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:

**it does what it does because it is the way it is**, and that**to make sense, an equation must be homogeneous in dimensions.**

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