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Why mass is additive?

  1. Jul 14, 2013 #1
    The picture of mass I have is as follows :- A particle, say an electron resists some force because of some intrinsic property it has called mass. Now this property is caused by the higgs boson particle (which I know nothing of).

    When two such particles, in this case when two electrons are present the total mass will be the sum of the individual masses? Why is this?
  2. jcsd
  3. Jul 14, 2013 #2

    Simon Bridge

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    You mean why do two particles resist twice as much as one ... (note: the property is called "inertia" - it happens to be the same as mass.)

    If the inertia did not add, the it would take a different amount of work to move two masses separately as to move them together. How would the two masses know if they were "together" or not?

    Maybe because they see each other's electromagnetic field?
    Technically, two charged particles will have energy in their mass and in their electric field ... if the charges are bound (i.e. electron and proton) then there will be a mass discrepancy (mass of an atom is not the sum of the masses of it's components).

    What adds up is energy and mass is a kind of energy.
    It's the superposition principle all over again.

    So the next is "why do the energies add up?"
    Nobody knows - they just do. That's the rule.
    If two identical objects are stacked one on the other, then the whole is twice as high as any one.
    why is this?
    why is 1+1=2?

    Note: "why" questions like this are endless ...
    Last edited: Jul 14, 2013
  4. Jul 14, 2013 #3


    Staff: Mentor

    The best you can say here is that it is borne from experiment where adding the masses is consistent with getting the results that match the theory and so it is defined in the math of the theory.

    Its similar to why do forces add vectorially in classical mechanics. Its consistent with CM experiments and so is defined as a legal operation in the math of the theory.
    Last edited: Jul 14, 2013
  5. Jul 14, 2013 #4
    I think you need to know the origin of inertia mass. Inertia mass comes from Newton's law.

    First, according to Newton's second law, ƩF=ma, you know mass is the ratio of net force and its acceleration. But what is net force? We can't say net force is its inertia mass times its acceleration because that's a circular argument. So, how to answer this problem? Or, I should ask "how to not to use the unknown "net force" to get the idea of inertia mass?".

    Let's use Newton's third law, suppose that there are two ball colliding each other.

    ƩF1 = m1a1
    ƩF2 = m2a2

    if we could neglect the frictional force, then the net force of each ball comes from each other.

    ƩF1 = F21 = m1a1
    ƩF2 = F12 = m2a2

    Next, using Newton's third law,

    F21 = - F12

    m1a1 = - m2a2

    m1/m2 = - a2/a1

    Finally, let's define mass of the one ball is 1 kg, then we can know the other ball by the above equation. So the newton's 2nd law is well-defined.

    Second, we can consider two mass points system.

    ƩF1 = m1a1
    ƩF2 = m2a2

    then sum them up

    ƩF1 + ƩF2 = (F21 + F1,ext) + (F12 + F2,ext)

    ∴ƩF1 + ƩF2 = ƩFsys,ext = m1a1 + m2a2 = (m1 + m2)ac

    ac is the acceleration of center of the mass(system)

    So, the reason why mass is additive is that's how we defined the center of mass.

    If the acceleration of every particles in multiple-mass-points system is the same, then

    you'll see the mass-additive property is guaranteed by Newton's second law.

    PS. Newton's law is only suitable to single particle system. Euler's law of motion is advanced newton's law and it can be used in multiple particles system.
  6. Jul 14, 2013 #5
    Yes, that's what I meant

    You say that "maybe" they see each other's electromagnetic field. So we are not sure about this?

    Now that I asked this question many other similar questions are coming up like - Why is distance additive? Why is force additive? and more.
    So the correct answer to all these questions is that because this is what we observe or is there a more deeper answer?
  7. Jul 14, 2013 #6
    If you want to know the reason why forces is vectorially additive, then maybe you can check this:

    law of equilibrium on the inclined plane

    The story of Simon stevin

    And The Science of Mechanics by Ernst Mach
  8. Jul 14, 2013 #7


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    We live in an essentially linear world. To within our limits of measurement, we usually find that we can apply simple arithmetic (Addition and Multiplication) when we combine quantities. Even when we impose extreme conditions, we nearly always find that the world is 'Monotonic', so twice as much of something will usually increase and effect, even when it's not actually proportional. There are notable exceptions to this, of course, particularly in complex systems (many biological and electronic examples).
    To give us half a chance of getting a grip on Science, it's a good idea to start with the linear examples or where would we be? (The realms of Magic, or worse, I think).
  9. Jul 14, 2013 #8


    Staff: Mentor

    This isn't generally the case in relativity. In relativity the mass of a system is often greater than the sum of the masses of the individual particles.
  10. Jul 14, 2013 #9
    I never gave a thought before on why the physical quantities follow simple arithmetic when we combine them but now it's truly amazing to realise that they follow something so neat and precise.
    I suppose there is no answer to my question, right? Mass is additive because it is, that's how nature is, right?
  11. Jul 14, 2013 #10


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    Did you miss Ethan0718's excellent answer in post 4?
  12. Jul 14, 2013 #11


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    There are a couple of pretty good answers above, but if you want something less mathematical and more intuitive, you might think about what happens if you take a an object and break it into two pieces. Do you expect the two pieces together to weigh the same as the original object? Of course you do, and that's what mass being additive means.
  13. Jul 14, 2013 #12
    Nature is like that but it is also crafty !!
    If you ever find that 1 + 1 does not equal 2 you should hope that it equals 1.414 you have then found a crafty way that nature can behave......it is not difficult ! To get to grips with.
  14. Jul 14, 2013 #13
    Suppose here's a charged particle, for example, an still electron.

    We know that the charged particle will emit radiation (photon) when it's accelerating.

    So, let's push this charged particle and make its velocity be non-zero velocity from zero.

    Assume its mass is m.

    The change of its kinetic energy will be ΔEk = [itex]\frac{1}{2}[/itex]mvfinal2 - [itex]\frac{1}{2}[/itex]m02


    However, it's not the energy you lose. Don't forget the emitting radiation. Let's be some energy,Ephoton.


    So the total energy you lose (or, the work done by you) during this process is

    ( [itex]\frac{1}{2}[/itex]mvfinal2 - [itex]\frac{1}{2}[/itex]m02 ) + Ephoton = [itex]\frac{1}{2}[/itex]mvfinal2 + Ephoton

    So, it's like you push a new-mass charged particle:

    [itex]\frac{1}{2}[/itex]mvfinal2 + Ephoton = [itex]\frac{1}{2}[/itex]m'vfinal2 - [itex]\frac{1}{2}[/itex]m'02

    Obviously, m' > m

    That's why you'll feel its mass greater than its still mass.
    Same argument can explain the case you mentioned, the scenario that two electrons are present.

    ps. actually, I prefer to think the electric field has some mass.
    Last edited: Jul 14, 2013
  15. Jul 15, 2013 #14

    Simon Bridge

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    No - we are sure that they see each others electric field ... I was proposing an answer to my question.

    You should also see that if mass were not additive, then a balance would not work - a single 20g mass would not balance two 10g masses. You'd also be able to generate energy out of nothing ... but why conservation of energy? etc. etc. etc.

    It more that any answer anyone can possibly give for such a question just leads to the question being applied to the answer - you know, like when a child asks "why?" to everything, only more grown up?

    Why is X like that?
    Because Y is true.
    Then why is Y like that?
    Because of Z.
    But why...

    I could say it is because of the interconnectedness of physical laws - laws of locality and causality ... which are derived from the early Universe being quite small - but then you'd just ask: why was the early Universe so small? See?

    This is the sort of question I am talking about when I say, 'Science does not do "why" questions - ask a philosopher.' (It seems to upset some people.) Empiricism can tell you what is, and how it comes to be, but does not tell you why it is like that.

    The standard answer is to ask you to rephrase your question in terms of "how" or "what" to make it explicitly something that can be handled by science tools. Mind you, Ethan0178's earlier post is pretty good.
  16. Jul 15, 2013 #15
    Yes I am constantly reminded not to ask the "why" questions in physics here on PF but it's very tempting for me to know the "why" even though I might not get anywhere. Anyways I'll replace "why" with "how" now onward.

    Doesn't that assume that force is additive (I don't need to know why is that)? Anyways thanks for your answer.
  17. Jul 15, 2013 #16
    Yes, it assumes that force is vectorially additive!

    I've give you some suggestions about this question at post 6.
  18. Jul 15, 2013 #17


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    Some historian can maybe be more precise but I think this argument goes way back. I think Galileo argued, and even people before him, that if all objects did not travel at the same rate under gravity, well they wouldn't, and so composite objects would separate out as they fell. And they realised all objects they knew of were composite.
  19. Jul 15, 2013 #18


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    As DaleSpam already pointed out, mass is not additive in special relativity. Maybe the most relevant example is the mass defect in the fission of uranium nuclei where the sum of the masses of the fission products is less than the mass of the original uranium nucleus.
    The point is that in special relativity, mass is the energy of a system in it's rest frame but for a composite system, the particles bound together can still have kinetic (and potential) energy due to their relative motion. This extra energy contributes to the mass in addition to the mass of the parts of the system, i.e. the energy they would have if they were at rest and widely separated.
  20. Jul 15, 2013 #19


    Staff: Mentor

    I wouldn't call that an assumption. It is a definition. Forces are defined to be additive by Newtons 2nd law.
  21. Jul 15, 2013 #20


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    yes. although to be specific, this is the 'invariant mass' which is not additive. Conversely, the 'relativistic mass' is additive.
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