## It is F=ma, you know

“It is cheese. (Caerphilly.)”

Main research finding of the first manned Welsh mission to the Moon, as reported by Max Boyce c. 1974.

### A Brief History of F=ma

When writing A Brief History Of Time, it is said that a literary agent warned Stephen Hawking that each mathematical equation that he included in the final draft would halve its eventual sales.

And one of the complex equations that Hawking wished to include? A summary of Newton’s 2nd Law: F=ma.

In other words, the force F acting on an object is equal to the mass m of the object multiplied by the acceleration experienced by the object.

(In the end, Hawking opted to include only Einstein’s E=mc2 in what turned out to be the ultra bestselling A Brief History of Time.)

### F=ma is Newton’ s 2nd Law: NOT!!!

@SciByDegrees wrote an interesting post arguing that the old Physics teacher’s standby of summarising Newton’s Second Law of Motion (N2) as F=ma is wrong.

The gist of his argument (and it’s a hard argument to counter) is that F=d(mv)/dt is a far better expression of the law than the F=ma version because it covers a wider range of circumstances.

This states N2 in terms of momentum, where momentum is the product of mass m multiplied by the velocity v. More exactly, it says that force acting on an object is equal to the object’s rate of change of momentum: or, if you prefer, the change in momentum divided by the time taken for the change is equal to the force.

This is the version of N2 stated in most dictionaries of Physics. For example, the Oxford Dictionary of Physics (2015) p. 383.

I know, because on reading @SciByDegrees’ post I immediately looked up N2 in the Dictionary with the express intention of countering the argument. Imagine my consternation and horror when I found that I was wrong. (Actually, not that much consternation and horror: I am fairly inured to being wrong as it happens fairly often…)

The argument suggests that, just like V=IR is not a statement of Ohm’s Law unless R has a fixed value (like a fixed length of wire at a constant temperature), F=ma is not a sufficient statement of N2 unless the mass m is constant.

For example, if we consider a rocket capable of producing a steady 1000 N of thrust; at t=0 its mass is (say) 10 kg so its acceleration is 100 m/s2. However at t=1 s its mass has decreased by 1 kg so the acceleration is now 111 m/s2 even though the thrust is still 1000 N so obviously F is not proportional to m so F does not equal ma in this situation.

### Feynman as the new Aristotle

Richard Feynman (1965) wrote along similar lines in his justly famous Lectures on Physics:

Thus at the beginning we take several things for granted. First, that the mass of an object is constant; it isn’t really, but we shall start out with the Newtonian approximation that mass is constant, the same all the time, and that, further, when we put two objects together, their masses add. These ideas were of course implied by Newton when he wrote his equation, for otherwise it is meaningless. For example, suppose the mass varied inversely as the velocity; then the momentum would never change in any circumstance, so the law means nothing unless you know how the mass changes with velocity. At first we say, it does not change.

However, I think Feynman is considerably oversimplifying what Newton said here. Dare one suppose that Feynman, who had an enviable natural facility for talking intelligently and arrestingly about nearly any subject under the Sun, had perhaps skimped a little on his background reading?

Incidentally, does anyone think that physicists (especially physics educators — myself included) are beginning to treat Feynman as the medieval scholastics are reputed to have treated Aristotle? That is to say, he is regarded as the final word on everything; or, at least, everything physics-related in the case of Feynman.

### What Would Newton Do?

George Smith (2008) points out that:

The modern F=ma form of Newton’s second law nowhere occurs in any edition of the Principia [ . . . ] Instead, it has the following formulation in all three editions: A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. In the body of the Principia this law is applied both to discrete cases, in which an instantaneous impulse such as from impact is effecting the change in motion, and to continuously acting cases, such as the change in motion in the continuous deceleration of a body moving in a resisting medium. Newton thus appears to have intended his second law to be neutral between discrete forces (that is, what we now call impulses) and continuous forces.

This, I think, supports my contention that F=ma is as good a modern reformulation of Newton’s 2nd Law as any other.

If we go back to the rocket example, the instantaneous acceleration at t=0 and t=1 s can be calculated using F=ma (provided we take account of the change in m, of course). In effect, we are considering the change in motion due an instantaneous impulse here.

Please note that I would cheerfully concede that F=d(mv)/dt would yield a better and more productive analysis of rocket motion if we are considering the continuous action of the force over time rather than at isolated instants.

The analogy with V=IR is useful here. V is always equal to I times R but V is only directly proportional to I over a continuous range of values of I for a limited set of conductors we call Ohmic conductors whose resistance R is fixed over a range of physical conditions. Likewise, F is always equal to m times a but F is only directly proportional to a for a continuous range of values of a when we are considering a system whose mass is fixed.

As V=IR is neutral with respect to whether R is fixed is not, I believe that F=ma is neutral with respect to whether m is fixed or not.

### Will the Real Second Law Please Stand Up?

What is Newton’s Second Law? Is it a definition of force? Is it a definition of mass? Or is it an empirical proposition linking force, mass and acceleration?

Brian Ellis (1965) argues that it partakes of all three:

Consider how Newton’s second law is actually used. In some fields it is unquestionably true that Newton’s second law is used to define a scale of force. How else, for example, can we measure interplanetary gravitational forces? But it is also unquestionably true that Newton’s second law is sometimes used to define a scale of mass. Consider, for example, the use of mass spectrography. And in yet other fields, where force, mass and acceleration are all easily and independently measurable, Newton’s second law of motion functions as an empirical correlation between these three quantities. Consider, for example, the application of Newton’s second law in ballistics and rocketry [ . . .] To suppose that Newton’s second law of motion, or any law for that matter, must have a unique role that we can describe generally and call the logical status is an unfounded and unjustifiable supposition.

In some senses, I suppose we might like those unfortunate nations in Gulliver’s Travels who fought a long and bitter war over the question of whether one should eat a boiled egg from the pointed or rounded end:

During the course of these troubles, the emperors of Blefusca … accusing us of making a schism in religion, by offending against a fundamental doctrine of our great prophet Lustrog, in the fifty-fourth chapter of the Blundecral (which is their Alcoran). This, however, is thought to be a mere strain upon the text; for the words are these: ‘that all true believers break their eggs at the convenient end.’ And which is the convenient end, seems, in my humble opinion to be left to every man’s conscience.

REFERENCES

Ellis, Brian. “The origin and nature of Newton’s laws of motion.” Beyond the edge of certainty (1965): 29-68.

Feynman, R. P., Leighton, R. B., & Sands, M. (1965). The Feynman Lectures on Physics; vol. 1 (Accessed from http://www.feynmanlectures.caltech.edu/I_09.html on 11/4/19)

Smith, George, “Newton’s Philosophiae Naturalis Principia Mathematica“, The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2008/entries/newton-principia/&gt;.ewto