Fear of Physics (2 page)

This old joke, if not very funny, does illustrate how—at least metaphorically—physicists picture the world. The set of tools physicists have to describe nature is limited. Most of the modern theories you read about began life as simple models by physicists who didn’t know how else to start to solve a problem. These simple little models are usually based on simpler little models, and so on, because the class of things that we
do
know how to solve exactly can be counted on the fingers of one, maybe two, hands. For the most part, physicists follow the same guidelines that have helped keep Hollywood movie producers rich: If it works, exploit it. If it still works, copy it.

I like the cow joke because it provides an allegory for thinking simply about the world, and it allows me to jump right in to an idea that doesn’t get written about too much, but that is essential for the everyday workings of science:
Before doing anything else, abstract out all irrelevant details!

There are two operatives here: abstract and irrelevant. Getting rid of irrelevant details is the first step in building any model of the world, and we do it subconsciously from the moment we are born. Doing it consciously is another matter. Overcoming the natural desire
not
to throw out unnecessary information is probably the hardest and most important part of learning physics. In addition, what may be irrelevant in a given situation is not universal but depends in most cases on what interests you. This leads us to the second operative word:
abstraction.
Of all the abstract thinking required in physics, probably the most challenging lies in choosing how to approach a problem. The mere description of movement along a straight line—the first major development in
modern physics—required enough abstraction that it largely eluded some pretty impressive intellects until Galileo, as I’ll discuss later. For now, let’s return to our physicist and his cow for an example of how useful even rather extreme abstraction can be.

Consider the following picture of a cow:

Now imagine a “supercow”—identical to a normal cow, except that all its dimensions are scaled up by a factor of two:

What’s the difference between these two cows? When we say one is twice as big as the other, what do we really mean? The supercow is twice the size, but is it twice as big? How much more does it weigh, for example? Well, if the cows are made of the same material, it is reasonable to expect that their weight will depend on the net amount of this material. The amount depends
upon the
volume
of the cow. For a complicated shape, it may be difficult to estimate the volume, but for a sphere it’s pretty easy. You may even remember from high school that if the radius is r, the volume is equal to (4π/3) r
³
. But we don’t have to know the exact volume of either cow here, just the ratio of their volumes. We can guess what this will be by recalling that volumes are quoted in cubic inches, cubic feet, cubic miles, and so on. The important word here is
cubic.
Thus, if I increase the linear dimensions of something by 2, its volume increases by the cube of 2, which is 2 × 2 × 2, or 8. So the supercow actually weighs 8 times as much as a normal cow. But what if I wanted to make clothes out of its hide? How much more hide would it yield than the normal cow? Well, the amount of hide increases as the surface area of the cow. If I increase the linear dimensions by 2, the surface area—measured in
square
inches, feet, miles, and so on—increases by the
square
of 2, or 4.

So a cow that is twice as “big” actually weighs 8 times as much and has 4 times as much skin holding it together. If you think about it, this means that the supercow has twice as much pressure pushing down on its skin as the normal cow does, due to its weight. If I keep increasing the size of a spherical cow, at a certain point the skin (or organs near the skin) will not have the strength to support this extra pressure and the cow will rupture! So there is a limit to how large even the most gifted rancher could breed his cows—not because of biology but because of the scaling laws of nature.

These scaling laws hold independent of the actual shape of the cow, so nothing is lost by imagining it as a simple shape like a sphere, for which everything can be calculated exactly. If I had tried to determine the volume of an irregularly shaped cow and to figure out how it changed as I doubled all the dimensions of the
animal, I would have gotten the same result but it would have been harder. So for my purposes here, a cow is a sphere!

Now, as we improve our approximation to the cow’s shape, we can discover new scaling relations. For example, picture a cow slightly more realistically, as follows:

The scaling arguments are still true not only for the whole cow but also for its individual parts. Thus, a supercow would now have a head 8 times more massive than that of a normal cow. Now consider the neck connecting the head to the body, represented here by a rod. The strength of this rod is proportional to its cross-sectional area (that is, a thicker rod will be stronger than a thinner rod made of the same material). A rod that is twice as thick has a cross-sectional area that is 4 times as large. So the weight of a supercow’s head is 8 times as great as that of a normal cow, but its neck is only 4 times stronger. Relative to a normal cow, the neck is only half as effective in holding up the head. If we were to keep increasing the dimensions of our supercow, the bones in its neck would rapidly become unable to support its head. This explains why dinosaurs’ heads had to be
so small in proportion to their gigantic bodies, and why the animals with the largest heads (in proportion to their bodies), such as dolphins and whales, live in the water: Objects act as if they are lighter in the water, so less strength is needed to hold up the weight of the head.

Now we can understand why the physicist in the story did not recommend producing bigger cows as a way to alleviate the milk production problem! More important, even using his naive abstraction, we were able to deduce some general principles about scaling in nature. Since all the scaling principles are largely independent of shape, we can use the simplest shapes possible to understand them.

There’s a lot more we could do with even this simple example, and I’ll come back to it. First I want to return to Galileo. Foremost among his accomplishments was the precedent he created 400 years ago for abstracting out irrelevancies when he literally
created
modern science by describing motion.

One of the most obvious traits about the world, which makes a general description of motion apparently impossible, is that everything moves differently. A feather wafts gently down when loosened from a flying bird, but pigeon droppings fall like a rock unerringly on your windshield. Bowling balls rolled haphazardly by a three-year-old serendipitously make their way all the way down the alley, while a lawn mower won’t move an inch on its own. Galileo recognized that this most obvious quality of the world is also its most irrelevant, at least as far as understanding motion is concerned. Marshall McLuhan may have said that the medium is the message, but Galileo had discovered much earlier that the medium only gets in the way. Philosophers before him had argued that a medium is essential to the very existence of
motion, but Galileo stated cogently that the
essence
of motion could be understood only by removing the confusion introduced by the particular circumstances in which moving objects find themselves: “Have you not observed that two bodies which fall in water, one with a speed a hundred times as great as that of the other, will fall in air with speeds so nearly equal that one will not surpass the other by as much as one hundredth part? Thus, for example, an egg made of marble will descend in water one hundred times more rapidly than a hen’s egg, while in air falling from a height of twenty cubits the one will fall short of the other by less than four finger-breadths.”

Based on this argument, he claimed, rightly, that if we ignore the effect of the medium, all objects will fall exactly the same way. Moreover, he prepared for the onslaught of criticism from those who were not prepared for his abstraction by defining the very essence of
irrelevant:
“I trust you will not follow the example of many others who divert the discussion from its main intent and fasten upon some statement of mine which lacks a hair’s-breadth of the truth and, under this hair, hide the fault of another which is as big as a ship’s cable.”
¹

This is exactly what he argued Aristotle had done by focusing not on the similarities in the motion of objects but on the differences that are attributable to the effect of a medium. In this sense, an “ideal” world in which there was no medium to get in the way was only a “hair’s-breadth” away from the real one.

Once this profound abstraction had been made, the rest was literally straightforward: Galileo argued that objects moving freely, without being subject to any external force, will continue to move “straight and forward”—along a straight line at constant velocity—regardless of their previous motion.

Galileo arrived at this result by turning to examples in which the medium exerts little effect, such as ice underfoot, to argue that objects will naturally continue at a constant velocity, without slowing down, speeding up, or turning. What Aristotle had claimed was the natural state of motion—approaching a state of rest—is then seen to be merely a complication arising from the existence of an outside medium.

Why was this observation so important? It removed the distinction between objects that move at a constant velocity and objects that stand still. They are alike because both sets of objects will continue doing what they were doing unless they are acted upon by something. The only distinction between
constant
velocity and
zero
velocity is the magnitude of the velocity—zero is just one of an infinite number of possibilities. This observation allowed Galileo to remove what had been the focus of studies of motion—namely, the position of objects—and to shift that focus to
how
the position was changing, that is, to whether or not the velocity was constant. Once you recognize that a body unaffected by any force will move at a constant velocity, then it is a smaller leap (although one that required Isaac Newton’s intellect to complete) to recognize that the effect of a force will be to
change
the velocity. The effect of a
constant
force will not be to change the
position
of an object by a constant amount, but rather its
velocity.
Similarly, a force that is changing will be reflected by a velocity whose
change
itself is changing! That is Newton’s Law. With it, the motion of all objects under the sun can be understood, and the nature of all the forces in nature—those things that are behind all change in the universe—can be probed: Modern physics becomes possible. And none of this would have been arrived at if Galileo had not thrown out the unnecessary details in order to
recognize that what really mattered was velocity, and whether or not it was constant.