All that we have done so far is not Dimensional Analysis per se. The invention of the concepts of measurement and equations were necessary to make definite statements at all, and the selection, among all the characteristics that a system may have, of those that are important to encoding the pattern of interest was one of trial and error. Through these we first developed the ability to state physical laws, which was necessarily a precursor to using them to do anything. Now it is time to show how the concepts of units and their rôle in equations, beyond enabling us to predict the behavior of objects from the form of a single law, in fact predict the forms of new laws from what we know about existing ones. That form of prediction is what we refer to as Dimensional Analysis. Much as we discovered the way to form laws by examining what we needed them to communicate, we will uncover the methods of dimensional analysis by `inventing' them to generate answers to certain new questions.

Our goal in the next exercise is to understand the relation of the concept of mass to that of weight. When we considered a rock in free fall, we found as the first observation that the particular ``amount of stuff'' comprising the rock, whether considered as mass or as weight, did not seem to matter. Then, when we considered the oscillitory motion of some object placed on the end of the spring, we found that the amount of stuff did matter, which led us to the concept of mass, but that in that case the notion of weight did not seem directly relevant to the oscillations, since the time they took to complete was the same whether the object was hanging under the influence of gravity or moving sideways in a way that made gravity irrelevant to the motion. In that way we were led to an identification of mass as the property of something that characterizes its resistance to being accelerated.

Yet from common experience we associate mass and weight. The understanding that objects do not like to be accelerated is in fact a common one, because when one places one's head in the path of a flying object, what the head does is deflect the object into a new direction from the one it had, in other words, the head accelerates it. And we understand that it is more unpleasant to use the head to accelerate large rocks than small ones, because they have a greater resistence to acceleration. At the same time, we recognize that the larger rocks are the harder ones to lift, which affords heads a measure of protection.

So from common experience, we not only recognize but also associate the characteristics of mass and weight, though from the oscillation experiments with the spring we observed that they are not the same property, because in particular we found a system that behaved the same way whether the earth was nearby or not. So, to understand weight, we choose a system that does the same thing a person does when considering weight, which is hold some object from falling freely. Since we are familiar with the spring already, we make a spring scale. In other words, we hang the object from the spring toward the earth. But this time, rather than making it oscillate and measuring the characteristic times, we recognize that the more mass is hung on the end of the spring, the further it hangs down in its steady state, and we ask whether there is a way to associate the amount of mass with the amount the spring stretches. Since it does much the same thing as an arm, the spring, by stretching, seems likely to be responding to the weight the same way as we do.

Now certainly, we could be as experimental with the hanging object as we were with the oscillating object, and measure its properties until we found the relation between the amount of stretch and the amount of stuff hung on the end, at which time we would have still a third physical law, applicable to objects that hang on springs in the presence of gravity.

But instead we will do something different. We have already made one law for gravity, which encodes the obvious pattern we see in the way things freely fall. And we have made another for springs, which encodes the way things attached to them are allowed to move. Now in this new circumstance, we have an object in the presence of gravity, and the only thing that is keeping it from moving freely is the spring attatched to it. So we could almost expect that there is nothing intrinsically new to this system beyond the content of systems we already understand, and that this case is simply a new application. Another way to say this is that, in having already identified our previous laws by measurement and trial and error, we have arrived at an understanding of what each of the parts of the system is, that is, what numbers represent each of their important properties. In this new circumstance, we would like to know what the parts do.

We reason our way to the answer, guided by what we have already done, as follows. Suppose that, one way or another, we had already found the law describing objects that hang from springs. What form would it have to have? In the last section, simply by construction, we arrived at an equation, having on one side a combination of characteristics of the motion that we could measure, and on the other side a number with the same combined dimensions which encoded the pattern or consistency described by that law. The important point here is that, in the equations that represent physical laws, numbers with dimensions must be compared to other numbers with the same dimensions. Any other kind of ``equation'' would make no sense, because it would not describe any actual measurement and comparison that could be made. That is why it was important before to notice that some dimensions are independent, or in other words that knowing one, such as a mass, was not a substitute for knowing another, like a volume. We describe this property, that the expressions on both sides of an equal sign have to have the same units for the equation to actually make any sense, by saying that an equation must be homogeneous in its dimensions. Obviously, the same thing that applies to statements that two things are equal (comparable) also applies to any statements that they can be added or subtracted, because that too involves just another form of comparison (one cannot subtract a weight from a length any more than one can substitute a weight for a length and get the same information).

So we conclude that, if we had the answer, it would have to have the form of a prediction:

The form of the equation follows from the requirement that it be homogeneous in its dimensions. But there is a far more fundamental restriction to which it is subject, which provides the central theme of Dimensional Analysis.

The whole advertized purpose of Dimensional Analysis, and the first step we have performed in all that has preceded, is the sorting out of the important from the unimportant. We found that neither mass nor weight mattered in the law for free fallers, which was represented in the fact that the equation encoding that law had no units of mass in any of its expressions. Similarly, since for the spring we could bump it into oscillations of any magnitude and still obtain the same time period, there was no measure of size, no unit of length, in the law for the spring. In some sense, right away, we worked to identify the characteristics of the system that were not relevant and did not need to be considered.

But implicitly long before this we assumed something much more sweeping. We did not consider the day of the week, or the positions of the stars, or the attitudes of a certain population of people. Implicitly, from experience we assumed that the behavior of the system should be determined by the properties of the system. This idea (which again seems obvious until one reads the history of alchemy, astrology and so forth, indicating that it has not always been considered so) has great power when combined with the property of dimensional homogeneity. This is because, if we assume that we understand the behavior of things influenced by gravity when we have formulated our gravity law, and things influenced by springs when we have formulated our spring law, then we do not expect anything new to be introduced to the laws describing the behavior of the system beyond what we already have, if the construction of the system includes nothing but the springs and gravity we have already addressed.

More concisely, we assume that ``It does what it does because it is the way it is''. This requires that, whatever the expression is in our new law, having the dimensions of a length, it must be built only from the numbers appearing in the previous equations, if indeed in them we have encoded all of the relevant patterns of the parts. Thus we must have as our law:

(Here M refers to the particular mass of the object, and m to the name of the dimension.)

Recalling from our previous expressions the units of each of these predictive numbers, we see that has units of , and has units of . Yet the answer we want can have only units of length, and in particular no units of time, so whatever combination is allowed must cancel all units of time, which is true of the units for the product , which are . Similarly, since the answer has no units of mass, we recognize how the mass of the object can be factored in to achieve this, since has units of . So we have arrived at a guess for the form of the new law, which is that:

where M is the mass of the particular object we consider, and and are the numbers we discovered previously as the content of the systems we refer to as (earth's gravity) and (spring). These three numbers are the ``what it is'' part of the description, and the predicted value of the length is the ``what it does'' part.

The first thing we can note is that according to our new law, when we put on twice the mass, the object should hang down twice as far. While simple, this was not obviously true, since in the case of fallers alone, when we dropped the object from twice the height it did not take twice as long to complete the fall. This is an easy relation to check experimentally, and if we do so we find that it is indeed correct. The second thing we note is that our new law also predicts the scale factor of the deflection, in terms of the experiments we have already done. In other words, the factor of proportionality between how much mass is hung and how far it hangs is not a new dimensionful number, but merely a combination of numbers we already claim to know. This prediction too can be checked, because we can simply directly measure the factor of proportionality and compare to the one predicted by the new law. Doing so again verifies that the new law is correct.

So at this point we can summarize Dimensional Analysis as the combination of the two ideas, that:

We now return to the question of weight with which we started. We now understand the deflection of the spring scale, and we have the law that encodes our understanding. So we ask if there is a way to write this law that identifies for us what is the characteristic of weight, as represented by measureable quantities, which is so familiar intuitively. We have noted that weight seems to increase when mass increases, and we have also noted that the difference between weight and mass is the influence of gravity. But presumably the spring itself has nothing to do with the weight of the object hung on its end, since we consider weight all the time in the absence of springs. The same is true of the amount of the deflection.

From these considerations, we note that it is possible to simply re-arrange the previous equation so that, rather than make a prediction about the value that will be observed for a measured length by placing it on one side of the equal sign, we instead isolate the properties of the system that depend only on the mass and the earth's gravity, equated to an expression that depends on whatever is left. We thus write:

and propose that all along what we have been identifying as weight is the combination Mg. In this new form for the law, we have expressed on one side the weight that characterizes the attraction of the object to the earth due to gravity, (much as we feel it when lifting things), and on the other side the properties and response of the scale that is being used to measure it. The property of the spring is encoded in the number k, and the response in the length of deflection L.

Now we have, not only an understanding of the concept of weight, along with a precise expression for it and the units () with which it is measured, but also another prediction, about how the same spring scale would function in a different gravitational environment, as on the moon. This is a deceptively powerful step we have taken. If you were asked, out of the blue, after having watched the pictures of the astronauts walking and dropping things on the moon, to estimate quantitatively how much things weigh on the moon, you might have thought you did not have enough information to do it. Yet we know the height of a person, whether he is on the earth or on the moon, and so we can estimate the height from which a hammer is dropped. And, if we can ascertain that the film is being played at the correct speed, we can simply watch and time the fall from any stopwatch, after which time we can make our own law for the moon's gravity by checking first whether the combination is the thing preserved on the moon as it is on the earth (it is), and then finding out which number, analogous to g but different in numerical value, is the correct one to describe the gravity of the moon. But then, since from our experiments with the spring hanging or supported on tables, we know that the special number k characterizing the spring does not depend on gravity, we infer that if the same spring scale were taken to the moon, it would obey the same law, but in place of the Mg = kL of the earth, we would have to expect . From nothing but the estimated heights and timed falls of the hammer, we have thus extracted an estimate of the weight of things on the moon relative to that on the earth, in this case as measured on a spring scale, but because of the way we have written the law, obviously our conclusion is independent of whether it was a spring scale or anything else that was used as the example in arriving at it.

This example is typical of the power imparted by Dimensional Analysis. We start with a set of statements and observations that seem so simple as to have almost no content at all. But by being careful of how we make our statements, we suddenly realize that the world is FULL of information that can be used to figure many things out, and that it is available to be used if we are prepared to notice it.

One final point about our last form for this law is worth noting, especially for the developments two chapters hence. This form of the equation provides a separation between the concept of weight, as an intrinsic property of the object in gravity, from the properties of the system affected by it, in this case the spring and its deflection. The fact that such separations are possible makes the various concepts useful, like that of weight which has a meaning independent of what is used to measure it. This particular combination, with the units of a mass times an acceleration, has proven so useful that it is given its own name, Force (often denoted F). The units of force are therefore . Force is important because it can very often be isolated in this way, as a circumstance on one side of an equation which is separable from the response it creates in some measuring part, like a spring. In this case, (Mg = F) = (kL = response) is another way to say that the force ``causes the deflection'' of the spring. We, as experiencers of events, are apt to recognize force because we are usually in the role of the spring, as the thing acted upon by a force (such as a weight) and thereby made to respond.

We have now seen examples of some of the ways in which a physicist uses what he can measure to define what he knows, by formulating laws, and then uses what he knows to predict what he does not, through Dimensional Analysis, and finally checks his predictions to see if his original assumptions about what was or was not important were correct. And along the way, with no more effort than that required to take notice, we found that we can predict the weight of objects on the moon by watching a picture on TV.

We close this discussion with a simple treatment of one more system, omitting the commentary, so that the reader can see the path of ideas from beginning to end. Most people have seen a grandfather clock or cuckoo clock at one time or another. In fact, these are the oldest kinds of clocks, because they are controlled by the motion of pendulum, a massive object hung on the end of a rod. Why do these clocks work? More importantly, how was it possible for people ever to design such clocks, which required that they understand ahead of time, how the parts should be made in order to do a certain job? The answer requires two facts.

First, it is possible to make a pendulum clock at all, because a pendulum is our second example of a system that possesses oscillitory motion (and a pendulum is a much simpler system to make than a good-quality spring, which is why these were the first clocks). Second, it is possible to predict what that period will be, because all of its characteristics are things that we already know, which means that we can apply our methods of Dimensional Analysis to it. Specifically, the object on the end of a pendulum is simply a mass that is influenced by gravity. Furthermore, if we use a very heavy mass on the end of a very light rod, the fact that the rod even has a mass can approximately be ignored. On the other hand, the length intrinsic to the pendulum itself is entirely determined by the rod, because that is the length from the pivot point to the massive end, which is not permitted to move.

In one way, the motion of the pendulum is almost as simple as that of the mass on the spring, and in another way it is much simpler. While it is not obvious from guessing, we can find from experiments (playing around) that, as long as we push the pendulum only a small angle to the side of its preferred resting position and let it go, its motion is cleanly cyclic, and the cycled period does not depend on how large the cycle is. This is just what we found for the spring. (Actually, this result is not quite as accurate for the pendulum as it is for a good spring, which is one reason pendulum clocks are not the common ones today. If we create very large oscillations, the cycle period does depend somewhat on the size of the oscillation. However, the fact that this dependence becomes ignorable faster than the oscillation becomes ignorable, as we go toward smaller and smaller oscillations, enables us to ignore this complication for the moment). The way in which the pendular motion is simpler is that the cycle time also does not depend on the mass hung at the end (as long as the mass of the light rod can be ignored). This fact could have been guessed, from dimensional considerations. For practical purposes, the only property that the rod has is length, and we already know that the motions of objects in gravity is encoded entirely in the constant , which has no dimensions of mass in it.

Since our wish it to predict a cycle time, and the only properties of this system are gravity and the length of the rod, the only guess we can make for the cycle time, built only from these properties, is

where is the length of the rod, from the pivot to the position of the heavy mass.

In this particular case, we can see one of the limitations of Dimensionl Analysis, which is its inability to provide exactly the right constants of proportionality. The relation we have just predicted gives just the right scaling behavior for the swinging of pendula. If we increase the length of the rod to twice what it was, the period becomes, not twice as long, but the square root of 2 times as long (about 1.4 times). Similarly, if we were to take the pendulum to the moon, where the value of is different, we could find that the resulting differences in period were again correctly predicted.

However, if this had been the first experiment we had done, rather than a prediction made on dimensional grounds, we would have found that the measured cycle period is proportional to the number , but the proportionality constant is not . Rather, it is almost 40 times as long. The exact number happens to be . This can be seen as telling us, in secret, something about the qualitative difference in the way these scale factors are expressed in different systems, which information we have chosen to discard (indeed, which we had no good way to quantitatively keep). The constant was originally measured in context of the ever-downward accelleration of fallers. In this application, it is expressed in the context of the back-and-forth cyclic motion of oscillations. The difference in context is responsible for the fact that it arises with different scale factors. If we had first measured through its action on pendula, we could as easily have defined it to include the factor of . In that case, the relation for fallers would have been the one to differ.

(Now compare spring clocks to pendulum clocks, for their dependence on presence of the earth, etc. Set up various HW's about walking men, etc.)


Thu Aug 31 12:01:42 CDT 1995