January 3, 1997
PHYSICS, Ex Chao Ordo
Cognoscere REAL Physics
(Physics for Amateurs)
Copyright 1997 by Donald G. Shead
MECHANICS and THE MEASURING PROCESS
Mechanics is the branch of physical science that deals with the
study and measurement of matter, force and motion.
While we actually deal with these every day, we don't consciously
think very much about them. The majority of us leave these things to
teachers, engineers and physicists, thinking that only they need to
know, and are able to fully comprehend them.
In fact even most of the engineers that I've worked with leave
the real understanding of 'physics', per se, to physicists. This is
probably so with many teachers of physics too. This neglect has
caused the subject to fall into the hands (or minds) of a relatively
few deep thinkers, and philosophers. Instead of finding simple
answers to complicated questions, that simplify it, they have
compounded the questions with ingenious theories; pyramiding it into
an entire philosophical empire of abstraction that today is largely a
realm of their own; beyond the ken of most but themselves. Let's see
if *we* can make a clean start; regarding ordinary everyday things and
phenomena, at least:
Asside from such as its color and shape, there are two basic
things that most of us first notice about an object - a body of
"matter"; its size or extent, and its weight or heaviness. We also
notice that these can vary from body to body.
Then we will notice that bodies, if not too large, are movable;
sometimes it takes effort to move them, and sometimes they move,
and/or are moving by themselves. This movement is sometimes slow, and
Fundamentally then, before thinking very much about what they
consist of and what makes them heavy and/or why they move, there are
three basic measurement concepts involved here:
1.) The size of a body is a measure of its three dimensional
spatial extent; how much space it occupies. This "volume" consists
of the product of three perpendicular "length" measurements. These
are commonly known as length, width and depth. Where length is a
concept of one dimensional linear distance in space; which also
applies to spatial (in space) separation of points places and things,
as well as the progressive changes in spatial positions of movement
and/or "motion" (which is a time rate of movement).
2.) The weight of a body is a measure of the force it exerts upon
the ground, and or other support on which it rests, and is due to
gravity. For any body, anywhere, the ratio of this weight (w) to the
acceleration due to gravity (g) at this same place is a Constant!
This constant ratio (w/g) is a measure of the body's inertia - its
resistance to a change in its motion - and is what we call its "mass";
which represents the 'amount of matter' in it.
3.) In addition to measures of Length and Weight, one more
concept of measure is necessary to complete the list of the three most
basic fundamental measures:
When a body moves - changes its position - it does not jump
instantaneously from one place, or point, to another. It must
progressively pass over and/or by, all of the places in between.
When the change in position is substantial its easy to see that
there are many different 'in between' places. On the other hand when
the change in position is slight, or miniscule, it may appear that
there is no room for any such in between places. In either case
though, considering the nature of the concepts of 'points' and
'infinity', there is room for an infinite number of them.
When something moves then, it doesn't skip from one definite
place to another. It progressively passes by all of the in between
places without spending any time at any of them - it would take
forever if it did. This progressive movement occurs during the
continuous passage of the concept we call Time. Where duration is a
measure; portion, or period, of this continuous passage of time.
("Instants" aren't really short periods of time either. They are
points during time's continuous passage.)
So here we have it. The complete list of the three
"fundamentalest", most basic concepts, or "elements" of measure:
Essentially, with such exceptions as temperature and angular
measure, most of the quantity concepts of mechanics, such as size,
speed, linear acceleration and mass, are derived quantities: Derived
from and expressed with combinations of just these three ultimately
simple single measurements:
Which are: 1) length (linear distance in space), 2) force* (the
physical thrust exerted by matter against other matter), and 3)
duration (periods of time passage).
*(Where weight is just that particular centripetally directed
(toward Earth's center) force exerted by a body upon the ground or
other support on which it rests.)
We must contend with weight constantly, by exerting counter
forces with our legs and arms. From the time we get up in the morning
we must balance these counter forces to direct our activities so that
we can move around where we want to go, and avoid bumping into such
things as doors, steps and trucks.
Our "conventional" foot-pound-second (fps) system of measure -
even in its name - contains one basic standard unit for each of these
three measurements: The foot (') of length, the pound (#) of force,
and the second (") of duration. These are commonly supplemented with
the mile (5280') and the hour (3600") for such as travel trips and
With the recent trend toward decimalizing - using decimal
fractions - of these few units, instead of the many traditional
submultiples (such as inches, ounces, minutes, yards and rods) and
multiples or fractions thereof, a wide range of accurate measurement
can be quite easily attained.
While the metric systems also have units for force, the dyne and
the newton, they are inconveniently small and incorrectly derived
secondary (after-thought) units: The metric system's use of the gram
and kilogram of mass as 'weights' is confusing the important
difference between mass and weight. The plan here is to show that
these "absolute" metric systems are flawed for this reason, and are
unnecessarily complicating physics.
In fact the standard meter stick is also not a very convenient
length for ordinary laboratory and desk use either.
If we (continue to) use the pound (prototype) as a standard
weight, we will avoid most of the confusion between its mass (which is
1/32nd of a slug) and its weight (which, on Earth, is 1#). In fact
physics will be much simpler if we all reject the "absolute" metric
systems of measure, and use only one true "gravitational" system.
(Here on Earth where 'g' is a constant of about 32'/sec2, a
body's weight is (quite nearly) proportional to the quantity of matter
in it; so weighing is an easy (if not absolutely exact) way of
measuring quantities of matter for the international trade of
commodities: 'A sixteen ounce pint's a pound the whole world around.')
Derived Quantities and Units thereof
The measure, or quantity of such concepts as area, volume, speed,
acceleration, mass (inertia), and momentum; to name a few, are
compound combinations, or unifications of the three elementary
measures; being ratios and/or proportions derived by multiplying
and/or dividing the units of these fundamental quantity concepts:
Where such derived quantities, including mass, take on all of the
units of the fundamental quantities involved.
Area and Volume:
The quantity or measure of what we call the area of two
dimensionally extended surfaces is derived by multiplying a length (l)
in one direction by another length in a perpendicular direction (as
length x width): Area = length squared (l2), with units of feet2.
Volume is three dimensional extent (as length x width x depth);
Volume = length cubed (l3), with units of feet3.
Linear speed is the time rate of a body's motion; being the
ratio of a progressive change in its position, of some length (l),
divided by the period of time (t) during which it occurs. (A body at
"rest" has zero speed; a ratio of zero distance (length) divided by
the duration (t) during which it remains at rest): Linear speed
(abbreviated as 'v') = l/t, with units of feet/sec.
CHANGES in Speed
A change in linear speed is a change from one speed, call it an
'initial speed' (v`i = (l/t)`i), to another, call it 'final speed' (v`t =
(l/t)`t). Where this change in speed is the algebraic difference (increase
or decrease) between the final speed (v`t) and the initial speed (v`i). It
is important to realize that a change in speed is not (the same as) a
change in motion:
Forced (Change in) Motion, Displacement
(During) a change in speed (v`t-v`i), there is a (displaced) change in
position, a distance (s) from where the body would have gone, if it had
continued with its initial (inertial) speed (v`i), to where it actually is
displaced to; which is caused by a NET, resultant
impressed and/or frictional force. This distance (s) divided by the
period of time (t) during which it occurs (s/t) is a "forced (change
in) motion"; what I call "displacement"; which is equal to the MEAN
change in speed ((v`t-v`i)/2). Algebraically:
s/t = (v`t-v`i)/2 (Displacement)
Where for any given body, this displacement is proportional to
the product of the net displacing force (f) and its duration (t):
f:t::s/t; or ft^2/s = 2ft/(v`t-v`i) (Constants, for any given body)
Where this constant is equal to two times a body's mass (m):
First though, before regressing or skipping into the nature of
"acceleration" and "mass" by conventional methods, we'll consider the
resultant (actual) motion that we see as follows:
Resultant (Actual) Motion
The actual or resultant motion, that we see is the ratio of the
actual distance moved (d) divided by the period of time (t) during
which it occurs. Resultant motion consists of two different
theoretical motions: Initial (inertial) motion (v`i), and forced (change
in) motion - displacement (s/t), of the inertial motion. It is (equal to)
the algebraic sum of the initial (inertial) motion (v`i) and the forced
(change in) motion ((vt-vi)/2) - or the displacement (s/t). Algebraically:
d/t = v`i + (v`t-v`i)/2 = v`i + s/t (Resultant (Actual) Motion)
Linear acceleration is a Time Rate of Change in (a body's) Speed:
The quantity of (this concept of) linear acceleration (a) is the
quotient of a change in a body's speed (v`t-v`i), as was described
above, divided by the period of time (t) during which it occurs, and
takes on the units of both speed and time (feet/sec2).
a = (v`t-v`i)/t (Acceleration)
(The quantity of that particular acceleration due to gravity (g)
varies depending on the location. At Earth's surface it is about 32
feet/sec^2. On the moon it is only about one sixth as great.)
g = (v`t-v`i)/t (Approx.32'/sec^2 @ Earth's surf.)
Mass: Inertial and Gravitational
Inertia is the resistance of matter to changes in its motion.
The quantity of (the concept of) inertia is called mass! Where it
follows that the mass of a body is (also) a measure of the "quantity of
matter" it contains:
Although Einstein said something to the effect that Inertial Mass
and Gravitational Mass are two manifestations of the same thing, his
general theory was quite complicated and it didn't do much to help
clarify their "quite different attributes".
The "difference" between inertial mass and gravitational mass is
really quite simple:
So called Inertial Mass is derived under theoretically "inertial
conditions"; where friction and gravity are avoided by conducting
experiments on 'horizontal frictionless surfaces', or somewhere out in
'deep space' where the effect of gravity would be negligible.
Under such conditions - ideally achieved with thought
experiments, and closely approximated with wheels, air tracks, and
clever devices such as dry ice pucks - the ratio of an impressed force
(f) divided by the acceleration (a) that it causes is a Constant.
This constant ratio (f/a) takes on all of the units of force (pounds)
and acceleration (feet/sec^2): pounds/(feet/sec^2); or more concisely,
pounds x seconds^2/feet. Where for any given body of matter this ratio
(f/a) is (also) equal to the body's so called Gravitational Mass:
Which is the ratio of that particular force that it exerts upon the
ground or other support on which it rests; that we call its weight
(w), divided by that particular acceleration due to gravity (g),
measured at the same location. This ratio (w/g) too, takes on, or has
the units of (pounds x seconds^2/feet).
This equality of inertial mass and gravitational mass can be
expressed as equal ratios; in an Algebraic Formula:
f/a = w/g (Constants, for any given body)
With this simple formula, and algebraic transposition, we can
derive equations to solve the value of any one unknown when the others
f = wa/g (w/g replaces m in f=ma)
w = fg/a (f/a replaces m in w=mg)
a = fg/w (a/g is equal to f/w)
g = wa/f (g/a is equal to w/f)
Where, for such theoretical inexact work as engineering, 'g' is
often approximated as 32'/sec^2.
Putting in their units:
f# = w#.a('/sec^2)/g('/sec^2) -- '/sec^2 cancel
w# = f#.g('/sec^2)/a('/sec^2) -- '/sec^2 cancel
a'/sec^2 = f#.g('/sec^2)/w# -- # cancel
g'/sec^2 = w#.a('/sec^2)/f# -- # cancel
Frictional and Gravitational Restraint
In determining the inertial mass, above, we avoided and/or
reduced frictional and gravitational restraints to an extent that they
could be neglected.
In actual practice though, it is virtually impossible to avoid
them completely. The effect of forces are reduced by these restraints;
so that in the equation: f/a = w/g, the effective, force (f) is
reduced by them: It is this reduced, Net, Resultant force that is
proportional to the acceleration that it causes.
The net resultant force is the total impressed force (f) minus
any resisting forces; which may be handled as follows:
The most familiar resistance to force and the change in motion that it
produces is friction. The magnitude of a frictional force is the product of
coefficient of friction (u) and the normal force holding the two
sliding surfaces together. For level surfaces this normal force is the
weight of the body, so that the net force is the total force (f) minus
the frictional force (uw). so that:
(f-uw)/a = w/g
Other resisting forces can be handled in a similar fashion by
adding to and/or modifying this "coefficient of resistance". We'll
dwell on this in greater detail later. For now it is enough to say
that this coefficient can be made a permanent part of the formula
because if its zero (no resistance) its presence in the formula will
not cause a problem.
The 'coefficient of friction' (u), can also be expanded to
include the effects of gravitation: Where for vertical lifting, the
gravitational resistance is equal to an object's weight; or ONE (1)
times its weight; so that for lifting straight up, where the 'slope'
is ninety degrees, the coefficient of gravitation is equal to the sine
of ninety degrees which is (equal to) ONE. For other slopes it is
equal to the sine of their angle of inclination.
For direct lifting then, the "coefficient of (gravitational)
restraint" (u) is equal to one, so that the full force (f) used to
counteract the object's freefall, and lift it, is, and must be, a
little greater than its weight. Where the impulsion is (likewise)
eased off at the top of the lift, and the object being lifted coasts
to a stop (in the last fraction of a second, usually; to a shelf,
ledge or tailgate) by its own inertia.
In these cases, where the object's final position, but not its
final speed or direction of motion, is changed, the excess force,
beyond that required to balance the counter-effect of friction or
weight (uw) is maintained for less time than required for the object
to move. Therefore the greater force (f) for (multiplied by) the
shorter period (t`a) is the effort (ft`a) which is equal to the product of
the opposition (uw) for (multiplied by) the full period of time (t)
throughout which it reacts (uwt).
If there is one thing that we must emphasize here: it's that the
net or resultant force (f-uw) is what does the "work" of accelerating
and moving a body! The product of this net force and the period of
time (t`a) during which it acts is the net impulse ((f-uw)t`a) that
causes, or increases, momentum (v`t-v`i)w/g.
We can exert force, thereby expending energy, all day long, but
if we don't, or can't, exert a force (f) somewhat in excess of the
opposing and/or restraining force (uw), there will be no net force
(f-uw), and consequently the (theoretical) displacement is
counteracted or balanced by the opposing restraint so that there is
just a lot of stress and strain (deformation) and heat losses due to
increased "molecular" motions, but no (actual or visible) "molar"
displacement; no relative movement. We can generate a lot of
muscular fatigue without accomplishing anything useful.
This makes it imperative, when doing work, to accomplish it as
quickly as possible; to avoid working against the relentless and
wasteful opposing restraint any longer than we have to: The weaker
the force, the longer it takes; the longer we must fight this
constant restraint: Ideally, we should use the strongest force
available to cut down on the length of time it must be maintained.
Realizing this, we should now understand the recent success of
the trend to use heavier construction equipment, et al; which has
been made possible largely through development of improved diesel and
electric motors. Not to mention envying the physically stronger
person's ability to more easily accomplish various everyday tasks.
(But remember too, that "success" also depends on the persistent
efforts of such as woodpeckers and rams.)
Since exerting force requires muscular effort, or the expenditure
of chemical, or heat energy, things which generally require
considerable effort to come by, it is vitally important to know how to
best achieve a maximum desired effect with a minimum of precious
While the measurement of "work", then, is usually made only in
terms of the magnitude of the force and the permanent change in
position that results, the actual effort, and energy consumption to do
work is dependent on the force and its duration. Some work - and even
"no work," such as pushing on a solid wall or holding something up for
a long time - can require a lot of effort and consume considerable
energy. Doing work as quickly as possible - by using as great a force
as is available - will avoid excessive wasting of effort. Without
the ability to apply a force slightly in excess of the restraining
force (uw), either directly or with leverage of some sort, no work
will be done.
While 'the feeblest motor will,' in theory, 'raise the Sphinx, if
given enough time,' it would have to run a very long time, and use an
awful lot of energy and overcome an awful lot of friction; so we
might just as well forget about trying it.
AND finally, don't let anybody tell you that you can move
anything without accelerating it. It just isn't so. Once you get it
moving, then you can reduce the force to equal that of the opposition
and still keep the thing going.
Back to Fundamentals
By substituting 'v`t-v`i/t' and '2s/t^2' for the shorthand symbol
(f-uw)/a = w/g
We come up with:
(f-uw)t/(vt-vi) = (f-uw)t^2/(2s) = w/g (Unified Formula #1)
f = w(vt-vi)/(gt)+uw = 2ws/(gt^2)+uw (Force)
(f-uw) = w(vt-vi)/(gt) = 2ws/(gt^2) (Net Force)
t = w(vt-vi)/(gf) = sqr(2ws/(gf)) (duration-time req'd.)
vt-vi = fgt/w = 2s/t (change in speed)
vt-vi/2 = fgt/(2w) = s/t (Displacement)
vi = vt-fgt/w = vt-2s/t (init. speed)
vt = vi+fgt/w = vi+2s/t (final speed)
l = vt*t-fgt^2/w = vt*t-2s (init.chg.in pos.)
s = fgt^2/(2w) = (vt-vi)t/2 (dist.displaced)
d = vt*t-(vt-vi)t/2 = vt*t-s = l+s = (vt+vi)t/2 (dist.moved)
w = ftg/(vt-vi) = fgt^2/(2s) (Weight)
g = w(vt-vi)/(ft) = 2ws/(ft^2) (Accel.of Grav.)
This unified formula, and the equations derived from it show how
the three basic concepts and measures thereof combine to unify and
describe the variables of linear motion.
Other aspects of mechanics, such as changes in direction
(curvilinear motion), the nature of force (and friction) and the force
of gravity, will be taken up on a contingency basis, as the needs
By using only systems of measure with the three truly fundamental
measures, physics will be much simpler. Where for now anyway, the
foot-pound-second (fps) system seems to satisfy most ordinary
To avoid their becomming further entrenched in physics and our
minds we'll now discuss the fallacies of the metric systems:
The Metric Systems
Since first beginning to study physics, as a prerequisite for
engineering, I've been trying to figure out why it is such a difficult
and forbidding subject. Recently, while contemplating some of the
reasons for the United States being so slow in converting to the
metric system, one answer has quite suddenly come to me:
The main problem with physics is still just a simple
misunderstanding of the difference between weight and mass. Nowhere
is the misleading misuse of mass (the ratio w/g), instead of weight
(w), as a fundamental quantity, or measure, any greater than in the
Back about two hundred years ago, after the French Revolution,
when the United States was quite young, the world was beginning "to
settle down". Enough anyway so that politicians and scientists began
to contemplate the need for an international system of weights and
measures so that international trade could be carried on more easily.
Prior to this time there were many different systems of measure.
Since no one was consistent in all respects it was determined that an
international system should start all over with fresh ideas. Such an
"absolute" system should also be based on the decimal system.
Instead of length units based on human anatomy, which vary from
person to person, it was decided to use length units pertaining to the
size of the Earth. Considerable work was put into this and without
too much regard as to its convenience, the final unit of length was
determined to be the meter. To provide for longer and shorter
measures, instead of using decimals of it, multiple and submultiple
units based on decimals of it were invented. That is, each decimal
place of a metric unit also has a name of its very own. "Simply" by
using Greek or Roman prefixes to designate its relation to the basic
'Mass' was chosen for determining quantities of material goods,
for their international, world wide, exchange; because weight varies
depending on the acceleration due to gravity at the location where it
is measured, and the mass of a quantity remains unchanged. The same,
The chosen unit of mass was to be (equivalent to) a volume of
water (a liter) measuring one cubic decimeter. A metallic prototype
of this unit of mass was made, and was (is) called a 'kilogram'; the
French name for weight. Herein lies a problem: Is the kilogram a
measure of weight, or a measure of mass?
As a unit for measuring quantities of commodities it doesn't
matter: Merchants and politicians are satisfied because, by mutual
consent, the gram and kilogram apply universally to goods in
commercial trade. Why physicists have not come to realize their
mistake in respect to physics is the real question:
Then of course this wasn't too long after Newton, so the
importance of the difference between these two concepts was probably
not realized: I've read remarks by physicists decrying the confusion,
but the feeling seems to be that 'What's done is done'.
If not the original perpetrators of it, the "absolute" metric
systems are at least perpetuating the confusion between mass and
weight: The gram and kilogram of mass are commonly called weights,
and are inconsistently defined in various texts and dictionaries:
Sometimes they are defined as mass, sometimes as weight, and sometimes
as mass OR weight.
For most people, including many scholars, this ambiguity keeps us
in a state of doubt and confusion. Clearly distinguishing between the
mass and the weight of a body is very important in beginning the study
and comprehension of physics.
Giving names to "units of mass", the gram (gm) and kilogram (kg),
and definitions so that: 1 gm = 1 dyne sec2/centimeter; 1 kg = 1
newton sec2/meter, does not allow these connections to be arbitrarily,
or otherwise, rewritten as: 1 dyne = 1 gm centimeter/sec2, and 1
newton = 1 kg meter/sec2: We cannot - in our striving for simplicity
- hide the fact that 1 gm has units of dynes.sec2/centimeter, and 1 kg
has units of newtons.sec2/meter. Here's how it really works:
A gram weighs (about) 980.6 dynes. Its mass (w/g = f/a) is 980.6
dynes.sec2/980.6 centimeters = (reduces to) 1 dyne sec2/centimeter.
THIS connection may be rewritten; so that 980.6 dynes.sec2/980.6
centimeters.1 centimeter/sec2 = (reduces to) 1 dyne; which is a
fundamental unit (of force, and/or weight) because all of the other
A kilogram weighs 9.806 newtons. Its mass (w/g) is 9.806
newtons.sec2/9.806 meters = (reduces to) 1 newton sec2/meter (= f/a).
This connection may be rewritten, so that 9.806 newtons.sec2/9.806
meters.1 meter/sec2 = (reduces to) 1 newton; which is a fundamental
unit because all of the other units cancel.
Mass cannot be arbitrarily, or otherwise, "... taken as
fundamental, and force as derived, ...", because it's the other way
The solution to this enigma, is to reject, *or modify,* the present
mass based metric systems, and any other 'absolute' systems of
measure. If one doesn't already exist, a meter-newton-second
"gravitational" system would more correctly serve in its place.
By using only systems of the three truly fundamental measures,
physics will be much simpler.
Better yet; for the time being at least, our existing
foot-pound-second system should be retained and/or reinstituted. This
system, where "a pint's a pound the whole world around", has been, and
still is, quite successful. For now and for some real scientific
comprehension and true progress, let's stick with our correct
foot-pound-second system, with real fundamental units.
Finally, the decimal system is applicable to any numerical
measure, including the foot, the pound, the degree of arc, and the second.
The metric system has no special claim to it.