A lot of attention has been given in this chapter to the power that Dimensional Analysis has in making the world a knowable place and in contributing to independently useful thought. As the reader works through the problems in these and other chapters, this point should be brought home far more effectively. On the other hand, physics does not end, and in fact only barely starts, with Dimensional Analysis. To understand why, we look at what we have achieved with this level of description, and what it leaves unaccomplished that we could want from more refined levels of description.
The Fermi problems of the last section were problems of estimation, and were therefore intrinsically plagued with imprecision, though by improving the quality of the numbers we used and testing the validity of our assumptions, we could reduce this imprecision, for some of the more concrete questions presumably to an arbitrary accuracy. On the other hand, the estimations of dimensional analysis include some matters of outright ignorance, where the methods themselves are incapable of providing, even approximately, certain aspects of the answers. Often these can be estimated themselves in the manner of Fermi problems, but that estimation is a separate process outside of the methods of Dimensional Analysis itself. We illustrate with an example.
Two sections ago, we used the laws for springs and for free fallers to predict a law for the deflections of spring scales. We did this on the basis of the units that had to appear for the law to make any sense. Plain counting numbers (1,2,3,...), though, have no units, so this constraint of dimensional homogeneity, while it could tell us in what combination the relevant dimensionful constants of the system had to appear, could not tell us whether that combination appeared multiplied overall by a factor of two, or 0.15, of 10, 000 or just 1. The same problem inflicted our prediction about pendulum clocks. While we could predict the relative behavior of such clocks on the earth and on the moon, we could not predict, up to such things as factors of two, the absolute behaviors of either of them without measuring it at least once. This is an intrinsic limitation of Dimensional Analysis that is built into its method for generating answers, and it provides a consequent limitation in the usefulness and confidence we can obtain in any of the answers we generate. Laws of this kind, that tell how the constants of the system have to be combined to predict that characteristics of the motion, and which therefore tell how those characteristics scale with the changes we make, but cannot tell about the absolute magnitudes, are called scaling laws. Dimensional Analysis is therefore a method for predicting the scaling laws relevant to a particular system.
Dimensional Analysis has a more severe limitation, though, which we have not been forced to acknowledge so far because the systems we have considered were relatively simple. However, it is easy to expose it simply by combining two of our previously considered systems.
Suppose we take our pendulum clock and attach the spring to the mass, as shown in fig. 3.6.
Now suppose we ask the same question; what period do we expect for the oscillations that we can excite. Certainly, since either the pendulum or the spring by itself oscillates, and since both of them do so with frequencies that do not depend on the magnitude of the oscillation, we expect the same to be true of this combined system, and would be surprised if it were not. And in fact, we can check this and find that it is a correct guess.
But now we have a problem in predicting the period of oscillation. Using either the constants of the system for the pendulum or for the spring, we could generate an expected period, and in general the two will not be the same. Yet, the real system has just one period of oscillation. How then should we predict it? From the forms of the results for the two individually, we could imagine many things, all of which are consistent with the requirements of dimensional homogeneity and the maxim that `it does what it does because it is the way it is'. For example, we could expect that:
or in fact many more things than that. Mathematically, the kind of trouble we are in can be seen as follows. Because both the combinations L/g And m/k have the units of , the combination (L/g)/(m/k) = (Lk/mg) has no units at all. Therefore, starting with any expression, like , (which would assert that the spring does not matter at all), we could insert any power, or in fact any expression at all, made from the dimensionless number (Lk/mg) into our previous expression and obtain another that satisfies the criteria of Dimensional Analysis. We have no way to choose among them.
This too is an intrinsic limitation of Dimensional Analysis, and it stems from the fact that a system can have several influences which are not entirely independent in their units, in which case the resulting motion can be a complicated function of any or all of them. Often this is precisely the case, and this is the biggest single reason that Dimensional Analysis is not an adequate description of nature, and much additional insight will be required to create a description that is.
In a remarkable fraction of cases, though, in spite of the fact that there is no reason they should be, physical systems nonetheless are simple enough that such ambiguities are not a problem. In some such cases, this may not be apparent from our initial observation, in which case Dimensional Analysis turns from being a predictor of outcomes into being a validator of assumptions. We obtain an example of this if we progress from our spring-pendulum system to a slightly more complicated one. We take now two pendula and hang them so that the spring connects, rather than either of them to a wall, the two of them to each other.
If we look at the resulting motions that are possible, we find that again they seem to be independent of amplitude as before, but this time they are much more complicated. In spite of this, if we are careful we can still isolate two simple and qualitatively different motions, as illustrated in fig. 3.6.
Not only do we find that the motions look different, in the sense that in one of them the two objects swing to the same side at the same time, whereas in the other they swing to opposite sides at the same time, but we notice that the two different kinds of swinging occur with different periods.
We then ask, is there a chance that we can predict either, or both of those characteristic periods. One way to argue would be ad hoc, based on cleverness. We can observe that in the motion with both objects swinging the same way at the same time, the spring neither stretches nor contracts at any point in the motion, so in a sense it may as well not be there. Thus, we can expect that for this motion the pendula will swing at the same rate as they each would individually. For the other motion, we could notice that the center of the spring never moves, so it may as well have been anchored to a wall, from the point of view of either pendulum, and we would expect that the resulting period is the same as it would be in the previous case with a spring of the appropriate length.
For the purposes of this example, that reasoning holds true, and to some degree deflates our punch line. On the other hand, understanding general methods is in a way better than cleverness, because one cannot count on seeing cleverly the way to break a system down as we did here, and it is easy to find systems which do not lend themselves readily to such analysis. Therefore, using our suspected answer simply to maintain orientation, we ask how this system looks, from the methodology of Dimensional Analysis.
Of course, we have the same problem we had for the single pendulum/spring system, in that there is an overall dimensionless number (Lk/Mg) that we can produce from the characteristics of the system, which permits an ambiguity in our predicted cycle periods. So we ask the question in the other order. Recall that the whole reason all of these terms appear is that we have assumed that they are relevant characteristics to the motion of the system. If, however, we can find some class of motions of the system that can be predicted without some of those numbers, we will have learned from Dimensional Analysis itself which parts of the system are important for which motions.
In other words we do the following. Being careful to excite only the collective motion, and then doing this with several different masses on the ends, or different lengths of the pendula, we see if we can find a scaling law that fits them. What we find (not surprisingly, having given the answer away) is that the scaling law for this motion is the same as it was for a single pedulum, that
and that the mass does not matter. Similarly we could, by trying various springs, find that the spring does not matter. However, notice that once we have found that the mass does not matter, we must conclude from the dimensions alone that the spring also cannot matter, because the constant that describes springs contain a factor of (m) in its units, and if the masses themselves do not matter to the motion, there is no other M against which to cancel the units of (m) in the spring constant, since neither g nor L can provide it. Thus by testing the motion and trying different scaling laws against it, we use the agreement of one and the disagreement of the other to discover that certain parts of the system are ignorable when producing certain kinds of motion.
This is a very important idea. Our clever analysis of the system hinted to us beforehand that for this motion the spring did not matter, so the result from Dimensional Analysis seems anticlimactic. However, it is not hard to think of systems where such a conclusion is not at all obvious. In such cases, the insight provided by Dimensional Analysis in distinguishing the important from the unimportant is invaluable.
Finally, we look at the other motion to see what we can discover. First of all, we can change the masses and the lengths, and observe that the resulting period for this motion does depend on both, so in general we are stuck with the result that the motion depends on both the spring and on gravity, at which point Dimensional Analysis alone is simply inadequate to tell us how to combine the various properties to produce the correct answer. We can notice that, when the pendula are made very long, the period of the resulting motion becomes independent of the length entirely, and depends only on the masses, which indicates that in this limit of `ignorably long' pendula, the frequency moves toward the result predicted for a system of object and spring alone, that . This final result does point out an interesting application of this system as a way to measure the strength of a planet's gravity, because in a sense it is two clocks in one.
The fast clock is made by the opposed swinging motion, which can be made effectively entirely dependent on the spring if we choose long pendula. Therefore it stays the same wherever the motion is made, on the earth, moon or wherever. On the other hand, the collective motion depends only on the gravity of the place and the length of the pendula in a way that we understand, so by taking this device from place to place we could use one clock, which never changes, to measure the behavior of the other, and hence the local value of little-g for that place.
And, it serves as a useful reminder that there simply are cases in which the behavior of the world is just complicated, because many competing effects work together to make it what it is.