes up to compensate for the increased energy and not the mass. no?

 

[pmb_phy]

I don't think that's true learningphysics. E=K+mc^2. Melting the ice increases the internal kinetic energy of the water molecules. So K goes up to compensate for the increased energy and not the mass. no?

learningphysics is correct. The relation you posted, E=K+mc^2, is the inertial energy of a single point particle which is moving. He is not speaking of such a particle. He is speaking of water. One can think of the water as balls connected by springs. The balls are the H and O atoms and the springs represent the mutual electric force between the balls. All the balls are moving and the springs are compressing and expanding. There is a total inertial energy for each ball and a potential energy for the springs (the potential energy well is not symmetric as I recall). An increase in energy will then cause an increast in the total energy (kinetic energy of all balls + potential energy of all springs).

Simply put - The ice must absorb energy for it to melt. Any change in energy of the ice must result in an increase in the mass, and hence weight, of the H₂O.

Pete

 

[pmb_phy]

For instance, when a charge A moves away from an opposite charge B, the rest mass of A remains unchanged, likewise for B. However the mass of the system as a whole, A+B, increases due to an increase in potential energy.

The mass of the system as a whole remains unchanged. The magnitude of the potential energy decreases with seperation. Whether potential is negative or positive and whether the totak kinetic increases or decreases depends on the signs of the charges. The sum of kinetic and potential energy remains the same. This, of course, neglects the radiation due to acceleration of the charges.

What has not yet been addressed is the mechanism of the increase due to potential energy. Nor has the mechanism due to the increase in weight been addressed. I wrote a paper on all of this and will post it on my web page tommorow. It will explain these mechanisms.

Pete

 

[learningphysics]

learningphysics is correct. The relation you posted, E=K+mc^2, is the inertial energy of a single point particle which is moving. He is not speaking of such a particle. He is speaking of water. One can think of the water as balls connected by springs. The balls are the H and O atoms and the springs represent the mutual electric force between the balls. All the balls are moving and the springs are compressing and expanding. There is a total inertial energy for each ball and a potential energy for the springs (the potential energy well is not symmetric as I recall). An increase in energy will then cause an increast in the total energy (kinetic energy of all balls + potential energy of all springs).

Simply put - The ice must absorb energy for it to melt. Any change in energy of the ice must result in an increase in the mass, and hence weight, of the H₂O.

Pete

But rest mass only increases with an increase in potential energy right (not KE)?

 

[learningphysics]

The mass of the system as a whole remains unchanged.

But this is relativistic mass that you're referring to right? Doesn't rest mass change with potential energy?

 

[learningphysics]

I don't think that's true learningphysics. E=K+mc^2. Melting the ice increases the internal kinetic energy of the water molecules. So K goes up to compensate for the increased energy and not the mass. no?

Both the water and the ice are at 0C... so the kinetic energy is the same. Water at 0C has more potential energy than ice at 0C. When you heat up ice the temperature goes up until you get to the melting point 0C... then all the energy goes into potential energy which leads to the melting of the ice.

Increased potential energy, I believe means increased rest mass.

 

[learningphysics]

And then I read to the bottom: "As a result, we might expect the rest mass of a spaceship to be slightly larger after it leaves the Earth than it was on Earth, simply because it has left the "gravity well'' of the Earth. This is the case! However, the mass change is imperceptibly small in this case. " D'oh! Now I go back to confusion again. Woe is me!

Yes... I really don't know where exactly the increased rest mass resides.

 

[pmb_phy]

But rest mass only increases with an increase in potential energy right (not KE)?

No.

But this is relativistic mass that you're referring to right?

It makes no difference.

Doesn't rest mass change with potential energy?

No.

The above assumes that "rest mass" referres to the mass of a closed system as measured in the zero momentum frame.

The following may help (what you call rest mass others call "invariant mass". The usage is not really consistent in the literature.)
http://www.geocities.com/physics_world/sr/invariant_mass.htm

This is a perfect worked example of what this thread is about
http://www.geocities.com/physics_world/sr/nuclear_energy.htm

Pete

 

[El Hombre Invisible]

The mass of the system as a whole remains unchanged. The magnitude of the potential energy decreases with seperation. Whether potential is negative or positive and whether the totak kinetic increases or decreases depends on the signs of the charges. The sum of kinetic and potential energy remains the same. This, of course, neglects the radiation due to acceleration of the charges.

What has not yet been addressed is the mechanism of the increase due to potential energy. Nor has the mechanism due to the increase in weight been addressed. I wrote a paper on all of this and will post it on my web page tommorow. It will explain these mechanisms.

Pete

Hi Pete. First off, if the system is closed then the relativistic mass of the radiation is part of that system and so would be counted towards the relativistic mass of the system as a whole, so it should not be neglected.

In my example, I specified two opposite charges - let's say a proton and an electron. As the electron moves away from the proton, the strength of the field, or the intensity of the photons interacting with each, should increase and the kinetic energy of the moving charge will decrease. This will lead to a decrease in the relativistic mass of the electron compensated for by the increase in the energy of the field, no? If this equal potential energy were stored within the electron itself, rather than in the field, there would be no observable decrease in its relativistic mass, as ΔEpot - ΔEtrans would be 0, and so E would be constant. If you then open this up to all interactions the electron takes part in, then in any frame of reference the relativistic mass of the electron would remain constant, as any change in position would be compensated for my a corresponding change of potential energy, and the relativistic mass would no longer be effected by velocity, which would seem inconsistent with SR. It would also appear to defy the laws of conservation of energy, as if the relativistic mass of both the proton and the electron remained constant during separation, and the intensity of the photons in between increased, where does this increase of energy come from?

 

[pmb_phy]

Hi Pete. First off, if the system is closed then the relativistic mass of the radiation is part of that system and so would be counted towards the relativistic mass of the system as a whole, so it should not be neglected.

I neglected if for simplicity assuming that it is neglegible compared to the kinetic and potential energy. Its simple to add it in but it was easier to speak of. Not to mention it seemed what you were doing since you also never mentioned the radiation in your example and it was your example I was addressing.

In my example, I specified two opposite charges - let's say a proton and an electron. As the electron moves away from the proton, the strength of the field, or the intensity of the photons interacting with each, should increase and the kinetic energy of the moving charge will decrease. This will lead to a decrease in the relativistic mass of the electron compensated for by the increase in the energy of the field, no?

As the distance between the charges increases to infinity the total potential energy of the system increases towards zero and the kinetic energy decreases toward zero. The total energy of the system remains constant. Therefore the total mass of the system remains constant. It'll be easier for you to see all this by the principle of the conservation of energy. Since E = mc^2 then since E = constant then so does m. However when you actually calculate this it is the potential energy which comes into the energy calculation.

If this equal potential energy were stored within the electron itself, ...

No. That is not true. The mass is in the system itself and is not "stored" in one of the particles.

It would also appear to defy the laws of conservation of energy, as if the relativistic mass of both the proton and the electron remained constant during separation, and the intensity of the photons in between increased, where does this increase of energy come from?

You've added in a new complication when you started to take into account the radiation. This is a very tricky question and I believe that the answer is related to the self force acting on each charge so you no longer have just the Coulomb force acting (and the associated potential energy). I'm only moderately familiar with the physics so I'll be quite on this point for now (stuff this complicated I forget a week after I figure it out/learn it! ). I believe I can dig that up somewhere though. If this is related to the problem called "mass renormalization" then that is really really really tricky and requires some really advanced stuff to give an answer to. That seems to go far beyond what you're looking for since you don't care about that - you care about potential energy and its relationship to mass. Recall Einstein's first derivation - A body can absorb radiation and when it does the mass increases. What happens inside the body is the electromagnetic energy is changed to internal potential energy. This then is the energy associated with the increase in mass.

If you consider two like charges at rest - find the mass - then move the charges closer together and then let them be at rest once again then the mass of that system increases. The mechanism which causes the increase in mass can seen by weighing the system. Each charge exerts a force on the other charge and the force has a negative component, i.e. in the direction of the g-field. This added force gives an added weight.

Here's a tricky one for you - Consider a point charge. What is the mass? You'll have to address the mass of the field and the intrinsic mass of the particle itself. Note that the mass of the field is infinite.

If you're truly interested in find and answer to your questions then try to imagine an experiment which will measure the mass you're speaking of.

Pete

Note; Speaking about the field itself as if it had a mass is very tricky and will mess you up big time. The mass associated with an EM field of a chared particle will not transform in the same way as the mass of a particle.

 

[learningphysics]

No.
It makes no difference.
No.

The above assumes that "rest mass" referres to the mass of a closed system as measured in the zero momentum frame.

The following may help (what you call rest mass others call "invariant mass". The usage is not really consistent in the literature.)
http://www.geocities.com/physics_world/sr/invariant_mass.htm

This is a perfect worked example of what this thread is about
http://www.geocities.com/physics_world/sr/nuclear_energy.htm

Pete

Thanks for your help Pete. Your website is great.

It seems like I never really "get" special relativity. Every time I think I've understood, there seems to be something I've missed.

What I've learned now... Invariant mass of a system is not necessarily the sum of the invariant masses of all the particles within the system.

Energy of a system is equal to the sum of the energies of the constituent particles... Momentum of a system is equal to the sum of the momentums of the constituent particles... is this right?

Is this right: If energy is added to a system... and the total momentum of the system remains the same (in whatever frame you're using), then the invariant mass of the system increases... whether or not the energy went into increasing the kinetic energy of the constituent particles... or the potential energy of the constituent particles.

What is still bothering me is the weight mechanism (the force of gravity)... I'm unsure as to why the "weight" of a system would be dependent of the invariant mass of the system (as opposed to the sum of the invariant masses of the constituent particles)... It was no problem before when I thought that invariant mass was additive... but it isn't clear now. Can you elaborate? Thanks.

 

[pervect]

I think that you need some familiarity with GR and the stress-energy tensor to really understand the weight issue (or gravity itself, for that matter - you need Einstien's field equations before you can start dealing with gravity seriously). Steve Carlip, for instance, has a paper on the issue of gravitational mass of a system (which is trickier than it first appears BTW), but it definitely takes more than simple SR to understand the paper. The abstract, though, at least clearly states that kinetic energy does contribute to the gravitational mass of a system of particles - so you may have to just take this on faith a bit, unless and until you get the needed background.

Kinetic Energy
and the Equivalence Principle
S. Carlip∗
Department of Physics
University of California
Davis, CA 95616
USA
Abstract

According to the general theory of relativity, kinetic energy contributes
to gravitational mass. Surprisingly, the observational evidence for this
prediction does not seem to be discussed in the literature. I reanalyze
existing experimental data to test the equivalence principle for the
kinetic energy of atomic electrons, and show that fairly strong limits
on possible violations can be obtained. I discuss the relationship
of this result to the occasional claim that “light falls with twice the
acceleration of ordinary matter.”

 

[pmb_phy]

I think that you need some familiarity with GR and the stress-energy tensor to really understand the weight issue (or gravity itself, for that matter - you need Einstien's field equations before you can start dealing with gravity seriously).

"really" understand? If one understands it without GR does that mean neccesarily that he "really" doesn't understand it? The first derivation I ever did to determine whether a moving body weighed more I did before I learned GR to any great extent. All one requires is the equivalence principle. From that one can extrapolate to a more complex object such as a 2-d gas in a 2-d box. If one recalls Einstein's 1905 SR paper then in it Einstein states that to measure the transverse mass of a particle one can use a balance. Since the transverse mass = relativistic mass one obtains the weight of the particle. Its always best to try to figure these things out before one puts pen to paper. I've rarely made any progress in anything I've ever done by starting with the math.

Pete

 

[pmb_phy]

Thanks for your help Pete. Your website is great.

Thank you.

It seems like I never really "get" special relativity. Every time I think I've understood, there seems to be something I've missed.

Join the club.

What I've learned now... Invariant mass of a system is not necessarily the sum of the invariant masses of all the particles within the system.

That is correct.

Energy of a system is equal to the sum of the energies of the constituent particles... Momentum of a system is equal to the sum of the momentums of the constituent particles... is this right?

Sure. As long as you're talking about non-interacting particles. You don't even need to employ energy conservation. This fact can be derived without ever employing the principle of the conservation of energy. Using the theory of the conservation of mass makes it trivial.

Is this right: If energy is added to a system... and the total momentum of the system remains the same (in whatever frame you're using), then the invariant mass of the system increases...

Brigtht boy!

For a derivation of what you just stated see
http://www.geocities.com/physics_world/sr/mass_energy_equiv.htm

What is still bothering me is the weight mechanism (the force of gravity)... I'm unsure as to why the "weight" of a system would be dependent of the invariant mass of the system (as opposed to the sum of the invariant masses of the constituent particles)... It was no problem before when I thought that invariant mass was additive... but it isn't clear now. Can you elaborate? Thanks.

See -
http://www.geocities.com/physics_world/sr/weight_moving_sr.htm
http://www.geocities.com/physics_world/gr/weight_move.htm

Pete

 

[El Hombre Invisible]

Pete - I now realize that it made no sense to address my post to you. We seem to be in complete agreement, bar the point about the rest mass of the system where I temporarily went insane - you are of course 100% correct.

 

[pmb_phy]

Pete - I now realize that it made no sense to address my post to you. We seem to be in complete agreement, bar the point about the rest mass of the system where I temporarily went insane - you are of course 100% correct.

I will remind you of this when it becomes my turn and my head explodes and messes up the place.

Pete

 

[pmb_phy]

What is still bothering me is the weight mechanism (the force of gravity)... I'm unsure as to why the "weight" of a system would be dependent of the invariant mass of the system (as opposed to the sum of the invariant masses of the constituent particles)... It was no problem before when I thought that invariant mass was additive... but it isn't clear now. Can you elaborate? Thanks.

I too thought about this. I knew that E = mc^2. But what I wanted to know was Why? People rearely ask themselves this question since they seem to assume that there can be no answer to "Why?" questions. I'm not one of those people. Please see

On the concept of mass in relativity, Peter M. Brown

I placed this online today at
http://www.geocities.com/physics_world/mass_paper.pdf

See page 52 section - XI Why does E = mc²

I addressed the mechanis for 3 different kinkds of energy as I recall. This should be the best answer you're going to get on the interent.

Pete

 

[learningphysics]

I too thought about this. I knew that E = mc^2. But what I wanted to know was Why? People rearely ask themselves this question since they seem to assume that there can be no answer to "Why?" questions. I'm not one of those people. Please see

On the concept of mass in relativity, Peter M. Brown

I placed this online today at
http://www.geocities.com/physics_world/mass_paper.pdf

See page 52 section - XI Why does E = mc²

I addressed the mechanis for 3 different kinkds of energy as I recall. This should be the best answer you're going to get on the interent.

Pete

Hi Peter. The paper was very useful. I'm wondering... mass in a gravitational field... If we have two masses in a gravitational field at rest... then we take the two masses further apart and keep them at rest. The way I understand it, the total mass has increased now. Is the mechanism involved in increasing the mass analogous to that of the electrical force? What about the other fundamental forces?

The mechanism that creates mass associated with electrostatic potential energy... it almost seems like the mass of E/c^2 comes out coincidentally. On page 55 you write "Utilizing electrodynamics Boyer has shown that... Comparison with proper energy gives us the expected result"... Could the result have been otherwise?


Is there any book that you can recommend on Special Relativity that will help give a good solid foundation (hopefully give enough to prepare you to learn about GR) ? More rigorous the better.