- #26

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There's no stuff gets heavier at high speeds term in Newtons equations.

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- #26

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There's no stuff gets heavier at high speeds term in Newtons equations.

- #27

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Exacty - Newton's second law: "Compared to F=dp/dt

What turned out to be inexact in Newton's theory, at high speeds, were his assumptions about mass, time and length.

- #28

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Which assumptions about mass do mean?What turned out to be inexact in Newton's theory, at high speeds, were his assumptions about mass, time and length.

- #29

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Symon has a nice treatment of this in Chapter 14.

- #30

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Yes. For me, this was all I needed to be satisfied that the relationship was recovered intact in SR (even if the terms in component form were not the same). Seeing it satisfied in vector form made me very happy.p/dt =Frelativistically, where bold indicates 3-vectors. This can be derived from the more general dp_{μ}/dτ=F_{μ}, which gets us into the 4-force that Chet was talking about. However, none of these terms is what it is in Newtonian mechanics.

Symon has a nice treatment of this in Chapter 14.

Chet

- #31

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I understand Chet's point now. However I am still under the impression that this formal similarity between the equations for force has more to do with the way we define forces, i.e.with the way we represent interactions in the mathematical formalism of physical theories. This is regarding the right side of equations of the type "F= (some def. of force)".Yes. For me, this was all I needed to be satisfied that the relationship was recovered intact in SR (even if the terms in component form were not the same). Seeing it satisfied in vector form made me very happy.

Chet

I am not sure why the form of the force law for a charged particle interacting with an electromagnetic field is so formally similar though, I should take another look at Chet's reference.

- #32

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Personally I prefer using the variational approach when it comes to relativistic mechanics.

- #33

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"DEFINITION I.Which assumptions about mass do mean?

Thus air of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction; and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter."

With SR this simple and straightforward definition of mass had to be abandoned. According to SR, hot water weighs more than cold water. On top of that, a water molecule does not even weigh the same as the sum of its atoms.

- #34

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That depends on what you mean with "these terms". F=dp/dt applies both for classical mechanics and relativity but the the different transformations result in different expressions for F.It's true that dp/dt =Frelativistically, where bold indicates 3-vectors. This can be derived from the more general dp_{μ}/dτ=F_{μ}, which gets us into the 4-force that Chet was talking about. However, none of these terms is what it is in Newtonian mechanics.

I don't see why. It is not very helpful and today we rather use it in reverse to define density (as mass per volume) but that does not mean that it is wrong."DEFINITION I.

The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.

[...]"

With SR this simple and straightforward definition of mass had to be abandoned.

If the volume remains constant then heating the water will increase its density and its mass by the same factor. If you keep the density constant than the volume will be increase in the same way as mass. If nothing remains constant then the situation gets complicate but it will be always full consistent with definition 1.According to SR, hot water weighs more than cold water.

How does this collide with Newton's concept of mass?On top of that, a water molecule does not even weigh the same as the sum of its atoms.

- #35

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I don't follow you. According to SR, a constant number of water molecules (amount of matter) will increase in mass when heated due to increased kinetic energy. According to Newton's mechanics the mass is fixed. Of course, a discussion of m=E/c[..] If the volume remains constant then heating the water will increase its density and its mass by the same factor. If you keep the density constant than the volume will be increase in the same way as mass. If nothing remains constant then the situation gets complicate but it will be always full consistent with definition 1.

According to Newton's mechanics, the mass of all particles together ("condensed" or other) equals the sum of all particles separatelyHow does this [a water molecule does not even weigh the same as the sum of its atoms] collide with Newton's concept of mass?

- #36

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That does not result from definition 1.According to SR, a constant number of water molecules (amount of matter) will increase in mass when heated due to increased kinetic energy. According to Newton's mechanics the mass is fixed.

By replacement of Galilean transformation by Lorentz transformation Newton's quantity of matter turns into relativistic mass and relativistic mass is additive.According to Newton's mechanics, the mass of all particles together ("condensed" or other) equals the sum of all particles separately.

That's another topic. Mass defect is the difference of the total rest mass of a system (including binding energy and kinetic energies) and the sum of the rest masses of its sub systems (excluding binding energy between them). That doesn't contradict Newton because he didn't make corresponding claims.The fact that this is not exactly the case is therefore called mass "defect".

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- #38

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It is stated that mass equals "amount of matter" and it is implied that it is not a function of temperature or speed. You will search in vain for any such a relationship in classical mechanics.That does not result from definition 1.

We disagree about how to present the same facts; I'm afraid that we will have to agree to disagree. Newton's quantity of matter is not a function of speed. The Newtonian definitions and laws resulted in (or gave an explanation for) the "Galilean transformations" which Newton's mechanics assumed to be correct. The "relativistic mass" and "invariant mass" concepts came about because Newton's mass concept - as well as his time and length concepts - could not be maintained in relativity theory.By replacement of Galilean transformation by Lorentz transformation Newton's quantity of matter turns into relativistic mass and relativistic mass is additive. [..]

In order to stay within the bounds of this forum, I won't comment on claims about relativity theory in this thread. However, if you can show mass increase due to temperature increase in Newton's mechanics, I'll be happily corrected!

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- #39

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Also the definition of mass is often used differently in relativity. Most authors now prefer to define the mass of an object the invariant mass while others use variable mass.

- #40

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Quantity of matter is Newton's name for his concept of mass but not its definition. You must not confuse it with amount of substance or similar modern concepts. I totally agree with brainpushup that quantity of matter is a measure for inertia.It is stated that mass equals "amount of matter" and it is implied that it is not a function of temperature or speed.

Not in classical mechanics, but in special relativity.Newton's quantity of matter is not a function of speed.

That does not apply to definition 1-2 and lex 1-3. They work with Lorentz transformation as well.The Newtonian definitions and laws resulted in (or gave an explanation for) the "Galilean transformations"

In special relativity the relativistic mass directly results from Newton's quantity of matter (as defined by definition 2, lex 2 and lex 3).The "relativistic mass" and "invariant mass" concepts came about because Newton's mass concept - as well as his time and length concepts - could not be maintained in relativity theory.

Than you must not post in this thread at all. The break down of Newtonian dynamics at the speed of light is well outside classical mechanics.In order to stay within the bounds of this forum, I won't comment on claims about relativity theory in this thread.

Why should I do that? It is sufficient that Newton's mechanics does not exclude an increase of quantity of matter due to temperature increase. In addition it is in agreement with historical concepts of heat (e.g. phlogiston).However, if you can show mass increase due to temperature increase in Newton's mechanics, I'll be happily corrected!

That's why it is better to use Newton's term quantity of matter to avoid confusions with mass (zoki85 already got caught in that trap).Also the definition of mass is often used differently in relativity. Most authors now prefer to define the mass of an object the invariant mass while others use variable mass.

- #41

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DrStupid, are you just arguing for the sake of arguing?

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