# What exactly is mass? [duplicate]

I was looking for a definition of mass and most of the time what I got was that "it is the amount of matter". Now that is very vague. And the way we define matter is "anything that has "mass" and occupies space". so... what exactly is mass? Can you please answer it to the level where a highschool junior/11th grader can understand?

When we ask about such a basic property as mass, it can happen that we find it hard to put into words in a succinct way, so what we do is offer a group of ideas and equations and thus show the role that mass plays in physics.

In the case of mass, there are two main roles: one to do with inertia, the other to do with gravity. Accordingly we may speak of "inertial mass" and "gravitational mass" but, interestingly, it turns out that they are the same.

In classical physics (as opposed to quantum physics), inertial mass is defined ultimately by its role in Newton's second law. This is the one which says that rate of change of momentum is equal to force. The idea is that first we find various force-providing systems, such as a rocket motor in a given state, or an electric plate with a given amount of charge (attracting some other charge), or a rubber band at a given temperature and stretched by a given amount. Next, we imagine attaching each of these force-providing systems to a variety of objects, and we measure the acceleration of each object. It is found that the ratio of force to acceleration is the same, for a given body, independent of what kind of force was provided. So we write the equation $$f = m a$$ and we define inertial mass of the body as $$m = f / a$$.

There remains a difficulty, however, in that we need to calibrate the whole approach. We can decide when one kind of force (say from a rubber band) is equal to another (say from a rocket motor) by applying them in opposite directions to a given object and checking whether the acceleration is zero. So this enables us to compare different types of force. But we still need an overall scale to say what one unit of force is, or what physical situation creates one unit of force. If we knew that then we could deduce what one unit of mass is by finding a body whose acceleration is one metre per second-squared when one Newton of force is applied. However in practice this is done the other way around: we come up with an agreed way of saying what kind of object has unit mass, and then we can find out the size of a force using $$f = m a$$.

Notice that in the above there are two issues: first, what kind of property mass is, and then how we pin it down quantitatively. The kind of property is "that which sets how much acceleration a body will have when subject to a force." The quantitative definition then employs experimental ingenuity and Newton's second law.

• This is a great start. But I think it would be appropriate to also present an equation with gravitational mass (Newton's law of gravity), and mention how mass relates to energy and how a large amount of mass is due to binding energy. No description of mass can be complete without these points (if ever). Jan 27 at 20:03
• Wouldn't it be more accurate to say that no experiment has yet detected a difference between inertial mass and gravitation mass? IIRC, there's no theory that explains why an object's resistance to change in motion should be related to the gravitational force it exerts. Jan 27 at 21:40
• @chepner actually, the whole theory of general relativity is based on the equivalence principle. Find an experiment that shows a difference between the two types of mass, and GR breaks into pieces. Jan 27 at 22:43

The mass of an object is a fundamental property of the object; a numerical measure of its inertia; a fundamental measure of the amount of matter in the object. Definitions of mass often seem circular because it is such a fundamental quantity that it is hard to define in terms of something else. All mechanical quantities can be defined in terms of mass, length, and time.

italics mine

One has to start with the everyday concept of force, momentum, energy etc as known before they were codified in Newton's laws of motion. At that time, weight was measured for everything, and it was a given that the weight is an invariant of the object. Even now we talk of a one kilogram mass, giving the units of weight.

Newton's laws codified this in order to develop classical mechanics. The laws are axiomatic assumptions but they worked well in theoretically fitting the data at the time and predicting new situations.

Law 1. A body continues in its state of rest, or in uniform motion in a straight line, unless acted upon by a force.

Law 2. A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force.

The $$F=m*a$$ where $$a$$ is acceleration and $$p$$ is the momentum $$mv$$

The $$m$$ is an axiomatic assumption that it describes the object under consideration.

Law 3. If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

The mass m, given by the weight definition at the time, and modeled with $$F=ma$$ was adequate both to model kinematics and gravitation and to introduce the difference between gravitational mass and inertial mass, see this question and answers.

In the study of the high velocities where Lorentz transformation and four vectors are necessary the characteristic mass for all objects is given by the invariant mass of the four vector, the same in all inertial frames.

Andrew Steane's answer is good, but I would like to add a point.

The universe is full of stuff. Some stuff is matter and some is energy. If you look carefully at the universe, you can see patterns of behavior. If you push on matter, it changes speed. That was more or less the way physics was done in the time of Aristotle. In his view, rocks roll downhill and stop because they have a lot of earth in them, and it is the nature of earth to be low and still. By contrast, the stars and planets were full of fire. It is their nature to be high and to move.

Galileo changed that. He applied math to the patterns of the universe and found that you could match the behavior of the universe with mathematical rules very well indeed. Newton extended this idea a great deal.

So physics is a mathematical model of the universe. Different from the universe itself.

Mass is a number, a physics thing. Matter is stuff, a universe thing. So you might want to ask what mass is used for instead of what it is.

Andrew has answered that. It turns out to have two main purposes. Mass helps describe exactly how matter changes speed when you push on it. And Mass helps determine exactly how strong is the force of gravity generated by an object.

There is no obvious reason why the same number serve both purposes. So people have gone looking for any difference. It has been shown that if two different numbers are needed, the difference between them is less than one part in a trillion. More precise experiments are being planned.

This might oversimplify, but ---

Anything not travelling at speed $$c$$ and which produces momentum in response to a force is mass. Of course this defers the question, philosophically, to what momentum is.

There is also a small school of physics which claims there is no such thing as mass, just, basically, highly localized energy or probability.

• "There is also a small school of physics which claims there is no such thing as mass, just, basically, highly localized energy or probability.", can you please give me a link to that school's works? Jan 27 at 21:47
• @ÁrpádSzendrei it's pretty much limited to the writings of the lunatic fringe (and of course the engine running the spaceship Heart Of Gold) Jan 27 at 22:15
• @CarlWitthoft Not so lunatic fringe as you might think. In quantum mechanics, some 85% of what we normally call mass is thought to be binding energy of the strong force. Jan 28 at 9:05
• @fishinear sure, but we can, and do (in the case of all binding energies) view that field energy as having been converted to mass. Otherwise $E = mc^2$ wouldn't be useful. Jan 28 at 12:03

I suggest you look into Kleppner's An Introduction to Mechanics. In Section 2.5, he gives an operational definition of mass without ever resorting to the concept of force (thus avoiding circularity).

The definition is as follows. Take an airtrack with a cart holding whatever object you want to measure, tie a rubber band to it, and pull on the rubber band in such a way so that the band is always elongated at a fixed length $$\ell$$. Then you should see a constant acceleration. The ratio of the acceleration to the acceleration of a reference object (pulled by the same band elongated also at $$\ell$$) gives you what we define as inertial mass.

Interestingly enough, this definition goes all the way back to James Maxwell.

Note that the cart itself has some mass so you have to amend your definition. There are two ways you can do this.

• Conceptually simple but impractical: you can imagine doing this in space with no airtrack and no cart messing things up.
• Conceptually messy but practical: you take into account the fact that the cart itself has some mass in your calculations, so in the definition of mass you no longer take a simple ratio of accelerations.

In any case, this should give a proof of concept that it is possible to define mass in an operational manner.

A historical approach might do it.

Suppose you have two balls made of the same material, one bigger than the other. One finds the bigger is the heavier. The weight is proportional to volume. Three of the same size ball is heaver than one ball, another proportionality property. Balls of different material of the same size can have different weights. Four balls of one material can have the same weight as ten balls of another material. There is some fundamental property that follows a rule of proportionality held in common by both materials that is not dependent on size. A heavier object has more of whatever this is than a lighter object of the same shape and size, yet that same quality follows a strict proportionality relationship with volume for the same material. Mass was an amount of some constituency of the object. This quantity was often reckoned according to a common standard. The Ancient Roman's used Carob seeds. There are more seed-weights in this collection of items than that one. So there was some universal amount associated with weights.

In the 17th century, researchers like Galileo and Newton noticed some other features of this quantity. The more massive an object, the harder it was to change its motion. It was more difficult to start it from rest and more difficult to stop it from moving. Of two objects moving at the same speed, it was easier to stop the lighter one. This gave rise to the notion of Inertia.

While Aristotle thought heavier objects fell faster the lighter objects, Galileo realized that objects did not fall at a constant speed, but accelerated under the influence of gravity, and they accelerated at the same rate whatever the material.

Not only are heavier objects more difficult to move up and down, they are more difficult to move side to side. They have more heft. Newton realized doubling an applied force on an object doubled its acceleration. Acceleration was proportional to force. He discovered that the proportionality constant was the object's mass. So he discovered Inertial Mass.

When Newton turned to Gravity, he discovered that the Force between objects was proportional to the product of their masses and inversely proportional to the square of the distance between them. One object can be considered to cause the field that works on objects of different mass, the more massive, the stronger the force. That was passive gravitational mass. There was the mass generating the gravity which was the active gravitational mass.

More thorough experiments in Chemistry led to other considerations of mass on the microscopic level. Certain conservation principles were developed.

In the 20th century Einstein discovered a mass energy equivalence. Other radical considerations led to the concept of anti-matter, where objects are identical except for having opposite charge. Mass can be observed to convert to energy and vice versa. Massless particles were discovered. How can an object exist if mass is the amount of material and it has no mass?

In his book The Structure of Scientific Revolutions, Thomas Kuhn argues that the "mass" referred to by Newton was not known to convert to energy so was radically different from the concept of "mass" as it was referred to in the latter half of the 20th century. While the same syllable was used to refer to both concepts, they weren't actually the same term, the two notions of mass were "incommensurable". Physicists take issue with this.

So mass began as a reckoning of the amount of a substance independent of its shape or size. It was closely tied to motion and gravity. Finally, it was found not to be all that different from energy. Some question whether the 17th Century term and the 21st Century term actually refer to the same thing.

Physics isn't mathematics. Physical quantities are defined by the methods used to measure them. So, mass may be defined by what a balance measures: given two bodies, they have the same mass if they balance. Put them together, and balance an object of twice the mass. By this means you measure gravitational mass.

Once you have a tested theory, you may use that theory to extend your suite of measuring devices. So, given the well-tested assumption that inertial mass is equal to gravitational mass, you may then do something like apply a known force to an object and measure its acceleration to infer its mass. And on it goes. But in the end, the theory cannot tell you what mass is, it can only tell you how mass behaves.

Mass is the property of an object, that

1. resists acceleration by requiring a force (inertial mass, $$m_i$$)
2. attracts another object (gravitational mass, $$m_g$$)

Once you start quantifying these observations, you arrive, respectively, at

1. $$F = m_ia$$
2. $$F = G m_{g1} m_{g2} / r^2$$

It was a deep insight that these two concepts could be treated as equivalent, i.e. $$m_i = m_g$$.

«I was looking for a definition of mass and most of the time what I got was that "it is the amount of matter". Now that is very vague.»

No, that is wrong.

Chemists and physicists agree the «amount of matter» is not mass $$m$$ (e.g., in the unit of kg), but about a counting number n (in the unit of mol). The mole is tied to Avogadro's constant ($$\approx 6.022 \times 10^{23}\,\mathrm{mol}^{-1}$$), but conceptually similar to e.g., a dozen of items; it is a more practical one in chemistry when counting electrons, atoms, molecules, etc. when balancing reaction equations (keyword stoichiometry).

A testimony about amount of matter is the standard established by BPIM. Though the concept is older, since 1971, it is one of the SI base units and may be found e.g., in the very first section of the four-page summary of IUPAC's Green Book (link to pdf (open access)) Quantities, Units and Symbols in Physical Chemistry, which was prepared jointly by chemists (by IUPAC) and physicists (by IUPAP).

Much of IUPAC's nomenclature is summarized in one of IUPAC's Color Books (by tradition, the color of the printed editions' cover is specific to the topic) the section References about Nomenclature the sibling site chemistry.se compiled; many of them accessible freely for anybody interested (open access).