## 4.3 Basic Force Laws

The “stage” on which inorganic or organic processes take place was thought before 1905 to be the ordinary three-dimensional space of Euclidean geometry, with change occurring in a medium called “time.” Einstein, in his theory of relativity, demonstrated the need for us to reshape our notions of space and time, since in different frames of reference (moving at different velocities with respect to each other), lengths and times will differ. Events which seem to occur simultaneously in one frame will not appear to be simultaneous to an observer in another frame. Relativity is of course a rather heady subject, and one which we won’t discuss further, but it points up the difficulties which arise in defining precisely and appreciating the most basic quantities in physics.

A notion of what “time” is, for example, is so ingrained in our everyday experience, that we hardly think it necessary to define it. (PROBLEM: Try to define time!) The best the dictionary seems able to do is to define time as the “*period* during which an action, process, etc., continues; measured or measurable *duration*.” (Emphasis added.) Thus “time” is a “period,” or a “duration.” What, pray, is a “period” or a “duration?” We may invent some more synonyms, but eventually we find ourselves caught in a circular definition…time is time (and then you wave your hand and say “You know what I mean!”). We are just going to have to content ourselves with this, provided we can give a fairly precise means of measuring that which we call “time.” One way is to divide the period of rotation of the earth into 86,400 equal parts and call it the “second.” This has been found to be insufficiently accurate for many reasons (one is that the earth is **not** that constant a clock). In 1960, the “second” was defined in an international agreement to be a certain fraction of the particular year beginning on the vernal equinox of 1900, ending on the vernal equinox of 1901. This being a rather difficult standard to move about from laboratory to laboratory (you can’t even go to Paris for it!), a recent definition has been provisionally accepted based on the number of oscillations of the cesium atom.

Without attempting to give a verbal definition of length (“You know what I mean!”), we give you the standard length of one meter, which was formerly defined to be the distance between two scratches on a platinum-iridium bar kept in Paris and measured under standard conditions of temperature and pressure. This standard was based on an old measure of the earth’s circumference, thought to have been 40,000 kilometers on a great circle passing through Greenwich, but the standard is now based on the wavelength of a particular line in the emission spectrum of krypton 86. This new primary standard is a more reproducible standard than the Pt-Ir bar (How wide is the scratch on the Pt-Ir bar? How closely can standard temperature and pressure be held?), and a more convenient measure since it can be maintained in one’s own laboratory for about the cost of three or four round trip fares to Paris.

The “actors” on our “stage” of three-dimensional space plus time are particles—atoms, molecules, protons, neutrons, electrons, etc. The particles are characterized by several fundamental properties, which when fully understood should, in theory at least, allow us to comprehend the most complicated of life processes. These properties (expressed as the *forces* which particles exert on each other) are surprisingly few in number; there are four basic force laws, (1) gravitational, (2) electromagnetic, (3,4) weak and strong nuclear, which are used in conjunction with the law of “inertial force” and which embrace all of what the physicist understands of the universe today. (There may be more force laws forthcoming, e.g., a law describing the force holding the constituent parts of a proton together, but it’s not likely that any newly discovered force laws will have a strong bearing on life processes.)

The atomic theory in conjunction with a knowledge of the force laws will allow us to view a gas as a collection of moving particles, whose pressure is the result of collisions of these particles with the walls of its containing vessel, or perhaps with your eardrums. We will be able to calculate pressure in terms of the “inertial force” (i.e., the change in momentum of the molecules as they bang into the wall and reverse their direction). The drift of the particles, if they’re all moving in one direction, will be called *wind*. If the motion of the particles is *random*, we shall call it *heat*. Ii the motion is in waves of excess density occurring at a regular frequency, we will know it as *sound*, whose pitch we’ll discover depends on the frequency.

The understanding of these things based on so few underlying principles, is a remarkable achievement. We shall proceed by discussing the notions of mass and force, and then by describing the force laws.

### 4.3.1 Inertia

In just the same way that “time” and “length” were so difficult, in fact impossible, to provide true definitions for (in the mathematical sense of definition), we shall find that “mass” and “force” must also be defined with some hand-waving. The student should not despair—this is not the fault of physics, nor of this presentation—it’s just the way life is. Even the mathematician must face up to a fundamental imperfection in his otherwise perfect discipline. For mathematics must eventually trace all of its definitions back to some primitive concept, the universally accepted primitive concept being that of the “set.” What is a “set” of objects? Well, it’s a “collection” of them. What’s a collection? It’s a “group.” Et cetera. Eventually we use up all our synonyms for “set,” and return to…“set.” The definition is circular. Thus mathematicians rely on grasping intuitively, without precise definition, the notion of set. Once that is accepted, of course, mathematics is on sound footing.

Thus we shall simultaneously introduce the ideas of force and mass (and hence, inertia) by presenting Newton’s first and second laws:

Newton’s first law (law of inertia): Every body will remain in a state of uniform motion unless acted on by external force.

Newton’s second law: The acceleration of a particle is directly proportional to the resultant external force acting on the particle, is inversely proportional to the mass of the particle, and has the same direction as the resultant force.

The second law is usually written:
\[\begin{equation}
F=ma
\tag{4.1}
\end{equation}\]
where

\(a\) = acceleration (change in velocity per unit time)

\(m\) = mass

\(F\) = force

If we know the meaning of position and time, then velocity (time rate of change of position, meters/sec) and hence acceleration (time rate of change of velocity, meters /sec^{2}) give us no problem. Let us say for the moment that we have some intuitive sense of what “mass” is. In fact, let us “define” one kilogram of mass to be the “quantity of matter” in a certain cylinder of platinum-iridium alloy preserved at the International Bureau of Weights and Measures in Paris, and let us measure unknown masses by balancing them opposite this standard (both masses presumably being acted on by the same “force” due to gravity).

We then might be inclined to “define” force in such a way that Newton’s second law holds. That is, if we observe that a body is either at rest or moving in a straight line at constant velocity (what’s a straight line?), we will say that no net force is acting on the body. Or contrariwise, that if the body is accelerating, then a net force must be acting on the body. This has the effect of rendering Newton’s second law as a mere definition with no physical content, and hence not an experimentally verifiable law of physics. However, the real content of Newton’s laws is supposed to be this: “that the force is supposed to have some *independent properties*, in addition to the law \(F = ma\); but the specific independent properties that the force has were not completely described by Newton or by anybody else, and therefore the physical law \(F = ma\) is an incomplete law. It implies that if we study the mass times the acceleration and call the product the force, i.e., if we study the characteristics of force as a program of interest, then we shall find that forces have some simplicity; the law is a good program for analyzing nature, it is a suggestion that the forces will be simple” (Feynman 1963, Ch. 12). Furthermore, there is the implication that forces are of material origin, that if a body is observed to accelerate, we will find some physical body nearby which is the source of that force. Thus we must consider simultaneously with Newton’s second law, the force laws associated with the nearby presence of matter.

This we’ll do, upon noting that if Newton’s second law holds, we may assign the units to force of the right-hand side of the expression, kg m/s^{2}, and since this is a clumsy unit, we shall call it the “newton”:

1 newton (nt) \(\equiv\) 1 kg m/s^{2}