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The mass is said to be matter content of an object. Is there any detailed definition of mass because the phrase , "matter content of an object" uses the word matter whose definition is itself " something that has mass and occupies space". Defining mass and matter in such a way gives rise to circularity. It is like saying "What is a box? Something inside which we keep things. What are things? Something we keep inside a box."

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  • $\begingroup$ This may help hyperphysics.phy-astr.gsu.edu/hbase/mass.html $\endgroup$
    – anna v
    Dec 17, 2023 at 8:16
  • $\begingroup$ This article is satisfactory. $\endgroup$ Dec 17, 2023 at 8:57
  • $\begingroup$ But then what is force $\endgroup$ Dec 17, 2023 at 8:57
  • $\begingroup$ The most basic definition of force is: A push or a pull which can change : the object's velocity, shape or size, state of rest or motion. If you're looking for something more, I suggest you post a new question. $\endgroup$
    – Stuti
    Dec 17, 2023 at 9:18
  • $\begingroup$ To define force as something which changes velocity of an object we need to involve mass which in turn is described by using the term "force". So again circularity $\endgroup$ Dec 17, 2023 at 9:28

4 Answers 4

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There is another definition of mass: mass is a measure of inertia. Inertia is the tendency of a body to stay in its current state of motion/ rest. Inertia is directly proportional to mass, i.e. greater the mass, greater is the force required to change its state of rest or of uniform motion.

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Mass is a property of atoms and is tabulated in the periodic table. Material objects are made of atoms. The mass of a material object is the total mass of the atoms it contains. The phrase "matter content" is an inexact way to say that mass is extensive, i.e., that the mass of a material object is the sum of the masses of its constituent parts.

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    $\begingroup$ Hmmm. but then fundamental particles - which are not made of atoms - also have mass. And the mass of a nucleus is not equal to the sum of the masses of its constituent parts because of binding energy. $\endgroup$
    – gandalf61
    Dec 17, 2023 at 10:31
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One way of defining mass is to look at collisions. To keep it simple, let's consider an object moving along at a known speed along a horizontal frictionless rail that collides head-on with a stationary object, and sticks to it. The combined object moves in the same direction but at a slower speed.

If the two objects are identical, then it moves at half the speed. If the objects are made of the same material, but the first is twice the volume of the second, then the combined object moves at two thirds the speed. It is as if moving objects had a quantity associated with the motion that gets "shared out" over the amount of stuff, determining the speed.

When objects are made of different materials, they still seem to have the same sort of quantity, but in a different (and fixed) proportion to their volume. Collisions where objects bounce off one another also have it, but this time it gets transferred from one body to the other. The velocity times the "amount of stuff" for all objects remains fixed.

We call this "motion" quantity the momentum and for any given object it is proportional to the velocity. We write $p=mv$ where $p$ is the momentum, $v$ the velocity, and $m$ a constant of proportionality that expresses the amount of material in the body, called the mass.

So to define the mass of any object, we take a standard object which we define to have unit mass, move it at 1 m/s towards our object to be tested, and measure the velocity of the two objects stuck together. The momentum prior to the collision was $m_1v_1$ (where $m_1=1$ and $v_1=1$). The momentum after the collision was $(m_1+m_2)v_2$. These are the same, so knowing $v_2$ we can work out $m_2=m_1(v_1/v_2-1)=1/v_2-1$.

Mass adds, so if we take two identical objects of mass $m$ and glue them together, the resulting object has mass $2m$. For a given uniform material, mass is proportional to volume, the ratio is the density. But different materials are observed to have different densities, so a given volume of cork has much less mass than the same volume of iron.

Mass and momentum are conserved quantities in classical mechanics, meaning their totals don't change. They can be moved around, but they cannot be made to increase or decrease.

Classical physics does not attempt to define what they are. It is simply that we observe that collisions act as if there was this quantity of "weighted motion" that can be transferred from one object to another, each object has a fixed ratio between this momentum and its velocity which we call mass.

Force is the rate of change of momentum - it is the rate at which momentum is transferred from one body to another. So bigger forces cause bigger changes in velocity, but heavier objects are harder to move, requiring more force to get the same change in velocity.

Note that if we want to use this to define mass, we can't define momentum using $p=mv$, because that becomes circular. We have to define momentum as the "quantity of motion" that is conserved in collisions, observe that it is proportional to velocity, and name the constant of proportionality the mass. Thus, the equation $p=mv$ is actually defining mass, not momentum.

This definition is sometimes referred to as inertial mass. There is also the gravitational mass which is proportional to the force applied to the object by gravity. (And hence depends on understanding momentum first.) We can most easily define this with a balance - we put various known standard masses in one pan of the balance, and the mass to be measured in the other, and stop when the forces balance - when there is no longer any transfer of momentum from one to the other. This is a lot like the collision method - we let the two objects interact, and transfer momentum, and the resulting changes in motion tells us the ratio of their masses.

Things change considerably when we get to relativity and quantum mechanics, and we have a much deeper understanding of mass arising from those theories. But that's a much longer explanation.

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The definition depends on the field of physics that you're targeting. In addition to the definitions given in the other answers, here are other, in actual use, definitions:

  • In special relativity, mass is defined as the norm of the 4-momentum vector.
  • In quantum field theory, mass is defined as the pole of the renormalized propagator for a free particle (taking interaction with the Higgs boson into account if appropriate).

So, when asking for such a fundamental definition, you cannot avoid specifying the context.

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  • $\begingroup$ I know nothing what is quantum field theory or special relativity. But a thing will always be the same thing in every field. When you are saying that the definition of mass depend on what firld we are targeting it is saying an apple kept in a room will become a banana when we take it to another room. I just want an universal definition of mass which no field or theory can contradict and which is complete $\endgroup$ Dec 17, 2023 at 9:41
  • $\begingroup$ Did people know what is mass when QFT or special relativity was not discovered $\endgroup$ Dec 17, 2023 at 9:42
  • $\begingroup$ @12jjsioe383 That's my point. There is no universal definition. The definition changes over time, as knowledge evolves. There are even cases where we don't know whether definitions are really equivalent (inertial mass vs gravitational mass, for example, if you want an classical example). $\endgroup$
    – Miyase
    Dec 17, 2023 at 11:25
  • $\begingroup$ Is there any field in mechanics by which every phenomenon can be explained? If there is such a field then what is the definition of mass there? $\endgroup$ Dec 17, 2023 at 14:40
  • $\begingroup$ @12jjsioe383 Precisely, such a field doesn't exist. In theory, taking the most foundamental theory should be enough to derive the equations of older, less foundamental theories, but in practice we rarely know how to do that. Hence the necessity for each field to provide its own definition of things such as mass, motion, space and so on. $\endgroup$
    – Miyase
    Dec 17, 2023 at 15:36

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