F ~ a.
This proportionality can be converted to an equality by introducing a constant of proportionality. For the car loaded in a particular way, the force may be written as a constant times the acceleration (now in vector form):
F = ma.
This is Newton’s second law.1 Here I want to discuss just its role in defining mass.
The constant of proportionality in this equation, m, is called the inertial mass, or often simply the mass, of the object under study, in this case the car plus its load. Before discussing the physical meaning of mass, let me note some facts about it revealed by its appearance in this equation. Its dimension must be the dimension of force divided by the dimension of acceleration, or force × (time)2/length. In standard units, the unit of mass, the kilogram, is equal to one newton (second)2/meter. With standard abbreviations, this equality may be written,
1 kg = 1 N s2/m.
Which of the two concepts, force and mass, is taken to be fundamental and which derived is arbitrary. So the connection above may also be written
1 N = 1 kg m/s2.
Notice also from the equation F = ma that mass appears to be a numerical quantity (or scalar quantity), not a vector quantity. The only thing that can multiply a vector (a) to give another vector in the same direction (F) is a number. However, the numerical or scalar character of mass requires experimental verification.
Experiments performed with cars loaded differently reveal different masses. Not surprisingly, the rule is: The greater the load, the greater the mass. From the definition of mass, it follows that a particular force gives more acceleration to a small mass than to a large mass. The greater is the load carried by a car, the more sluggishly it responds to a given force, and the less velocity it acquires in a given time. In short, it has more “inertia.” And inertia inhibits any change in velocity—to greater speed, less speed, or a change of direction. Mass, to put it generally, is a measure of an object’s resistance to a change in its state of motion.
Among charged particles, the least massive electron is the easiest to accelerate to high speed and the easiest to deflect into a curved path. It responds most readily to the pushes and pulls of electric and magnetic forces. Some particles—the photon and the graviton—are actually massless. If Newton’s second law remained valid in the particle world, these particles could achieve infinite acceleration and infinite velocity. They would have absolutely no resistance to a change in their state of motion. In fact, because of the existence of a speed limit in nature, they reach only the speed of light rather than infinite speed. However, they can be said to experience infinite acceleration. A photon spends no time reaching the speed of light, but starts immediately with that speed at the moment of its birth.
Returning to the loaded car on the track whose mass is defined as the ratio of the horizontal component of force acting on it to its horizontal component of acceleration, one must inquire about the self-consistency of the definition. If mass is truly a scalar quantity, the mass of two objects (or “loads”) is the sum of the masses of the individual loads. Numerous laboratory experiments confirm that this is true, leading to the conclusion that mass is indeed a scalar quantity. But as always in science, one must be cautious. It turns out that in the submicroscopic realm where relativity and quantum mechanics hold sway—in domains far removed from ordinary lab experience—masses do not necessarily add as scalar quantities. For example, the mass of a deuteron, composed of a proton and a neutron, is less than the sum of the masses of a proton and a neutron. The so-called “binding energy” holding the neutron and proton together plays a role. In keeping with E = mc2, the negative binding energy subtracts some mass from the total.
One more subtlety. Although, in modern physics, mass is not strictly a scalar quantity (in that the mass of a combination of two objects need not be the sum of the masses of the individual objects), it is still a numerical quantity. It is measured by a single number, has no direction, and, in the equations of physics, behaves as a number.
A solid law of chemistry—and indeed of physics, too, before the twentieth century, is the law of mass conservation. Mixing and matching atoms and molecules does not change their total mass—a law concordant with the scalar nature of mass. Where mass does not behave as a scalar (in modern particle physics), mass is not conserved. Mass conservation was a vital tool supporting advances in chemistry in the nineteenth century, and fortunately remains valid to a high degree in chemistry laboratories in the twenty-first century—and no doubt beyond. Not so in the physicist’s world of fundamental particles, where mass conservation is not even close to being valid.
1 Newton’s own version of his second law, although equivalent in content to this equation, was phrased differently.