On the surface mass, like space, seems like a simple and straightforward concept but there are deep subtleties involved here too. In order to bring out the subtleties about mass, I had a practice of asking students in my introductory physics classes to explain what they understood when they spoke of the mass of some object. The would usually give some vague formulation along the lines ‘the amount of stuff’ it has. When I pressed them by asking how that differs from their concept of volume, they would sharpen their answer, usually saying that volume was a measure of the amount of space that was occupied by the object but mass was a measure of how heavy it was. There are teachers who insist that we must emphasize that mass and weight are different but I am not one of them. After all, we all know from experience that objects with greater mass are heavier to lift. The intuitive idea that mass has a relationship to weight is a good enough starting point for learning about the subtleties of mass.

The question is how we quantify the mass of an object and this brings up the important insight about the need for operational definitions of physical quantities. An operational definition of something is a set of steps that one carries out so that at the end one gets a number that tells you the value of the quantity. So for example, rather that loosely talking about the location and time of an occurrence, the operational definition of the location of an object is given by the readings according to standardized rulers that form the three axes. That gives us its three coordinates. The time at which an event occurs is given by the reading on a standardized and calibrated clock that is located right at the point where the event occurred. These precise operational definitions become important when we deal with relativity when different observers will report different values for when and where an event occurs and we need to be able to understand the relationships between those values.

So what might be an operational definition of mass? There are many possibilities. One way is to start with an equal arm balance and place the object whose mass is desired on the pan on one arm. Then one places whatever we have chosen to be standardized units of mass in the pan on the other arm until the two arms balance. Then the mass of the object is given by the number of standardized units required. Another method might be to hang the object from a calibrated spring. Then the amount by which the spring gets extended, when compared with how much it gets extended when a standardized unit of mass is hung from it, will give you the mass of the object. Since both these methods depend upon the downward forces on the balance or the spring due to the effect of gravity on the object, it makes sense that they are considered to be due to the same property of the object and to which we give the same name ‘gravitational mass’. Both the balance and spring calibrations can be adjusted so that we get the same numerical value for the gravitational mass of any given object.

But we know from experiment that the same force applied to different objects can result in different accelerations. We explain the difference by saying that objects have a resistance to a change in their state of motion, with some objects having greater resistance and thus smaller accelerations than others. This resistance to change in the state of motion is what we give the name inertia to. The amount of inertia of an object can also be operationalized by applying known forces to the object, measuring its acceleration, and dividing the force by the acceleration. Note that in this case, the force due to gravity plays no role and so there is no reason to think that an object’s inertia is related to its gravitational mass since that mass as we have defined it is arrived at quite differently from the inertia as we have defined it. The gravitational mass is a measure of the source of gravitational forces while the inertial mass is a measure of the resistance to any force. For all we know, they could be quite different entities, like mass and volume, with no operational relationship.

But it is the case that experimentally, objects that have a greater gravitational mass also seem to have a greater inertia, so it is natural to explore the idea that the two might be related. What is the relationship, if any, between the two? This is where Galileo and Newton come into play. Although Galileo pointed out that all bodies in free fall do so at the same rate, and did some experiments that provided evidence in support of it, he did not give a dynamical reason for it. He and his predecessors however did give a logical reductio ad absurdum argument that I reproduce here from my book The Great Paradox of Science (p. 197).

The logical argument to prove that all objects in free fall must fall at the same rate starts by assuming the opposite to be true, that they would fall at different rates with a heavier object falling faster than a lighter one. We then imagine connecting the heavier and lighter objects with a string to form a third, even heavier, composite object. The slower moving lighter mass should now act as a drag on its heavier partner, making it fall more slowly than if it were falling on its own. But since the composite object has a greater mass than the heavier component, according to our starting assumption it should fall faster than the heavier object falling alone. Hence there is a contradiction that is resolved by assuming that all masses fall at the same rate.

It was Newton who provided a dynamical reason for why this might be so. Newton’s law of gravitation said that the force of gravity acting on a body was proportional to its gravitational mass mg. i.e., twice the gravitational mass will produce twice the gravitational force and so on. We can write this proportionality relationship as as F∝mg which can be converted into an equation F=kmg where k is a constant. But Newton’s second law of motion relates the force F acting on a body to its acceleration a, which is how we arrived at the definition of inertia as given by F/a. We more commonly refer to the inertia as ‘inertial mass’ mi and define it by mi =F/a or F=mia. So if we look at a body in free fall where the only force on it is gravity, then the two equations combine to give kmg = mi a. If (and it is a big if at this stage) mg = mi , then the two masses cancel out and we get that a=k. Thus the constant k represents the acceleration due to free fall in gravity (which on the surface of the Earth is usually denoted by g) and is independent of the mass of the object, the result that has been famous since Galileo’s time. Galileo’s experiments tested this and confirmed the result to a sensitivity of one part in 50.

Since both gravitational mass and inertial mass are arrived at by dividing a force by an acceleration, they have the same dimensions. It would have been tempting to simply assume that they must be the same quantity. But that need not be so. Consider for example torque and work. Both are obtained by multiplying a force by a distance and hence they have the same dimensions. But they are two very different concepts serving very different dynamical purposes. In order to maintain the distinction, we give them different names and the units of torque are given as Nm (Newton.meters) while the units of work are given as J (Joules) even though both units have the same dimensions.

Given the importance of the result that the inertial and gravitational masses may be equal, there have been careful experiments done to test it, the most celebrated of which were by Loránd Eötvös (1848-1919) (https://en.wikipedia.org/wiki/Eötvös_experiment). Without going into details, the basic idea is similar to that of the torsion balance that had been developed and used earlier to determine Coulomb’s law for the force between two electric charges. Eötvös’s experiments showed, to a sensitivity of about one part in 107, that the gravitational and inertial masses were the same for any object, irrespective of what the object is made of and even of the location where the experiment was done. Those experiments have been repeated many times since with increasing sensitivity and the equivalence of gravitational and inertial masses are now said to hold to one part in 1015. So the equality of the two masses has been established empirically to a high degree of confidence.

In order to deal with all the subtleties involved, we have taken what must seem like a long-winded route to arrive at the conclusion that in order to have all objects in free-fall fall at the same rate, the gravitational mass of an object must be equal (or at least proportional) to its inertial mass. Some readers might be saying, “Well, duh! Isn’t it obvious that they must be the same?” The answer is “Not in the least.” The reason we may think it is obvious is because in teaching introductory physics, this subtlety is ignored and right from the beginning the two masses are tacitly assumed to be the same, starting with using the same word ‘mass’ for both inertial and gravitational masses. Things do seem obvious when something that really should require an explanation is simply assumed right at the beginning. Hence, that all objects fall at the same rate is seen as a trivial consequence of applying Newton’s law of gravitation and the second law of motion, hardly worth remarking on, and results in this subtlety never even being perceived. When generations of students read scientific textbooks that flatly assert something, it becomes conventional wisdom and ‘obviously’ true.

But the equality of the two masses is far from trivial. While the experiments of Eötvös and others empirically supported the idea that they are equal, there was no theoretical justification for it until Einstein used that curious feature to develop his ideas of general relativity.

(To be continued.)

1. Rob Grigjanis says

Although Galileo pointed out that all bodies in free fall do so at the same rate, and did some experiments that provided evidence in support of it, he did not give a dynamical reason for it

Sometimes you don’t need dynamics. The acceleration of a body just can’t be a function of any single extensive property of that body, unless you introduce some highly contrived rules about what constitutes a body.

2. dogfightwithdogma says

I once thought, like so many others, that mass is simply the “amount of stuff” (atoms or molecules, for example). But I recently refined my understanding after watching a physics lecture series on Wondrium (formerly the Great Courses) by Richard Wolfson at Middlebury College. In one lecture in which he discussed Newton’s laws of motion, he described mass as the measure of inertia. More recently I’ve been looking into the subject of the Higgs field. In the past several days I have read a number of articles about the Higgs field and Higgs Boson as well as watched several videos in which several different physicists explained both. If I understood correctly what I read and watched in the videos, can we not define mass in terms of the degree to which particles interact with the Higgs field? So I guess what I’m asking is how, if at all, does the Higgs Field factor into the discussion about the definition of mass and the questions you have raised in these essays? Or am I confused and asking a question that is irrelevant to this discussion?

3. Mano Singham says

dogfightwithdogma @#2,

Your question is by no means irrelevant but it is asking deeper questions about the origin of the mass of particles than this series is going to delve into.

But you are in luck! Back in 2013, I wrote a 20-part series on the three aspects of the Higgs mechanism, the Higgs field, and the Higgs boson that you can read here.

While you are welcome to read the series from beginning to end, there are three posts that deal specifically with how the Higgs field gives rise to the inertial mass of elementary particles that address your question. Those posts are #9, #11, and #12.

4. dogfightwithdogma says

5. Mano Singham says