January 3, 1997

PHYSICS, Ex Chao Ordo

Cognoscere REAL Physics

(Physics for Amateurs)

Copyright 1997 by Donald G. Shead

(Connoisseur)

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

sometimes fast.

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

planetary orbits.

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.

Speed

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

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).

Algebraically:

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.)

Algebraically:

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

are given:

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

a

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.

COMMENTARY

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

effort.

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

'a', in;

(f-uw)/a = w/g

We come up with:

(f-uw)t/(vt-vi) = (f-uw)t^2/(2s) = w/g (Unified Formula #1)

From which:

f = w(vt-vi)/(gt)+uw = 2ws/(gt^2)+uw (Force)

Where:

(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

arise.

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

requirements.

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

metric systems.

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

unit.

'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,

anywhere.

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

units cancel.

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

around!!

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.