In order to illustrate the way Dimensional Analysis emerges from the patterns that are already familiar to all of us, but are often not useable as we most naively recognize them, and to establish a paradigm for the more general process of building physical theories, we will focus attention on one or a few physical systems and consider them repeatedly, describing them as they appear at the various levels of our understanding.

The system of interest is one of the oldest in human experience; a dropped rock. It is worth noticing that people have been watching things fall for a lot longer than the 6000 years on historical record, and yet the precise quantitative description that has finally made the phenomenon useful has existed for only the last four hundred or so. This is itself remarkable, and should provide a caution that in spite of the fact that the inventions of physics can seem obvious almost to the point of triviality once they have been learned, one should not underestimate their depth, since they were not obviously trivial to attain.

Everyone recognizes intuitively that falling objects have some consistency to their motions. Over and over, rocks fall `in the same way'. If a rock ever fell differently, it would suddenly seem glaringly out of place in the pattern of natural events. A nice example of this sense of dislocation comes while watching the television picture of a hammer dropped by astronauts on the surface of the moon, or of the walking of the astronauts themselves. If the same motion were observed on the surface of the earth in everyday life and not in a television image, it would have a certain shock value.

Yet at the intuitive level, how would such a violation of the
pattern of nature be described, and what practial use would it
have? Most likely, the intuitive description would be that the
moon hammer falls `too slowly'. But this by itself is not only
imprecise and therefore not very useful, it is also ambiguous.
What does `slowly' mean, and what aspect of the motion would
have to be changed to make it `normal' again? We can discover
the lack in this description by imagining using it to
communicate. We see immediately that we have not said `too
slowly' compared to what. This leads us to our first important
notion, that of a **standard**.
A standard is some reference
object, together with a prescription for comparing it to other
objects (such as a meter stick, with a prescription for
comparing its useful characteristic, length, to the lengths of
other things with the same characteristic). Obviously, if we
have to describe `too slowly' to someone who didn't experience
the event, we must compare it to something that he can
experience. So the standard should be either reproducable or
transferible to be useful.

(Tangent on arbitrariness of standards, which permits valuation based on convenience, and the fact that there is nothing indemic in the notion of a standard that requires that it not be arbitrary, except a certain convenience imparted by universality.

This important observation, that to make a useful description we
need comparison, leads us to the idea of **measurement**.
Measurement is the precise refinement of the idea of comparison,
because it defines the notion of *quantitative comparison*. It is
generally not enough to frame a useful description, to say that
something has the property of length, and that it is longer or shorter
than a meterstick which has the same property. What we want is to be
able to say, even if only approximately, that the object of interest
is as long as *a certain number* of metersticks laid end to end,
or if it is shorter, how we could break up a meterstick into smaller
equal pieces, in which case *a certain number of those* laid end
to end would have the same length. It is the notion of measurement as
the proper form of comparison that will cure our descriptions of their
vagueness, once we know how to construct descriptions at all.

Still, even among familiarly measureable objects, there is more than one characteristic available for comparison. Something may have a length, or some time may be be taken for a set of events to happen, or one thing may be heavier when compared to something else on a balance, and so forth, and among these different recognizeable characteristics it is not clear that any one or combination of them can be used as a substitute for the others in conveying necessary information.

Moreover, while length, weight and time are familiar characteristics to measure, it is not obvious what is the relevant characteristic to measure to encode the pattern we see as important to our falling rock. A tremendous discovery of Galilleo (which apparently took several tens of millions of years, following the recognizeable origin of man, to make) is that the `rate' at which something falls is independent of its weight, as long as it is compact enough that the influence of air can be neglected within the acuracy of the observation. And, obviously from experience, we can drop objects from any different collection of heights, from which they then take many different times to complete their falling. So it is not obvious that length or time have any direct bearing on the pattern we are trying to make explicit.

Yet here we proceed with caution. There are endless physical
characteristic that things can have, but some are not *as*
mutually different as others are. For example, the notion of
the volume of something is a different one from the notion of
the length of something else. So we might be tempted to say
that length and volume are different standards. Yet, certainly
for simple objects that happen to be shaped like solid
rectangles, if we have the ability to measure the lengths of the
sides, we can easily express the volume as the product of the
three lengths. And conversely, if we have some way of measuring
the volume and at least saying what ratio the lengths of the
sides have to each other, we can work backwards and deduce each
of the lengths themselves. Thus, the ability to measure one of
the characteristics is sufficient for us to determine both of
them. In that way they are not independent. And obviously,
even when the object in question is not simply square, with a
little resourcefulness we can still make the measurements
interchangeable, simply by immersing the object in a liquid and
measuring the amount of liquid displaced in a rectangular
container. The fact that there is indeed such a stuff as a
liquid is our evidence that the relation of the notion of a
volume to that of a length is not dependent on particular
shapes.

This brings up the notion of **units**.
Basically, the fact
that a volume can be either compared to a collection of other
volumes *or* to the product of three lengths allows us to
say that, whatever they are, the *units* of volume are the
same as the third power of the *units* of length, and it is
this relation that allows us to convert measurements of one back
and forth to measurements of the other.

The same is not obviously true for weight and length, or more
directly for weight and volume. Since natural objects can come
in many compositions, it is common for two things that have been
cut to the same volume to have very different weights, so
knowing the measure of the volumes tells us nothing directly
about the measure of the weights. Thus there is an important
notion of the **dependence**
or **independence**
of units,
which represents the fact that different characteristics may
indeed be related in such a way that one of them (like volume)
can be written as some power of others of them (like the third
power of length), while others simply cannot.

So while we have seen that weight seems to be totally irrelevant to describing or measuring the pattern characteristic of falling objects, and lengths and times are also not useful in their most directly measured forms, we examine the pattern again to see if this notion of dependence of units applies to it in a way we have overlooked. First, what characteristics are there? As noted, obvious ones for a dropped object are length fallen and time taken to do it.

There is also the notion of speed, familiar for millenia from
experience. Clearly speed, like distance, admits some ordering
in terms of ``more'' and ``less'', as there are very
matter-of-fact consequences arising because some things move
``faster'' than others. Another possibly relevant notion is
that of acceleration, familiar from the feeling one experiences
of being accelerated in a car, for example. (These notions are
being described vaguely on purpose, to make the point that this
is how intuition ``in its raw state'' recognizes them. It is
important to understand the difference between ways like these
of describing comparable properties, and the more concrete ways
that are about to be introduced, because the more concrete
descriptions will make it possible to *think* things that
simply cannot be framed well within this more vague language.)

We know that speed is not the relevant characteristic to encode the pattern of a falling rock, because just as we can drop it from different heights and cause it to take different times to fall, we can cause it to have different speeds (visibly so) at the end of the fall, and indeed the speed clearly grows from moment to moment on the way. Also, unlike length, which we can easily measure quantitatively by stacking meter sticks, or time by successively starting and stopping watches or counting heartbeats, it is not obviously so easy to measure quantitatively when one thing is ``twice as fast'' as another. From the way we have recognized acceleration so far, in terms of a feeling it produces in us when we are being accelerated, it is even less obvious how to make direct quantitative comparisons to the ``acceleration'' of a falling rock. That is why we were limited in our original statement to vague words like ``too slowly''. The notion of a special kind of mutual dependence that units can have will enable us to overcome these deficiencies.

For instance, for speed we know that there is a dependence of
its units on other things, because when something travels at the
same speed on different occasions, it always covers the same
distance in the same amount of time. When it travels what we
have come to call `faster' it covers more distance in the same
amount of time, and so forth. So we have come to identify the
intuitive notion which we recognize as **speed**
with the *ratio* of *distance* traveled to *time* taken to
travel it;

In this way we have created a definite notion of ``twice as fast'', meaning that twice the distance is covered in the same time. Thus, we not only can measure speed, but can measure it quantitatively in the same way as we do lengths and times, precisely because of our ability to incrementally measure lengths and times. This is an example of a different relation than that between length and volume. Rather than, as volume was, being a power of just one other characteristic, length, in this case speed is the ratio of a length and a time. Knowing a length and a time is knowing more than just a speed, so while a length and a time are sufficient to determine a speed, the converse is not true. In such a case, any two are enough to determine the third.

In just the same way as the intuitive concept of speed has been
made precise as the ratio of a measureable distance and a
measureable time, **acceleration**
has been made precise as
the ratio of a *speed* and a *time*.

Whether the speed is itself measured directly (by some means not yet considered) or expressed indirectly as a ratio of a length and a time, having the ability to measure those two, one has the ability to measure acceleration, and so also to say quantitatively when one acceleration is twice as big as another, or one half as big, or whatever. Because we have defined acceleration in this way, it follows that the units of acceleration are the same as the units of a speed divided by the units of a time, or equivalently as the units of a length divided by two powers of the units of a time.

This dependence of units is our first step deeper into
Dimensional Analysis. Notice that whenever something has a
given speed that does not change, it can cover many different
distances in many different times, and the speed alone is not
enough to determine which it does. On the other hand, as long
as the speed is constant the distance covered and the time taken
*are* enough to determine what the speed is, and moreover it
does not matter which distance or which time taken we choose,
because they are all equally useable. Obviously, since our
definition of the concept has been constructed in the same way,
the same must be true for acceleration.

So, simply by trial and error, we can drop many objects, and for
each case, measure the characteristics that are simplest,
weights, lengths dropped, and times taken to fall, and see if
any combination of them has a pattern that can be taken as the
precise *measureable equivalent* of the pattern we intuite.
First, checking Galilleo, we confirm that the weight does not
seem to matter. And, as noted, we see that for different
heights and different times, the numbers we get with the units
of speed by taking their ratios are different from one drop to
another.

But remarkably, for every drop we make, if we simply measure the
distance fallen, and the time taken to complete the fall, and
compute the ratio
, which we have learned to
call the characteristic of acceleration, we find that this
combination is the *always the same*. The distances can be
long or short, as can the times to fall, but any increase in the
distance is always accompanied by an increase in the time that
exactly preserves the value we find for this ratio.

In having made this discovery, we have just created our first
law of physics from a familiar pattern of human experience.
Understand what has been done: First we noticed that *there
was a pattern*. In other words, there were many things about
the way we perform the action, the dropping of a rock, that we
could change, such as the height, the weight of the rock, or
even the choice to throw it sideways rather than to drop it
straight down, through all of which *something recognizable*
yet remained the same. Something about the fact that it was a
compact object, that it was near the Earth, and that ``gravity
exists as a characteristic of nature'' leads to the consequence
that under a variety of circumstances with rocks and the earth,
something about the behavior of all of them is the same.

This idea of a thing that stays the same simply in consequence of what the system in question Is is important enough that we give it a name for future reference. We refer to such numbers as `constants of the system', because they are the things that, no matter how we change the particular circumstances that determine which particular motion is taken, remain as common, unchanged, or `constant' properties of all of the motions, simply because of the properties of the system for which the motion is created.

Then, starting with an inability to state anything precisely, we
learned that using the concept of quantitative comparison,
measurement, is the way to overcome this inability and obtain
precise statements. Finally, with the recognition that *something* is unchanging and the ability to talk about things
precisely (in terms of measurement), we simply *tried*
various combinations of the things we could measure, and
eventually found that *the reason that something we
recognize stays the same, i.e., that a pattern exists at all, is
that some combination of the things we can measure also stays
the same.*

This is the basic principle underlying all of physics. The very statement that a pattern exists is tantamount to the assumption that something quantifyable and definite can be found, which under all of the changes of circumstance, takes the same exact value. When we are able to discover what that quantity is, and how to relate it to characteristics of the system and its behavior that we can measure, we call the resulting statement a physical law. (It is not the time here to digress on matters of philosophy, but it can be argued that even the use of the word `thing' assumes a certain continuity of existence as a recognizable object or entity, and that it can indeed only be used consistently along with the assumption that the `thing' in question possesses certain patterns or consistencies, and that therefore it is describable eventually by a law of physics. In that sense, the entire structure of our language, as it has evoloved through our experience of remembering, imagining, choosing and acting and experiencing the consequences of our actions, contains implicitly the seeds of the assumptions made explicit by physics, and cannot even be used consistently in contradiction of them.)

The second system, which will be of interest to us over and over again in what follows, consists of some object attached to the end of a spring. The typical example we imagine is shown in fig. 3.2.

We suppose that one end of the spring is attached to a solid wall or
some other base that can be taken, for practical purposes, to be
``immovable''. Then the object of interest to us, like a rock, is
attached to the other end of the spring, so that it can bounce back
and forth. We will usually make the discussion as simple as possible
by only worrying about the motion of the object *toward or away
from* the wall. That way we only have to consider motion in one
direction.

This system is interesting because, in a certain way, it parallels the
system of the falling rock that we have just considered, while in
other ways it is different and even complementary. The rock fell
because the earth was nearby, though there was no obvious sense in
which it was ``attached to'' the earth. Similarly, an object on the
spring can bounce back and forth precisely because the spring is there
to move it, but in this case only because it *is* directly
attached to the spring. So the spring is like the action of Earth's
gravity in the way it makes things move, but different in the way it
requires a specific attachment to do so.

Like the falling motion, the bouncing motion has particular characteristics that we would like to encode. Also, as the property of a falling motion told us something about the earth (the answer would have been different on the moon), we expect whatever we discover about the bouncing motion to tell us something about the spring. Therefore, in exact parallel, we imagine taking a collection of objects, attaching each in turn to the spring, measuring aspects of the motion, and simply by trial and error finding out what combination of measurements encodes the pattern. As before, we assume all the objects are compact, so that we ignore any influences contributed by resistance from the air.

The first thing we notice is that the object has a preferred *resting place* at some definite distance from the wall. Second, when
we either pull it out or push it in from that place, and then release
it, it undergoes *oscillation*. That means that it repeats a
certain motion, toward and away from the wall, over and over again,
and any one *cycle* of the oscillation looks the same as any other
cycle. In particular, if we use a very good-quality spring, and if
the air really is ignorable, we find that, within each oscillation,
the object begins by moving from where we started it *back toward*
the preferred resting place. It then overshoots that place, goes *as far out of the way* to the other side as it was on the starting
side, and then comes back *precisely to where we started it*.
This is what makes oscillitory motion special: it naturally divides
itself up into identifyable units, the cycles, which are repeated over
and over again in identical form. Thus, any cycle is representative
of the whole motion, and we can talk of the properties of the whole
motion in terms of the properties of any cycle.

The fact that, at the beginning and end of each cycle, the object
returns to precisely the same place, means that there is a definite
notion of *length* associated with a cycle. The easiest one to
measure is the distance from the preferred resting place to the place
where we started the spring. Next, the fact that each cycle can be
cleanly assigned a beginning and an ending moment, by noting when the
object passes through that starting place, means we can associate a
precise *time* with the cycle, called its **period**,
which is the time we measure between its beginning and its end. (A
complementary notion to that of period is **frequency**,
which is the *number* of cycles completed in a given unit of time.
Thus frequency has units of (1/time).) Finally, and obviously, since
we can place different objects on the spring, they can have different
masses, which we can measure.

Simply by changing different aspects of the way we start the motion,
we can notice the following things: Whereas the mass of an object did
not affect its falling motion at all, it affects the motion on the end
of a spring very much. A very low-mass object on the end of the
spring will bounce back and forth very fast, that is, it will have a
very *short* period, while a very high-mass object will have a
very *long* period. Then, while the length through which
something was dropped clearly determined how long it took to fall,
with the spring we notice something rather remarkable. No matter to
what length we stretch the spring (*i.e.*, no matter what length
the cycle has), the *period* of the cycle stays *exactly the
same*. So whatever the pattern is for objects on springs, it seems to
have no dependence on length, even though the way we start the motion
can. Thus, the spring pattern involves only a relation between the
mass of the object and the period (characteristic time) of the cycle.
The particular combination that, in this case, always stays the same,
is . This is the constant of
the system for all motions of a particular object attached to a
particular spring. Since everything important about the object is
encoded in its mass, this remaining constant that characterizes all
possible motions must be a characteristic of the spring itself.

The process of Dimensional Analysis at every stage involves references
to these combinations of measured quantities that do not change, and
it becomes cumbersome to write out whole words each time. Therefore,
rather than write ``mass'', ``length'', and ``time'' (or ``cycle
period''), we will abbreviate these as , and , so that the
constant of the system for fallers becomes , and that for
objects on springs , even though the *kind* of time
interval we measured in the two cases was very different. Thus, this
kind of notation refers only to the *dimensional* content of the
pattern, and it drops all reference to *which* mass, length,
time,or whatever, was measured. That means that, in our law for
patterns, we are choosing to ignore all of the *qualitative*
information that makes the motion special, like the fact that objects
on springs have this special property of oscillitory motion, while
fallers do not.

At first, it might seem that the whole reason the dimensional content
of the pattern matters is that it describes a particular aspect of the
qualitative behavior, and so is given meaning in the context of that
qualitative behavior. The whole center of Dimensional Analysis, which
in fact makes it possible to define as a method, is that the *dimensional content* turns out to be the part that has re-useable
meaning. This may seem surprising, which is why it is important. The
particular qualitative behavior of motions can vary endlessly from one
system to another, and need not be repeatable. For instance, the
oscillitory motion of objects on springs is completely different from
the way the same objects can hang and stretch the spring *at rest*
if they are also acted on gravity. Yet, while the qualitative
behavior is not preserved at all, the *dimensional content* of
both of these properties turns out to be related. That is why it is
not only allowable, but in fact *the useful thing to do* to drop
reference to the qualitative context in which a particular dimension
can be measured (such the context of cyclic motion that makes it
possible to measure a cycle period), and keep only the characteristic
time value that arises from the measurement.

For now, consider just one spring, and regard the constant
as an equivalent ``characteristic of nature'' to the
constant of the last section. To the extent that we want
to compare *only* the motions of different objects, all
placed on the end of the *same* spring, the two constants
perform the same function (This will be important two chapters
hence, when we have reason to identify equivalently the earth's
gravity and the end of the spring as ``external worlds'' for an
arbitrary object. Later we can note that we could have changed
the spring and discussed a different aspect of the law and hence
changed its rôle from that of external world, but then this
is not different in principle from another change we could make,
by going to another planet, except that the latter takes more
effort. Thus, in whichever capacity, as external world or
accident of circumstance, the two really are properly regarded
in the same way).

Next, just by experimenting and finding out what happens, we can
notice that both the existence of oscillation as a
characteristic of the motion, and more remarkably also the
particular value we measure for the period of the oscillations,
does not depend on whether the object is hanging under the
influence of gravity or supported, for example on an air table,
so that gravity is irrelevant to its motion. It is worthwhile to
note *en passant* that this air-table motion serves as a
convenient model for the motions of things in space far from the
earth, because we would expect the only major difference in that
case to come from the fact that the earth would no longer be
very nearby, in which case its gravitational effects should
become ignorable. And on the air table, where we prevent all
motion in the only direction singled out by gravity (toward the
center of the earth), we have effectively ``removed'' its
influence from the motions left for us to observe. This
motivates the exceedingly important observation that *the
notion of mass is not intrinsically dependent on the motion
toward the earth that we most immediately associate with
gravity*.

The notion of mass, and what it signifies, is important enough
to merit a small digression from the analysis of measurement, to
address it as a topic in its own right. It seems clear that
mass, as the refinement of the notion of ``how much of something
there is'', has meaning independently of the presence of the
earth, because it creates the same kind of oscillations on a
spring without regard to where, or even *whether*, the earth
is located relative to it. But how can this notion be made
precise, and what is different in the notion of mass obtained
from the spring from the notion obtained from ``heaviness''? A
common way to compare masses that makes no explicit reference to
``heaviness'' is to put the the two things to be compared on
either side of a balance. The side with more stuff goes down
and the side with less stuff goes up. This defines the meaning
of more and less with mass. And as with meter sticks, we can
compare one block to several other blocks, to say how many of
them contain the same mass, by finding when the arm lies level,
or cut a larger block into several equal pieces to compare parts
of one thing to another, *etc*. Unlike the situation when
the mass is on a spring, there is no sense in which the balance
really ``feels'' the heaviness of the objects, because it is
simply comparing the influence on one side to that on the other.

One might say that it is still heaviness that is responsible for
way the side of the balance with more ``stuff'' on it moves down
and forces the other side up. But really there is no explicit
reliance on heaviness in this comparison, as we could see by
turning the balance sidewise and laying its arm, and the two
masses, on an air table. Of course, if they are just laying
there, they tell us nothing. But we could *push* against
the base of the balance, forcing its arm to move, and to try to
drag the masses with it. In this case, we would find that the
side with more ``stuff'' does less moving in reponse to our
push, forcing the other side to make up the excess. Or in a
variation of the same method, we could push and pull the base of
the balance back and forth, and the side with more stuff would
tend to remaing closer to still, while the other side would be
flung back and forth wildly. So the real property that we would
like to identify as mass can clearly be identified and even
measured without making any reference to anything having to do
with the earth or with motion toward the earth. The presence of
the earth was other thing we needed *in addition to* the
presence of mass to produce heaviness.

In this way we see the real essense of the notion of
mass. **Mass**
is
the property of an object that characterizes its
resistence to being pushed around (more precisely, to being *accelerated*). The fact that we can make the balance on the air
table (or one far out in space, for that matter) look just like
the balance right-side up in gravity, just by pushing up on the
base, will become very important to us when we discuss
Einstein's principle of equivalence in General Relativity.

So now we have obtained two physical laws, one that describes the
pattern of falling objects near the earth, and one that describes the
oscillating behavior of objects placed on the end of a certain spring.
Our statements of the laws have taken the general form: ``All of the
particular motions that these objects undergo share the pattern that
such-and-such a combination of the measureable characteristics of the
system and its motion always turn out to be the same.'' In its
essence, we have captured all of the content of *a physical law*.
On the other hand, our description of that law leaves a lot to be
desired ease of use. Recognizing this, we have just discovered the
reason physicists and mathematicians write equations; they prefer not
to exhaust themselves needlessly.

Thu Aug 31 12:01:42 CDT 1995