# Conceptual Question about the Change in Mass

raymondtco
My instructor continuously insists on the idea that I don't understand/believe. Therefore I hope the people here could verify/discuss with me.

Regarding the conservation of mass-energy, he took an example of the head-on-head collision of two identical clay lumps with same speed. What he asserted was that all the kinetic energy of two lumps will be converted into mass, in which the same element will be produced. If two lumps are not identical, then different elements will be produced according to the the ratio of the ingredients of the two lumps.

Also, same idea applies to the work done on a spring. The energy (work) is converted to the increase in mass of the spring. In addition, the increase in potential energy is just the increase in the mass of the object. Conclusively, there is no other forms of energy besides mass and kinectic energy. (He rahter insists that the kinetic energy is not the energy related to the increase in mass.)

Q:
Should all these be true, then what is the meaning of the rest mass of any elementary particle if it all depends on all sorts of the energy associated to those particles? (Gravitational potential energy, for example)
And, does it imply the mass is not intrinsic property of a particle?
That is to say, electron will be heavier after a collision?
When mass is produced, what if the expected delta mass cannot be divided by the mass of the expected elemetary particle?

Can anyone clarify a little bit for me? I somehow cannot understand how nature knows what mass to produce and according to what ratio. Why we all force ourselves to believe that all kinds of energy are nothing but increase in mass without any mechanism or rationale?

Thanks a lot for the time spent on my question in advance!
I'm really stuck at this point.

Raymond

## Answers and Replies

Mentor
An object such as a lump of clay or a spring is not an elementary particle, but a collection of particles which are bound together as a system. The mass of the object (the system of particles) includes the mass-equivalent of the kinetic energies of the particles as they move about inside the object, and the potential energies of the forces that bind the particles together, as well as the (rest-)masses of the individual particles.

Buckleymanor
Would have thought that most of the energy would be converted into heat after a collision.
Same with a spring when released.
When the spring is compressed though it would have more mass.

Dadface
The collision is inelastic ie there is a loss of ke and this energy will be used to distort the objects generate heat etc..I don't think fundamental particles can undergo inelastic collisions but I will search my memory with the aid of my old pal Mr Google.

Mr Google, can I have a word please?Damm he's busy.

p.s. It was one of Einsteins favourite illustration to say that an object when hot is more massive than the same object when cold.

Science Advisor
Inelastic collisions of elementary particles are fairly common in accelerator physics. You create all sorts of things in such a collision - and certainly not only the "same element", as the OP's instructor seems to claim.

raymondtco
jtbell: According to my instrucor, thermal and potential energy inside the clay is nothing but the increase in mass. Therefore, it's back to the increase in mass again. Is that right?

Ich: Yeah, that's what I believe. I thought the energy would go into some other sorts of paticles and their kinetic energy. It couldn't always be the the same elements right? (Or maybe not the same at all.)

Everyone:
How about the potential energy and other sorts, does all the energy really go into the increase in mass??

ImAnEngineer
Regarding the conservation of mass-energy, he took an example of the head-on-head collision of two identical clay lumps with same speed. What he asserted was that all the kinetic energy of two lumps will be converted into mass, in which the same element will be produced. If two lumps are not identical, then different elements will be produced according to the the ratio of the ingredients of the two lumps.

I've read something similar to this in Feynman's lectures of physics part I. As I recall, it says something like this. Consider an inelastic collision of two equal masses. The mass of the final mass is bigger than the two separate masses before the collision. (In a relativistic case)

I never understood this and it wasn't really explained... So I would really like to have this question answered as well

Science Advisor
Gold Member
When we talk of the mass of a lump of clay, or anything else made up of lots of particles, what we are really asking is, what would be the mass of a point-particle which behaved the same way as our lump of clay when we apply the same force to both of them. The only sorts of energy a particle can have (in the absence of an external potential) are mass-energy (via $E = mc^2$) and kinetic energy (due to the motion of the particle relative the observer measuring the energy). Whereas a lump of clay can have additional internal energy to the motion of its internal particles, or internal chemical or electrical sources, etc. When we consider an equivalent point mass, all that internal energy has to be accounted for somehow, and the only way to account for it is to add it to the mass.

A slightly more sophisticated is as follows. Because of conservation of energy, we can define the energy of a lump of clay to be the sum of the energies of all its constituent particles. Because of conservation of momentum, we can define the momentum of a lump of clay to be the sum of the momenta of all its constituent particles.

Energy E and momentum p are related by the equation

$$E^2 = (mc^2)^2 + |\textbf{p}c|^2$$​

where m is mass. We can then use this to define mass of our lump of clay

$$M = c^{-2}\sqrt{(\Sigma E)^2 - |\Sigma\textbf{p}c|^2}$$​

which need not be the same as

$$\Sigma m = c^{-2}\Sigma\sqrt{E^2 - |\textbf{p}c|^2}$$​

Note: throughout the above answer, "mass" means "rest mass" (a.k.a. "invariant mass"), not "relativistic mass".

Dadface
Inelastic collisions of elementary particles are fairly common in accelerator physics. You create all sorts of things in such a collision - and certainly not only the "same element", as the OP's instructor seems to claim.

Good but can elementary particles eg electrons transmute?

Science Advisor
Gold Member
Good but can elementary particles eg electrons transmute?
If you collide an electron with a positron (the positively charged equivalent of an electron) they can "annihilate each other" and produce two photons. If that's what you mean by "transmute", then yes.

Dadface
If you collide an electron with a positron (the positively charged equivalent of an electron) they can "annihilate each other" and produce two photons. If that's what you mean by "transmute", then yes.

Thank you DrGreg but I was thinking more of electron/electron collisions or electron/other particle collisions.I thought that electrons are stable and fundamental and although there can be photon/electron conversions during pair production and annihilation can there be other conversions?

Homework Helper
I believe so - for example, in inverse beta decay an electron interacts with one of the up quarks in a proton to produce a neutrino and a down quark.

The fact that the electron is stable only means that it will not "fall apart" (decay) into other particles when left to itself, and the fact that it's fundamental means that (as far as anyone knows) it doesn't have constituents. (Compare to the proton, which is not fundamental because it's made of quarks and gluons) Neither of those prevents the electron from producing other particles when it collides with something else.

Dadface
Thank you diazona,I wasn't aware that you could get an inverse beta decay but thinking about it it does seem to be a possibility.

raymondtco
The only sorts of energy a particle can have (in the absence of an external potential) are mass-energy (via $E = mc^2$) and kinetic energy (due to the motion of the particle relative the observer measuring the energy).
...
Note: throughout the above answer, "mass" means "rest mass" (a.k.a. "invariant mass"), not "relativistic mass".

Thanks for you detailed explanation. However, your answer brings me some more questions.

Relativistic Kinetic Energy is Ek= Gmc2 - mc2
I think of Gm as the new mass (relativistic mass) and therefore, compared with rest mass the energy gained is just the energy associated with the increase in mass. It makes perfect sense to me since Relativistic Kinetic Energy is proven analytically. On the other hand, all other sorts of energy are not proven to be related to mass increase, but why do we force them to be converted to the change in mass without any analytical proof?

Alright, if two particles will increase their masses after a collision, we can still do no work and separate them back into two pieces and have them rest, right? Each particle's rest mass increases before and after collision.
Then, what do you mean by rest mass? Doesn't it depend on how many collisions it has ever experienced since each collision will change its mass? It seems like the rest mass of a particle is no longer is constant nor the intrinsic property of the particle. Is that true? If so, why do we have the rest mass for all the elementary particles?

You specifically explude potential energy when talking about a particle's energy. Are you saying that the potential energy shoulde be considered separately and therefor it would not appear in the change in mass?

Thanks a lot for you time! It truly helps me, as a physics major, understand the concepts.

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Mentor
Alright, if two particles will increase their masses after a collision, we can still do no work and separate them back into two pieces and have them rest, right? Each particle's rest mass increases before and after collision. Then, what do you mean by rest mass? Doesn't it depend on how many collisions it has ever experienced since each collision will change its mass? It seems like the rest mass of a particle is no longer is constant nor the intrinsic property of the particle. Is that true?

For a composite object like a wad of putty, that is true. The "rest mass" of a wad of putty depends on its temperature.

If so, why do we have the rest mass for all the elementary particles?

The rest mass of a truly elementary particle such as an electron cannot increase after a collision, because it has (so far as we know) no internal structure that can move around randomly and cause it to "warm up." In an inelastic collision between elementary particles, the incoming particles change into different kinds of particles which have different masses.

For "elementary particles" such as protons and neutrons which are combinations of quarks, the situation is more interesting. Because of the internal structure, you can take the same three quarks and put them into different "energy levels" which have different rest-masses and are generally considered to be "different particles."

Mentor
the kinetic energy of two lumps will be converted into mass, in which the same element will be produced. If two lumps are not identical, then different elements will be produced according to the the ratio of the ingredients of the two lumps.
I apologize for repeating a point if you already understand, but this portion of your teacher's comments is wrong. The combined lump of clay does have more mass than the sum of the masses individual lumps of clay, but that mass is in the form of internal energy like heat, not newly created massive particles. The mass is extra mass in the system and cannot be assigned to an individual particle.

Remember, in relativity mass is the Minkowski norm of the 4-momentum vector. The 4-momentum of the system is the sum of the 4-momenta of the individual components, and the length of a sum of vectors is not in general equal to the sum of the lengths of the vectors.

ImAnEngineer
I apologize for repeating a point if you already understand, but this portion of your teacher's comments is wrong. The combined lump of clay does have more mass than the sum of the masses individual lumps of clay, but that mass is in the form of internal energy like heat, not newly created massive particles. The mass is extra mass in the system and cannot be assigned to an individual particle.

Remember, in relativity mass is the Minkowski norm of the 4-momentum vector. The 4-momentum of the system is the sum of the 4-momenta of the individual components, and the length of a sum of vectors is not in general equal to the sum of the lengths of the vectors.
I would say you're making perfect sense. But what I've read tells me otherwise.
I quote from the Feynman lectures, they're basically saying the same thing as the OP:

(...) yet they [the two masses] contribute to the total M, not the mass they have when standing still, but more. Astonishing as that may seem, in order for the conservation of momentum to work when two objects come together, the mass that they form must be greater than the rest masses of the objects, even though the objects are at rest after the collision.

So, who is right ?? Any experts?

Buckleymanor
Well I am not going to bother with clay instead I intend to purchase two Gold bars.
Have them thrashed together for the next twelve months then weigh them in.
Somehow I don't think they will weigh more but there temperature might be higher.
PS.I am no expert.

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Science Advisor
So, who is right ??
Both are right. When Feynman speaks of "at rest after the collision", he certainly does not mean the elementary constituents are at rest, but their center of gravity is.

Staff Emeritus
Science Advisor
Gold Member
I would say you're making perfect sense. But what I've read tells me otherwise.
I quote from the Feynman lectures, they're basically saying the same thing as the OP
Could you show me explicitly where Feynman states that the increased mass is caused by the creation of massive particles?

ImAnEngineer
Both are right. When Feynman speaks of "at rest after the collision", he certainly does not mean the elementary constituents are at rest, but their center of gravity is.
Which is the same thing because it's a perfect inelastic collision.

Could you show me explicitly where Feynman states that the increased mass is caused by the creation of massive particles?
He doesn't say that.

Staff Emeritus
Science Advisor
Gold Member
He doesn't say that.
I'm glad to hear it. In which case, what did you mean by your response to DaleSpam's post?
I would say you're making perfect sense. But what I've read tells me otherwise.
I quote from the Feynman lectures[...]

ImAnEngineer
I'm glad to hear it. In which case, what did you mean by your response to DaleSpam's post?
Oops sorry, you're right. Somehow I read something different, my mind must've been somewhere else. Never mind me

raymondtco
jtbell & DeleSpam:

Ahh... I get what you mean. Are you saying the mass for a composite system could be larger than the sum of all its elementary particles because some portion of mass is stored in the interatomic thermal and potential energy, which can be thought of as extra mass? That's why "DeleSpam" says the increase in mass cannot be assigned to each particle since it is "among" those particles?

DeleSpam:

Hmm.. When you mention about the sum of vectors, is that the same idea as what DrGrey is trying to say above by the equations with the summations?

All:

Is there any mathematical or experimental proof that the potential energy is also a form of increase in mass? I could understand how the kinetic and thermal energy can affect the mass of the particles, but not potential one.

Mentor
Are you saying the mass for a composite system could be larger than the sum of all its elementary particles because some portion of mass is stored in the interatomic thermal and potential energy, which can be thought of as extra mass? That's why "DeleSpam" says the increase in mass cannot be assigned to each particle since it is "among" those particles?
Yes.

When you mention about the sum of vectors, is that the same idea as what DrGrey is trying to say above by the equations with the summations?
Yes.

Mentor
Is there any mathematical or experimental proof that the potential energy is also a form of increase in mass?

The mass of an atomic nucleus (with two or more protons/neutrons) is always less than the sum of the masses of the individual protons and neutrons that comprise it. This corresponds to the fact that the potential energy of the bound nucleus is less than the potential energy of a collection of widely-separated protons and neutrons.

In order to separate a bound nucleus into its constituent protons and neutrons, we have to do work (supply energy), which increases the potential energy of the system.

Science Advisor
Gold Member
Well I am not going to bother with clay instead I intend to purchase two Gold bars.
Have them thrashed together for the next twelve months then weigh them in.
Somehow I don't think they will weigh more but there temperature might be higher.
PS.I am no expert.
If you raise the temperature of 1 kg of gold by 1 degree Celsius, by my reckoning its mass increases by 1.44 picogram (trillionths of a gram) which would be worth about 0.0000000042 US cents.

Or, to put it another way, if you had one million kilograms of gold and raised its temperature by 100°C, you'd have temporarily "created" half a cent of gold (to offset your huge energy bill for heating a million kg of gold).

Any gold dealers worried by this should take care to measure the temperature of the gold they are weighing.

Specific heat of gold = 129 J/kg K
c = 299792458 m/s
Gold price today = 29460 \$/kg

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raymondtco
Hmm.. I see. So as long as I account for the total mass of a composite system, I do not have to include any other forms of energy since they already considered in terms of mass.

Now everything is crystal clear! Thank you all for your effort and patience! It's greatly appreciated! =)

Science Advisor
Gold Member
Hmm.. I see. So as long as I account for the total mass of a composite system, I do not have to include any other forms of energy since they already considered in terms of mass.
Yes, except for the kinetic energy due to the motion of the composite centre of mass relative to the observer (if any), which is extra. (Or potential energy due to an external source.)

(Assuming that "total mass" means -- as it usually does -- "rest mass", not "relativistic mass" which would include the kinetic energy I mentioned.)

raymondtco
Oops.. You brought up another question of mine. Hmm.. Let say, we have two stars with gravitational potential energy (or two charges with electric potential energy). As a result, we do have higher mass of this composite system and the extra mass cannot be assigned to either one. Then what would happen if we weight each one and then both? Will we get different values? I think we do.

How come Feynman would say there is no meaning and no way to really look into a composite system. He took an example of Kaon that can split into two pions OR three pions to illustrate that an element(mass) does not really consist of a certain group of sub-particles.

Hmm.. How to reconcile these two ideas?