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The term "mass" is very common. But what does it depend on? How is it known?

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In classical physics mass has two definitions:

  1. It measures the amount of inertia that you have. In order to accelerate something you have to apply a force to it. The heavier your thing is, the less it will accelerate, $$ a = \frac{F}{m} \, .$$ If you know the force and can measure the acceleration, you have access to the mass.

  2. In the physics of gravity, the mass is the gravitational charge. Just like charged particles exert a force on each other that is proportional to the product of their charges, two massive particles attract each other with a force that is proportional to the product of their masses,

$$ F = m_1 m_2 G /r^2 \, .$$

What is fascinating about this (and lead to the discovery of general relativity) is the fact that these two definitions coincide. Inertial mass and gravitational masses are the same!

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What is physics? Physics is the modeling with mathematics of observations in the world around us. It is a way of creating a logical sequence that can be predictive and not only explanatory. It reduces the innumerable constants one would need to describe, for example , the trajectory of a ball with just space coordinates, to a simple parabolic function that can be used to give any one of those space coordinates.

To do this, find mathematical functions, it needs some basic assumptions, postulates, gleaned from the observations of falling apples and shooting arrows. These functions are described in the other answers. It was found that a unique mass assigned for each object allows a mathematical description of gravity with what is known as Newtonian mechanics.

We describe our observations within a system of units. Take the International system of units. It is the MKS, Meter, Kilogram Second . It defines the basic numbers for three independent postulates, that distance exists, mass exists and time exists. These are postulated in order to be able to model nature with mathematical functions.

But what does it depend on? How is it known?

Mass is a number unique for every physical body so that the mathematical model of Newtonian mechanics can describe and predict its position and reactions with the appropriate formulas . This number is measured by use of the formulas as explained in the other answers.

When physics started studying the microcosm of molecules atoms nuclei and particles the postulates were carried over , the mathematical descriptions of observations became much more complex, and still, each particle is assigned a mass measured and tabulated so that the mathematical models in use are descriptive and predictive of future behavior.

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What exactly is mass?

My answer will encompass four very different regimes each separated by a factor of 1027. I'll first look at things on the order of 10-27 kg, then 100 kg (1 kg), then 1027 kg, and finally 1054 kg.

Mass at the scale of 10-27 kg
This is the domain of atoms and elementary particles. A proton has a mass of 1.672621777×10-27 kg, to within a relative uncertainty of 4.4×10-8.

Physicists and chemists rarely use kilograms when looking at masses at this tiny scale. They instead use energy or atomic mass units. In terms of energy, the proton mass has a mass-energy equivalent of 938.272046 MeV (million electron volts), with a relative uncertainty of 2.2×10-8. In terms of atomic mass units, the proton mass is 1.007276466812 u, with a relative uncertainty of 8.9×10-11. Note the two to three order of magnitude improvement in precision when looking at mass as a ratio versus mass as energy versus mass at the ordinary human scale.

Another ratio of interest at this scale is the ratio of the mass the alpha particle to that of a proton. This is 3.97259968933 (unitless), with a relative uncertainty of 9.0×10-11. Note that this is less than four. A neutron is slightly more massive than is a proton, and an alpha particle comprises two protons and two neutrons. So how can the mass of an alpha particle be less than the mass of four protons?

This mass deficit gives a big clue with regard to what mass is: Mass is bound energy. In the core of our Sun, four protons combine via a chain of reactions to eventually create alpha particles. In the process, the mass deficit of about 0.03 proton masses per alpha particle is converted into photons and neutrinos. It's the photons released by that p-p fusion process that keeps our world warm.

Mass at the scale of one kilogram
This is the domain of cells (about 10-12 kg) to the total mass of humanity (about 1011 kg). We use three approaches to measure mass in our everyday, ordinary:

  • By weight (e.g., a spring scale),
  • By comparison with an object of a known mass (e.g., a balance scale),
  • By Newton's second law ($F=ma$).

The first two approaches work only because all objects fall at the same rate. This means that the mass in Newton's second law ($F=ma$) and the masses in Newton's universal law of gravitation ($F=(Gm_1 m_2)/r^2$) are one and the same. Galileo was the first to carefully observe this fact. This equivalence of gravitational and inertial mass has long been a mystery. It is an observed fact. It's an implicit assumption in Newtonian gravitation, and an explicit assumption in Einstein's general relativity. The universality of free fall is one of the key tenets of Einstein's equivalence principle.

Mass at the scale of 1027 kg
This is the domain of medium-sized asteroids to supermassive black holes. At this scale, gravitation is by far the dominant force. Astronomers observe how things interact gravitationally. They don't observe mass. They instead observe an object's gravitational parameter, the product of Newton's gravitational constant $G$ and mass $m$. Astronomers know the Sun's gravitational parameter to greater precision than physicists know the mass of a proton in atomic mass units.

What about mass?

There's a problem here. Newton's gravitational constant $G$ is one of the least well-known physical constants. It is known to a paltry four decimal places of precision. Expressing large masses in terms of our everyday, ordinary International System of Units is not a good idea. Since the gravitational parameter of the Sun is known to such a high degree of precision, astronomers use the solar mass instead of the kilogram as their gold standard.

Mass at the scale of 1054 kg
This is the scale of galaxies to the observable universe. Here there even bigger problems than relating mass on the atomic or stellar scale to mass on a human scale. On the galactic scale, the mass inferred from observed light (and hence stellar masses) doesn't match with the mass inferred from observed velocities. The way galaxies rotate doesn't make sense if one only considers mass inferred by light coming from galaxies. This is the galactic rotation problem. There are very few ways out of this problem:

  • The model of gravitation is fundamentally broken at these large scales, or
  • There's a lot more ordinary matter that astronomers can't see than is expected, or
  • There's some unknown kind of matter that is far from ordinary.

The second option has pretty much been ruled out, at least a complete explanation. The first option is fringe science. Most physicists and astronomers think that the third option is the right answer. That unknown matter is called dark matter, for two reasons. One is that it is dark; it doesn't interact electromagnetically (it doesn't absorb or emit light). The other is that whatever it is, it is beyond the standard model of physics. "Dark" in this context means "it's a mystery, at least for now."

On an even larger scale, galaxies that are far removed from one another are moving apart from one another (this is the Hubble expansion) and the rate at which they are doing so is increasing. This latter fact is a fairly recent discovery, and its origin is even more mysterious than dark matter. This is dark energy. What it is, no one knows.

Summary
A number of challenges remain for future scientists to solve regarding mass-energy.

  • Doing a better job of relating the atomic scale to our everyday, ordinary world.
    This is an ongoing effort. The ultimate goal of this program is a complete revamping of the International System of Units, with the elimination of the International Kilogram Prototype ranking as one of the highest priorities.

  • Doing a better job of relating the solar system scale to our everyday, ordinary world.
    This too is an ongoing effort, but improvements have been very slow. There has only been a two order of magnitude improvement in the precision of $G$ in the 218 years that have transpired between the Cavendish experiment and now.

  • Explaining the equivalence of inertial and gravitational mass.
    This is one of the goals of multiple programs that are attempting to merge general relativity and quantum physics. There have been some partial successes in this regard, but also a lot of non-successes.

  • Explaining dark matter and dark energy.
    The standard model of physics doesn't explain either of these. A number of hypotheses that go beyond the standard model have been proposed, but these have either been falsified or are unobservable at any humanly-conceivable energy level.

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Since your tags are "newtonian-gravity" and "mass" I will attempt to answer this question in a classical framework. In classical mechanics, mass is essentially defined as a measure of an object's inertia.

Let me explain further. We have Newton's second law $$F_\text{net}=ma$$ which is assumed to hold for all objects in classical mechanics. Suppose we apply a force of $1000\,\mathrm{N}$ to two objects. One is light, $m_1=10\,\mathrm{kg}$ and the other is heavy, $m_2=1000\,\mathrm{kg}$. Then the two objects have different accelerations ($a=F_\text{net}/m$) $$a_1=100\,\mathrm{m}/\mathrm{s}^2,\quad a_2=1\,\mathrm{m}/\mathrm{s}^2$$ So the object that has more mass is harder to accelerate, i.e. has more inertia.

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