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From https://en.wikipedia.org/wiki/Mass

Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. [...] Suppose an object has inertial and gravitational masses $m$ and $M$, respectively. If the only force acting on the object comes from a gravitational field $g$, the force on the object is: $$F=Mg.$$ Given this force, the acceleration of the object can be determined by Newton's second law: $$F=ma.$$

In theory, mass could be determined by the number of indivisible particles the object is made of. A better approach would be choosing unit mass and using law of conservation of momentum: $$\frac{m_1}{m_2}=-\frac{\Delta v_2}{\Delta v_1}.$$

If mass can be determined in the absence of any force, how could there (even conceptually) exist more types of mass?

Shouldn't we talk about "inertial and gravitational force equivalence" instead of about "inertial and gravitational mass equivalence"? Any kind of mass which is "not invariant" under different kinds of forces makes no sense to me.

The answers here (Why did we expect gravitational mass and inertial mass to be different?) and here (Question about inertial mass and gravitational mass) do not answer my question as mass is "determined" by force there.

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    $\begingroup$ What would an inertial force be? Note that inertia is resistance to being accelerated. It's very definition is $m=F/a$. $\endgroup$
    – garyp
    Jun 15 at 12:07
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In theory, mass could be determined by the number of indivisible particles the object is made of.

This would only be the case if all indivisible particles were the same mass and if the mass of a composite object were equal to the sum of the masses of the indivisible particles. Neither of those are true. Mass cannot be determined this way.

A better approach would be choosing unit mass and using law of conservation of momentum: 𝑚1/𝑚2=−Δ𝑣2/Δ𝑣1

Note that for this to work requires $\Delta v_1 \ne 0$. That in turn requires a force. So this method does not avoid the need for a force. However, what it does do is make it clear that the resulting measure is independent of the type of force, without eliminating the need for a force altogether. Of course, the force based definitions also do that, but not as clearly.

Shouldn't we talk about "inertial and gravitational force equivalence" instead of about "inertial and gravitational mass equivalence"?

Probably if we ever found gravitational mass to be different from inertial mass we would call it gravitational charge instead. So mass would continue to refer to the inertial mass. Then, just like the acceleration of an object in an electric field depends on the ratio of its electric charge and mass, so also the acceleration of an object in a gravitational field would depend on the ratio of its gravitational charge and mass.

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  • $\begingroup$ How about determining mass from the law of conservation of momentum? $\endgroup$
    – 1mik1
    Jun 15 at 11:18
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    $\begingroup$ I added a paragraph addressing that. That is what the usual force-based definitions do $\endgroup$
    – Dale
    Jun 15 at 11:20
  • $\begingroup$ So does it all boil down to the definition of mass we're using? If we use the "resistance to force" definition, then we suspect that gravitational and inertial mass could be different, but if we use the "conservation of momentum" definition, then there is clearly just one type of mass? $\endgroup$
    – 1mik1
    Jun 15 at 11:24
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    $\begingroup$ Sorry, the "yes" was to agree with the equivalence you stated. $\endgroup$ Jun 15 at 13:12
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    $\begingroup$ @1mik1 by the famous $E=mc^2$ the mass of a composite object also includes the internal energy, both internal potential energy and internal kinetic energy in the object's center of momentum frame (aka rest frame). $\endgroup$
    – Dale
    Jun 15 at 14:46
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Gravitational mass could be completely different quantity, why not? We have electrical charge, which behaves very similiar to gravitational mass, but is something different. We can have two bodies with same mass but dirrefent charge.

In other reality it could be more variants.

Suppose bodies A and B. You put them on rails and tried to push. You found that it is equaly hard to push them. This meant that these bodies have identical inertial masses. Now you put these bodies onto scales and saw, body A is heavier than body B. That meant that Earth attracts body A stronger.

This is rather possible picture in alternative reality, why not?

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  • $\begingroup$ You're determining mass by force and this is exactly what this question is trying to avoid. As you can see, there are other ways of determining mass. $\endgroup$
    – 1mik1
    Jun 15 at 11:17
  • $\begingroup$ What is the problem with force? You can try to avoid it, but how it relates with the difference between inertial and gravitational masses? $\endgroup$
    – Dims
    Jun 15 at 11:21
  • $\begingroup$ Note, that force doesn't mean acceleration. We can measure force with a spring. This is how some scales work. $\endgroup$
    – Dims
    Jun 15 at 11:22
  • $\begingroup$ The force-based definitions of mass seem to run into problems like having the conceptual difference between gravitational and inertial masses. That's the reason I'm trying to avoid it and looking for other ways of determining mass, e.g. see the conservation of momentum. $\endgroup$
    – 1mik1
    Jun 15 at 11:26
  • $\begingroup$ This is one method of physics: first you find which can be different, potentially, in alternative world, then you find that it is wrong in our universe. This way you find some "symmetry" and formulate it. Einstein did it with "equivalence principle" and found GR. By the way, THERE IS a unit of mass "number of indivisible particles", it is called "Dalton": en.wikipedia.org/wiki/Dalton_(unit) $\endgroup$
    – Dims
    Jun 15 at 11:34

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