~ TWENTY-FIRST CENTURY DYNAMICS ~

MOTION and FORCE

Introduction

In Chapter 12, of College Physics, by Weber, White and Manning,

3rd Edition of 1959, there is a statement in 12-3, Page 116, to the

effect that there is no displacement in the direction of the centripetal

force:

Being a retired engineer, and having vigorously studied force and

how it affects the motion of material bodies, I must take exception to

this. A "displacement", as will be hereinafter defined, does indeed

occur whenever a net (resultant) force, in any direction, acts upon a

body; moving, in any manner, or not. Apparently, the concept of

'displacement', and/or the result of a force or thrust, needs to be

clarified:

In order to better understand relative motion and forced changes in

it (displacements), we'll take an in depth look at this phenomena to see

just what does happen:

~ CHAPTER 1 ~

MOTION

All about us things are moving: It is not the movement in itself

that we call "motion" though. It is the progressive changes in the

relative spatial (in space) positions of various material bodies (of

matter) DURING the passage of time that is their motion.

First off, we must realize that all positions, directions and

motions are "relative" and vary according to the orientation and motion

of each individual observer of them: Our left may be someone else's

right. And so it goes with up, down, in, out, fast, slow and so forth.

What one observer sees as rest may even be seen as motion by another

observer. Motion is complicated by this relativity and there may be as

many different descriptions of a particular motion as there are

observers of it.

Thanks to Newton's First Law of Motion though we can set up and

illustrate simple "reference motions" to give us commonly understood

initial assumptions, which allow us to temporarily forget about

relativity:

With these simple (inertial) 'reference motions', including 'rest',

we can stipulate and present commonly understood starting speeds and

directions of motion for various problems of force and motion. From

these starting motions, including rest, we can then go on to determine

forced changes in them:

1.1 Free (inertial unchanging) Motion

In the theoretical (hypothetical) absence of frictional, or

gravitational opposition: A completely unrestrained body traveling

freely for a period of time (t), that starts with and maintains an

initial rate of change in position, an initial velocity (Vi), will move

in a single direction (along a straight line). It will move a distance

(l) that is equal to the product of that velocity (Vi) and that period

of time (t). Algebraically:

l = Vit. Therefore Vi = l/t.*

*[Velocity is usually reduced to a unit denominator such as (per)

second, or (per) hour: Like 10'/sec (10fps) or 60mi/hr (60mph),

etcetera.]

These motions (l/t) are the Inertial Motions of Newton's First Law.

In essence, this (theoretical) kind of free motion avoids some of the

immediate problems of the relativity of motion by providing us with the

very simplest of reference motions: These may be (designated as)

"forward" or "backward"; in any direction, and at any speed, including

"rest": From these simple unchanging motions, forced changes in motion

- changes in speed and/or direction - as will hereinafter be defined,

may then be demonstrated to proceed.

1.2 Displacement - Forced (change in) Motion

'Displacement' is an 'Accelerated Forced (change in) Motion'

similar to the "change in motion" of Newton's Second Law. It can be

directly and simply described and illustrated for both linear and

circular motions, and accordingly can be quite easily perceived and

understood:

1.3 Forced Change in Position

'Displacement' is not the progressive changes in the positions of

relatively moving inertial motion. They are relative. Displacement is

a forced change in the speed and/or the direction of a body - moving or

not; where it is forcibly displaced a distance (s) from where it Would

Have BEEN, GONE OR STAYED, if it was not displaced, to where it Actually

IS displaced to; during (per) a period of time (t). This ratio can be

written: Algebraically, as 's/t'.

Displacements (s/t), and (time) Rates of Displacement (s/t2) are

numerically related to acceleration (a) as: s/t = at/2 and s/t2 = a/2:

For circular acceleration, where a = v2/r: s/t = v2t/(2r) and s/t2 =

v2/(2r): Where 'v' is the initial and uniform speed of a body moving in

a circle, and is equal to the (increment of the) length of the arc (d),

per the period of time (t) during which it is traversed: v=Vi=l/t=d/t.

1.4 Forced (Mean) Change in Velocity

Displacement (s/t) is (also) a forced change in a body's

(otherwise) Inertial (free) Motion - moving or not, where the motion is

changed - physically altered - from an "initial" velocity (Vi), or rate

and direction, progressively through intermediate velocities

(V0.1,...V1,...V2, etc.) to another rate and/or direction, with a

"final" velocity (Vt); during a period of time (t). For any given

body, the MEAN (average) forced change in the velocity ((Vt-Vi)/2) is

numerically equal to, the ratio (s/t).

Algebraically:

s/t = (Vt-Vi)/2 (Displacements (in speed and/or direction))

Displacement is (just a little) simpler than Acceleration in that

it is a simple 'Change in Motion'; equal to the Mean change in

velocity: Where the velocity is changed from an initial velocity (Vi)

to another velocity (Vt), and is algebraically written as '(Vt-Vi)/2':

Whereas acceleration is a (time) rate of change in velocity '(Vt-Vi)/t'.

Although it may not seem so at first glance, this is a significant

simplification: ~

1.5 Resultant Motion

Inertial (unchanging) Motions, and Displacement (forced changes in

(inertial) motion) are theoretical: That is, while we can illustrate

(and imagine) them, they are (ordinarily) not

directly observed, or even observable.

It is the combination, or resultant, of these two theoretical

motions that we actually "see": A body's ongoing Inertial Motion

(Vi=l/t), and any Displacement (s/t) of it, results in the actual

"Resultant Motion" (d/t), which is the (ordinarily) observable motion

that we do see.

This resultant motion (d/t) is relative and will be seen as changes

in speed and/or changes in direction; which will be seen by variously

located and variously moving observers as a variety of different

motions; even as rest. Generally they will appear as variously

complicated elliptic, curvilinear motions.

To avoid these complicated "appearances", we'll limit our study of

displacement as follows:

Somewhat like the hypothetical ideal of simple Free (inertial

unchanging) Motion, that avoids some of the complications of relativity,

we can, in an effort to simplify this vast array of motions and

displacements, theorize and consider just two "ideal" kinds of

displacement: 1.) LINEAR Displacements, in line with, and along the

path of relative motion, and 2.) PERPENDICULAR (Centripetally directed)

Displacements, acting and maintained at right angles to the changing

direction of the path of the circular elliptical motion (that results):

1.6 Linear Displacements

Probably the simplest of these two (ideal) kinds to comprehend, if

not quite the easiest to illustrate, are Linear Displacements along the

path of motion; which change the speed of motion, but not its

direction. For these linear displacements, the mean change in velocity

((Vt-Vi)/2) equates to the mean change in speed: The mean difference

between the final speed (vt) and the initial speed (vi):

s/t = (vt-vi)/2 (Linear Displacement)

Which appears variously to various observers (relativity again) as

starting, speeding up, maintaining, slowing down, and stopping of a

body's motion.

1.7 Perpendicular (Centripetal) Displacements

Perhaps the next simplest displacements, and the easiest to

illustrate, are the Perpendicular, Centripetally directed, Displacements

(of circular motion). These displacements, maintained at right angles

to the changing direction of the path of the circular motion, and

consequently themselves, are continuously changing direction to remain

directed inwardly radial toward a central point, are continuous changes

in the direction of the motion, but (hypothetically) "neglecting"

friction, and in airless space, do not affect the speed. The

theoretical paths of centripetal displacement are illustrated as

follows:

According to Engineering Drawing, by French and Vierck, 8th

Edition, March 1953, page 88 (5.62): "An involute is the spiral curve

traced by a point on a taut cord unwinding from around a polygon or a

circle." The involute of a circular arc is a (simple) smooth curving

line that changes direction uniformly, along with that of its circular

evolute (the circular arc 'd') and has a length (s) equal to the square

of the length of this circular arc (d2) divided by the diameter (2r) of

the circle of which the circular arc (d) is a part of: Where 'r' is the

radius of the circular arc.

s = d2/(2r)

and, the displacement ((Vt-Vi)/2) equates to:

s/t = d2/(2rt) = v2t/(2r) (Circular Displacement)

The illustration of (uniform) circular motion proceeds (sort of)

like progressively rewinding, or bending a (taut) straight tangential

cord inward, into a circular curve:

The involutes from points along the tangent to points equidistant

along the circular arc, at various intervals during the motion,

represent the theoretical paths of this progressive displacement at

those points.

~ CHAPTER 2 ~

FORCE, IMPULSE and EFFORT

2.1 Force

The concept of force is based on the impenetrability of matter:

That property of matter which prevents two material bodies or particles

thereof from simultaneously occupying the exact same place. Any attempt

to (have them) do so causes them to thrust, push and shove against, and

deform and displace, each other:

What we (will) call Force (f) then, is the magnitude of the

physical thrusts that bodies (including humans) exert upon each other.

A thrust, push or shove is ordinarily the result of such as muscular

exertion, impingements, mechanical expansion, wind & water pressure,

gravitation, electrical and magnetic effects, etcetera. Where one body

"actively" exerts a thrust upon another body which in turn "reactively"

resists with a mutual, equal and oppositely directed thrust of its own.

In effect they exchange impetus, or momentum:

NOTE: "Tension" and "pulling" forces will be explained elsewhere,

at some later time and place.

2.2 Impulse, Effort and Energy

Now of course the thrust of bodies upon each other causes a certain

amount of deformation, and therefore is not instantaneous: Depending

upon their hardness, or softness, and their massiveness, bodies deform

accordingly. It takes longer to deform a soft massive body than to

deform a hard light one. Even billiard balls are not infinitely hard.

The duration of any thrust and the response to it lasts for a more or

less extended period of time; brief perhaps but for ordinary

macroscopic bodies, at least, thrust is not instantaneous:

The algebraic product of the magnitude of an exerted thrust, or

force (f), and its brief or sustained duration (t) is called an

"impulse" (ft); which is a measure of the (muscular, electrical or

mechanical) "effort", and/or energy, that must be expended to create

these brief or sustained thrusts; which cause, and are proportional to,

the hereinbefore defined displacements (s/t) of (friction) free bodies:

2.3 The Ratio of

The Effort Required to the Displacement Induced

The displacement (s/t) of a free body is in proportion to, and in

the direction of the impressed impulse, or effort (ft), that produces

it, and continues only as long as the effort endures. When it stops,

the displacement stops. For any given free body then, the ratio of its

displacement (s/t) to the effort (ft) causing it is a Constant:

ft/(s/t) = ft2/s = Constant

2.4 The Constants of Displacement

In other words: Whenever a force, or thrust, is exerted upon an

unrestrained (free) body, moving or not, it changes that body's motion.

This change is not instantaneous, nor is the thrust. As long as the

thrust continues, be it brief or enduring, the change in motion, or

Displacement, continues to occur. The duration (t) of the thrust (f) is

not just simultaneous with the duration of the displacement (s/t)

either; they are one and the same. At all times, the forced change in

position (s) that results is proportional to, and in the same direction

as the force (f) that produces it.

Continuous changes in motion (s/t) are proportional to and in the

same directions as the efforts (ft) causing them: Algebraically:

ft/(s/t) = f/(s/t2)

which are also equal to:

ft/((vt-vi)/2) = f/((vt-vi)/(2t))

and for circular motion:

ft/(v2t/(2r)) = f/(v2/(2r))

Where 'ft' is the impulse or effort imparted, 's/t' (which is equal

to '(vt-vi)/2', and'v2t/(2r)') is the displacement, 'f' is the

magnitude of the thrust, and 's/t2' (which is equal to '(vt-vi)t/2', and

v2/(2r)') is the (time) rate of displacement.

more concisely:

ft2/s = 2ft/(vt-vi) = 2fr/v2 ("Constants of Displacement")

Where these Constants are also equal to the ratio of the constant

force (weight (w)) of a body "resting" upon the ground or other support,

PER the (time) rate at which it is being restrained (displaced) from

freefalling further. Where this (time) "Rate of gravitational

Displacement" is equal to 'g/2'; which is about 16'/sec2 at Earth's

surface! So that:

ft2/s = 2ft/(vt-vi) = 2fr/v2 = 2w/g (Constants of Displacement, for any

free body, anywhere)

Where '2w/g' is the body's "Constant of gravitational

Displacement". Which is the ratio of its weight (w), to the

gravitational displacement (g/2). It is the measure of a body's "heft",

or heaviness; the measure of its @ rest "Static Inertia")

From this formula, with algebraic transposition, we can make other

formulas as well as equations for each variable.

First of all lets make a slight modification in this formula by

dividing each equality by two (2). The result gives us something that

is more familiar:

ft²/(2s) = ft/(vt-vi) = fr/v² = w/g (Equals a body's mass!)

See College Physics, by Weber, White and Manning, 3rd Edition of 1959:

Art. 4-8, and Table 1, page 37.

2.5 Inertia

Inertia is the persistence, or "inherent energy" of a body to

resist changes in its motion (including rest), whereby it has the

potential to exert force: Either by action or by reaction. Inertia is

the measure, or degree, of a body's perseverance to continue its present

rate of motion; where it takes energy, and/or will do work, in order to

change it.

Force (f) is the magnitude of the pressure of a thrust which

"results in" (algebraic symbol '=>'), or causes, a time Rate of

Displacement (a rate of change in the speed and/or direction) of a free

body's motion (s/t² = (vt-vi)/(2t) = v²/(2r)): Where the force is equal

to the product of a body's Constant of gravitational Displacement

(2w/g), and this Rate of Displacement; which represent the change in

its inertia; from the value it had (at Vi) to another value (at Vt).

Algebraically:

Force: f => 2w/g x s/t² = 2w/g x (vt-vi)/(2t) = 2w/g x v²/(2r)

more concisely:

Force: f => 2sw/(gt²) = (vt-vi)w/(gt) = v²w/(rg) (Change in

Inertia)

2.6 Momentum

Momentum is (sort of like) "dynamic" inertia. It is the energy of

a body's motion, whereby, through its impetus, it has the potential to

exert force; it takes work or energy to change it, and has kinetic

energy so it will also do useful work when properly directed. As

through striking hammer heads, and the impulses to the pistons and

crankshaft of internal combustion engines, and other mechanisms.

An applied impulse (ft), or effort, results in (=>) a change in a

body's momentum. Where this momentum is simply the product of a moving

body's dynamic inertia - in the direction of the motion - and its

duration. Where the terms of inertia: '2sw/(gt²)'; '(vt-vi)w/(gt)', and

'v²w/(rg)' represent the dynamic inertia, or impetus of a moving body;

the energy with which it moves.

Algebraically:

Effort: ft => 2sw/(gt) = (vt-vi)w/g* = d2w/(rgt) (Change in

Momentum)

Where the quantity of a body's momentum is changed from the initial

value that it had (at Vi), to another value (at Vt).

* Momentum is also known as a body's 'quantity of motion'; 'the

product of its mass and the velocity.'

NOTE:

Because the equation: 'f = 2sw/(gt²) = (vt-vi)w/(gt) = v²w/(rg)',

can also be written as 'f = 2w/g x s/t² = 2w/g x (vt-vi)/(2t) =

2w/g x v²/(2r); and Since mass (m) equals 'w/g', and acceleration (a)

equals '2s/t² = (vt-vi)/t = v²/r'; Newtons formula: 'f=ma', while

superfluous, and not as simple as it looks, can be applied here too:

2.7 List of Variables with (some) Equations

Note: Some of these formulas and equations apply to both linear

and centripetal displacements: Those equations containing linear speed

- 'vt' and/or 'vi' - apply only to linear motions with displacements

along the paths of motion, which may be along curves, circular or

otherwise. Those containing circular speed (v) and/or circular radii

(r) apply only to circular motion and centripetal displacements:

Applied Force: f => 2sw/(gt²) = (vt-vi)w/(gt) = v²w/(rg) (Change in

Inertia)

Distance displaced: s = (vt-vi)t/2 = fgt²/(2w) = d-vi x t = vt x t-d =

d²/(2r)

Displacement:* s/t = (vt-vi)/2 = fgt/(2w) = d/t-vi = vt-d/t = v²t/(2r)

Distance moved: d = (vt+vi)t/2 = vit+fgt²/(2w) = vit+s =vt x t-s = (vt=l=d)

=

sqr(2rs)

Result. Motion: d/t = (vt+vi)/2 vi+fgt/(2w)= vi+s/t = vt-s/t =

(v=l/t=d/t) = sqr(2rs)/t

Time period (Duration): t = sqr(2sw/(gf)) = (vt-vi)w/(gf) = d/v =

sqr(2rs)/v

Chg.in vel.- speed/dir.: Vt-Vi = 2s/t = vt-vi = fgt/w = d²/(rt)

Initial velocity: Vi = vi = vt-2s/t = vt-fgt/w = (v=l/t=d/t) (Inertial

motion (@ start))

Final vel.: Vt = vt = vi+2s/t = vi+fgt/w = 2d/t-vi = (v=l/t=d/t)

(Inertial motion (@ end))

Weight, due to grav.: w = fgt²/(2s) = fgt/(vt-vi)

Accel.due to gravity: g = 2sw/(ft²) = (vt-vi)w/(ft)

Displac.due to grav.: g/2 = sw/(ft²) = (vt-vi)w/(2ft)

Mass: ft²/(2s) = ft/(vt-vi) = fr/v² = w/g

Static Inertia: 2w/g (the meas. of a body's "@ rest" Inertia)

Acceleration: a = 2s/t² = (vt-vi)/t = fg/w = 2d/t²-2vi/t = v²/r

Applied Effort: ft => 2sw/(gt) = (vt-vi)w/g = d²w/(rgt) (Change in

Momentum)

These algebraic formulas and equations can also be expressed in

words.

~ Chapter 3 ~

Resistance, Restraint & Opposition to Free Motion

Note: For linear motion, some of the subject of this Chapter,

especially Section 3.7, is included in an article entitled "Unifying the

Concepts of Mechanics".

3.1 Reaction

Before going any further, and getting our minds set too deeply, it

must now be pointed out that most of what we've said so far pertains

only to motion that is theoretically free of frictional and/or

gravitational restraints. Such motion as would occur on 'frictionless

surfaces' or 'far out in space', away from the gravitational effects of

celestial bodies; where the only resistance to thrusts, impulses and

applied efforts would be those of a body's own inertia. Motions

occuring upon, or near, Earth however, are seldom if ever, friction or

gravitation free:

Here, on Earth, everything's being mashed toward its center by (the

force of) gravity; causing friction between everything. Here then, in

addition to the intrinsic resistance of a body's own inherent inertia,

virtually all of the thrusts and efforts that are exerted upon bodies

are also met with extrinsic resistance due to friction with other bodies

and/or gravitation, which acts toward Earth's center. Such extrinsic

resistance, or additional restraint, reactively opposes and reduces the

effectiveness of the actual forces (f), and efforts (ft) that are

exerted.

3.2 Coefficient of Reaction

The magnitude, or resisting force, of these reacting oppositions is

related to the weight of the body whose displacement is affected; as

being a coefficient, or portion of its weight; a variable (algebraic

symbol 'u'), that is the value, or coefficient, of this relationship.

It is the product of this coefficient (u) and the vertical component of

a body's weight (w) that is the magnitude of the reaction (uw). In

order to displace a body then, greater thrusts and efforts are required

to overcome this additional resistance. These additional opposing,

reacting forces (uw) can be included in the formulas like this:

The force (f) that is now required to change a body's inertia against a

reacting force is greater by that reacting force (uw).

Force: f = (vt-vi)w/(gt) + uw = 2sw/(gt²) + uw

Multiplying 'through', by 't', we get the full effort that is now

required:

Effort: ft = (vt-vi)w/g + uwt = 2sw/(gt) + uwt

Where the product of the applied force (f) and the duration of it

(t), is the effort (ft), or "Impulsion" acting to change the momentum

against any frictional and/or gravitational "Repulsion" (uwt); which is

the product of the reacting force (uw) and the duration of it (t).

This coefficient (u) is a combination of the coefficient of

friction (uf), AND a "coefficient of gravity" (ug), as explained below:

3.3 Frictional Force

Friction, or frictional force, is well known as an inhibitor to our

efforts to change, or create, and maintain motion: Coefficients of

friction (uf) are based on the roughness and slipperiness of surfaces in

contact with each other. The frictional resisting force of sliding

surfaces is considered to be the product of the estimated coefficient of

friction (uf) for those particular surfaces and the resultant pressure,

or magnitude of the force that is pressing normal (perpendicular) to

those surfaces; by which they are pressed together. On level surfaces

this frictional resisting force (uw) is quite simply the product of the

estimated coefficient (u), and the weight (w) of the body resting, or

sliding, upon that surface. Tables of these coefficients of friction

are found in most physics texts.

Friction on sloping surfaces, positive or negative, is complicated

due to the fact that the normal (perpendicular) pressure between the

surfaces is only a portion or component of the weight (w) depending on

the cosine of the angle of the slope: Thus (in effect) reducing the

"effectiveness" of the coefficient (u); so that this effectiveness is

reduced as the product of the cosine of the angle of the slope (oo) and

the coefficient of friction as given in various tables. The frictional

force on a slope is then the product of this reduced coefficient (uf)

and the weight (w) of the body.

uf = cosine(oo)x the coefficient (u) given in Tables.

3.5 Gravitational Force

The coefficient of friction, can 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 (ug) is equal to the sine of ninety

degrees which is (equal to) ONE. For other slopes, positive or

negative, the coefficient (ug) is equal to the sine of their angle of

inclination; including zero 'slopes'.

ug = sine(oo) (for gravity on slopes)

For direct lifting then, the coefficient of gravity (ug) is equal

to one, so that the full force (f) used to counteract the body's

freefall, serves only to hold the body at rest, and prevent its

gravitational rate of (freefall) displacement. To actually lift, or

displace it, a NET force a little greater than its weight is required:

To include both frictional and gravitational resistance, their

individual coefficients of resistance, or restraint, can be added, for

one (common) value (u), to be known as the "Coefficient of Reaction":

u = (cosine(oo)uf + sine(oo)) (as modified for slopes)

Where (uf) is the coefficient of friction (u) given in standard

Tables, and (oo) is the positive or negative inclination of any slope.

To determine their combined effect, the "Reacting Force" - positive

or negative - is the product of this coefficient of reaction (u), and

the body's weight (w):

uw = (cosine(oo)uf + sine(oo))w (as modified for slopes)

3.6 Net, or Resultant Force, and Work

While it is the full force that must be exerted, and the full

effort that must be applied, to displace bodies against opposing

reacting forces, it is only the NET force that actually causes, and is

equal to, the (dynamic) inertia and or momentum that is induced:

This Net force is the total force exerted minus the reacting force

(uw). Subtracting 'through' by 'uw', we get:

NET (Effective) Force: f-uw => (Vt-Vi)w/(gt) = 2sw/(gt2) (Chg.in

Inertia)

Multiplying this through by 't', we get:

Net (Effective) Effort: (f-uw)t => 2sw/(gt) = (vt-vi)w/g = (Chg.in

Momentum)

If there is one thing that must be emphasized here: It's that the

net or resultant force (f-uw) is what causes the displacement, and does

the "work" of accelerating and moving bodies!

We can exert force [F], 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 (f = F-uw), there will be no net force (f = 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 dissipative heat (losses) created 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.

3.7 List of Variables with (linear) Equations

For now, we'll forego, or neglect, the effect of frictional and

gravitational resistance on the changes in direction of circular motion,

and centripetal displacements. They are somewhat, if not considerably,

more complicated than for linear motion and linear displacements:

Linear displacements may act along curved and circular paths, where

changes in speed (vt-vi) will occur according to the equations

pertaining to linear displacements: In order for these 'linear formulas

and equations' to be complete, they must include the reacting, opposing

forces (uw) and/or "repulsions" (uwt):

Applied Force: f = 2sw/(gt2)+uw = (vt-vi)w/(gt)+uw

NET (Effective) Force: f-uw => 2sw/(gt²) = (vt-vi)w/(gt)

Dist. displaced: s = (vt-vi)t/2 = (f-uw)gt²/(2w) = d-vit = vt x t-d

Displacement:* s/t = (vt-vi)/2 = (f-uw)gt/(2w) = d/t-vi = vt-d/t

Dist. moved: d = (vt+vi)t/2 = vit+(f-uw)gt²/(2w) = vit+s = vt x t-s

Result. Motion: d/t = (vt+vi)/2 = vi+(f-uw)gt/(2w) = vi+s/t = vt-s/t

Time period (Duration): t = sqr(2sw/(g(f-uw))) = (vt-vi)w/(g(f-uw))

Chg. in velocity - speed: Vt-Vi = 2s/t = vt-vi = (f-uw)gt/w

Initial velocity: Vi = vi = vt-2s/t = vt-(f-uw)gt/w = l/t (Inertial

motion @ start)

Final velocity: Vt = vt = vi+2s/t = vi+(f-uw)gt/w = 2d/t-vi

(Inertial motion @ end)

Weight, due to grav.: w = (f-uw)gt²/(2s) = (f-uw)gt/(vt-vi)

Accel.due to gravity: g = 2sw/((f-uw)t²) = (vt-vi)w/((f-uw)t)

Displac.due to grav.: g/2 = sw/((f-uw)t²) = (vt-vi)w/(2(f-uw)t)

Mass: (f-uw)t²/(2s) = (f-uw)t/(vt-vi) = w/g

Static Inertia: 2w/g (the meas. of a body's "@ rest" inertia)

Acceleration: a = 2s/t² = (vt-vi)/t = (f-uw)g/w = 2d/t²-2vi/t

Coefficient of Reaction: u = f/w-2s/(gt²) = f/w-(vt-vi)/(gt)

u = (cosine(oo)uf + sine(oo)) (as modified for slopes)

Reacting Force: uw = f-2sw/(gt²) = f-(vt-vi)w/(gt)

Reaction, "Repulsion": uwt = ft-2sw/(gt) = ft-(vt-vi)w/g

Applied Effort, "Impulsion": ft = 2sw/(gt)+uwt = (vt-vi)w/g+uwt

Net (Effective) Impulsion: (f-uw)t => 2sw/(gt) = (vt-vi)w/g (Change in

Momentum)

All of these algebraic formulas and equations are expressible in

words of plain language.

Some final notes:

2.7b List of Variables with (some) Equations

Applied Force: f => 2sw/(gt²) = (vt-vi)w/(gt) = v²w/(rg) = ma

(Change in Inertia)

Dist.displaced: s = fgt²/(2w) = (vt-vi)t/2 = d-vit = vt-d = d²/(2r)

Displacement: s/t = fgt/(2w) = (vt-vi)/2 = d/t-vi = vt-d/t = v²t/(2r)

Dist.moved: d = (vt+vi)t/2 = vit+s =vtt-s = (vt=l=d) = sqr(2rs)

Result. Motion: d/t = (vt+vi)/2 = vi+s/t = vt-s/t = (v=l/t=d/t) =

sqr(2rs)/t

Time period (Duration): t = sqr(2sw/(gf)) = (vt-vi)w/(gf) = d/v =

sqr(2rs)/v

Chg.in vel.- speed/dir.: Vt-Vi = fgt/w = vt-vi = 2s/t = d²/(rt)

Initial velocity: Vi = vi = vt-fgt/w = vt-2s/t = (v=l/t=d/t) (Inertial

motion (@ start))

Final vel.: Vt = vt = vi+fgt/w = vi+2s/t = 2d/t-vi = (v=l/t=d/t)

(Inertial motion (@ end))

Weight, due to grav.: w = fgt²/(2s) = fgt/(vt-vi)

Accel.due to gravity: g = 2ws/(ft²) = w(vt-vi)/(ft)

Displac.due to grav.: g/2 = ws/ft²) = w(vt-vi)/(2ft)

Mass: w/g (Use 2w/g as the meas. of a body's "Static" Inertia)

Acceleration: a = 2s/t² = (vt-vi)/t = fg/w = v²/r

Applied Effort: ft => 2sw/(gt) = (vt-vi)w/g = d²w/(rgt) = mat (Change

in Momentum)

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, tortoises, and rams.)

Since exerting force requires muscular effort, or the expenditure

of chemical, or heat energy, things which generally require considerable

effort, and expense to come by, it is vitally important to know how to best

achieve a

maximum desired effect with a minimum of precious effort.

The full measure of this effort is not just the magnitude of the

force alone. The longer a force must be maintained, the greater is the

effort, or energy consumption; in proportion to that duration. So

effort (ft) is the product of the magnitude of the applied force (f) and

its duration (t); which when the force is equal to, or less than the

restraining force (uw) so that the repulsion (uwt) is the same as or

greater than the impulsion (ft) there will be no displacement, and

consequently no work.

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 started

moving, then you can reduce the force to equal that of the opposition

and still keep the thing going.