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 Donald G. Shead 08-13-2003 12:41 PM

21st Century Dynamics

~ 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

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
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.

 Donald G. Shead 08-13-2003 03:28 PM

21st Century Dynamics

have
motion,
The
Thanks Constantine; but the radius of an ellipse is not constant, and the
force _is_ centripetal.

 Constantine 08-13-2003 04:06 PM

21st Century Dynamics

centripetal

Indeed. There is something we are missing here...

Friendly, Kostas.

 Mark Mallory 08-13-2003 07:41 PM

21st Century Dynamics

You were NEVER an engineer, shitHead; you were a draftsman. You're an

For YOU to claim to be an engineer is to insult a great profession.

You're not even qualified for admission a high school physics class. You don't
have the ability. Your "vigorous" studies have been nothing more than
meaningless ruminations in your feeble mind. They have nothing to do with
physical reality.

Nobody cares what you think, shitHead. The rantings of an imbecile are of no
consequence.

<remaining garbage snipped>

 Donald G. Shead 08-14-2003 06:57 PM

21st Century Dynamics

Do you really think it's that good that I might get sued? I hope so; that
would be good publicity!

 Double-A 08-14-2003 11:43 PM

21st Century Dynamics

Why not put the entire textbook online and see what happens?!

Double-A

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