10,000 Words on:
Creating Universal Equations,
with Physics and Mathematics Formulas,
with the Application of a General
Theorem
This e-book demonstrates how to
create universal equations (generalized formulas) with the use of a theorem,
which I am calling theorem-U
This website is
under construction, and the e-book it contains is a work in progress. It may be difficult to understand some of the
sections of the book because it is incomplete, even if you have an adequate
background in mathematics and physics. However, the basic concepts will
probably be understandable, assuming you have the adequate technical background.
2/17/ 2009
By David Alderoty
Phone (212) 581-3740
Email Is RunDavid@verizon.net
Table of Contents
The Necessary Information Needed to
Understand this Booklet
The following paragraphs
contain information about this booklet, the notation and the words used in the
text. This information is needed to
understand the ideas presented in this booklet.
The booklet is primarily
written for people who have advanced knowledge in mathematics and physics. However, readers who are less knowledgeable
will probably be able to understand portions of the text.
The following words
are used as synonyms in this booklet:
paper,
manuscript , text, booklet,
formula(s),
equation(s),
derivation, proof, was determined
general equation, universal equation,
universal formula, general formula
constant, coefficient
Brackets, such as 
or [ ] are used along with parentheses to represent
multiplication. This is often done in
this booklet in a somewhat unusual way, such as the following:
Phrases, such as: the formula becomes, or the
equation becomes implies
that variables, constants, and/or values have been substituted into a formula,
resulting in a new equation.
There are many examples and derivations of equations in this
booklet. Some of these derivations are
extremely simple and others are complicated.
The examples marked AThe simplest cases@ can sometimes be
more confusing than the more complicated cases, because they are so
simple. That is, simplicity can be
confusing under certain conditions.
Thus, if a simple example seems confusing, proceed to the more
complicated examples, which will provide the needed clarification.
My terminology for
the technique I used to create the general equations: The most important
idea to take note of is a concept that I am calling: the theorem
of multiple constants. This can be thought of as a theorem, or the
methodology that I use to create general equations in this booklet. I call the theorem of multiple
constants throughout most of this paper theorem-C for
short Theorem-C is explained in detail in the following pages.
The summary that follows, presents some of the primary ideas in
this booklet. However, it does not
cover all the major concepts and formulas presented in the paper, but it does
cover a few additional ideas.
A Preview and Partial Summary
Over the years,
physicists have tried to find universal constants, such as Planck=s constant and the
gravitational constant. Such constants
have theoretical and practical utility in a number of formulas. However, this paper takes a different
approach, which in no way contradicts the above. Specifically, this booklet deals with the
derivation of general equations or universal equations, which have
coefficients, or constants that are not universal. Such equations can be derived from the
established formulas of mathematics and physics. That is universal equations (as the term
is used in this booklet) are formulas that apply to a number of situations, and
have a different constant or coefficient for each situation. For example, there are many equations that
deal with energy, such as for the kinetic energy of an object moving in a
straight line, the kinetic energy of a rotating solid sphere, the energy of a
photon of a specific frequency of light, the energy produced by a nuclear
reaction, etc. One of the equations derived
in this paper is a general energy equation ( 
) that deals with all of the above. The
general energy equation suggests that energy is the result of moving
particles. (Potential energy, relates
to potentially moving particles.) The concept
relates to a potential that might result moving particles.) This equation (
) works
because the value of 
is different for each
type of energy. Of
course, the final calculations are identical.
That is the numbers that come out of the generalized equations, are the
same that would be obtained from the conventional equations of physics and
mathematics, which should be obvious.
The above might raise the question, why derive general equations? There are many good answers to this
question, which include the following.
Generalized equations represent general principles. Calculations and mathematics by its self do
not necessarily explain what is actually taking place, in terms of a
process or in terms of dynamics, and the generalized equation might provide
more insight in this regard. This might
stimulate further insights, which might result in the derivation of new
equations and principles.
One of the ideas behind the
derivation of general equations is related to a useful principle of creativity,
which is to examine phenomena from different perspectives, or points of
view. Examining anything from multiple
perspectives can lead to a better understanding of the phenomena, and new
insights. When we do this in daily life,
or in the social sciences, we do not know for certain which perspective
is correct. However, with the concepts
that we are dealing with, in this booklet, we can be absolutely certain that
both perspectives are equally correct. We
can be certain of this, because we can easily check by comparing the calculated
results of the general equation, with the related conventional equations. The results will always be the same, unless
there is an error in the calculations or an error in the derivation of the
general equation.
Another advantage of the generalized equations is it is easier to master
and understand a single general formula, representing a general principle, then it is to master the related set of conventional
formulas. For example, the general area
equation derived in this paper is
, and it is for
calculating the area of two-dimensional geometric figures, and also for
calculating the surface area of three-dimensional objects. It is easier to remember this equation, than
the many formulas for the area of circles, triangles, ellipses, rectangles,
squares, and the surface area of spheres, cubes, cylinders, etc. Thus, a single formula may represent an
infinite set of formulas based on one general principle. The word infinite is no exaggeration,
and it becomes apparent when you think of all possible ways that an irregular
geometric figure can be scribbled on a piece of paper. Of course, to use this equation, it is
necessary to know the specific value of the value of 
, which can be
calculated for conventional geometric forms, from the standard formulas. For irregular geometric figures, it would be
necessary to determine the value of by some other means, such as by experimenting
or with the computer. However, the important idea here
is that sometimes the general equation can be used, where the conventional
formulas cannot be used.
The general equations are very useful for exploring mathematical
relationships, and the laws of physics.
It is very easy to establish general equalities between different
general equations. For example, with the
general energy equation, it is easy to derive other equations that relate to
energy at some level, such as equations for work, and power.
I have found from experience, that the general equations can be quite
useful and practical, when they are used to create specialized formulas for a
specific purpose. For example, if you
are operating a nuclear power plant, and you want to know how much energy you
can obtain from a specific quantity of uranium, you can use the general energy
equation. This can be done, because the
general energy equation, also encompasses Einstein=s equation
, This becomes obvious when
you exam the formulas. You can start
with the general energy equation (
) and workout a
specific value for 
that relates to the amount of energy obtain
from a given amount of uranium. Another
advantage is that it is not necessary to even use a standard set of
units, as long as the value is calculated
to use the mixed units you plan to utilize in the calculations. For example, the mass, could be measured in
terms of pounds, (in spite of the fact that the pound is a unit of force, not of
mass). V in this case is the speed of
light, and if you are only interested in the energy, it will be multiplied by into one numerical
coefficients.
The final answer, E, or energy could be in calories, BTUs, or any other
unit of energy.
All that was stated in the
previous paragraph is perhaps quite obvious, for anyone that has a reasonable mathematical
background, and could easily be obtained by using educated commonsense. If you use commonsense or
the theoretical framework of the general energy equation, you will end up with
exactly the same formula, which would look like this
. This formula obviously states the mass of
uranium multiplied by a constant is equal to the energy that can be
obtained. However,
theorem-K. and the general equations derived
with it, provide a theoretical or mathematical framework, which sometimes
provides additional utility and insight.
I have use the above idea when creating
electronic documents with MathCAD. This
involved one formula, but instead of using a single K value, I used a number of constants
with different values, such as for feet per second, for inches per second, for
miles per second, etc. This arrangement
require a single set of numerical inputs, but multiple outputs are produced as
result of the multiple K values, and related program design.
Thus, in this booklet, I applied theorem-C, area, volume, energy,
momentum, moment of inertia, etc. One of
the basic purpose of this paper is to explain the
durations of the general equations, with the application of theorem-K, which
will be seen in the following pages.
The above might raise the question: What is Theorem-C and how can it be
used to derive general equations? A
simplified, and somewhat incomplete, explanation is presented in the
following paragraph.
When Theorem-Cis applied to a set of formulas, that differ by the value of a coefficient, it is
usually only necessary to take one of the simpler formulas in the set and
replace its coefficient with a K. The
general equation that results will represent all the equations in the set, and
an infinite number of equations that were not in the original set. For example, a simple volume formula is 
This formula has a
coefficient of 1. If the 1 is replaced
by 
such, as 
, the equation that
results is a general equation that applies to the volume of most, if not all,
geometric forms. The value of the term will depend on the specific geometric
form.
Introduction
to Theorem-C
This paper
illustrates a method of creating universal or general formulas, which
may relate to general principles. I
am calling this method theorem-C, and I apply it, to a number of
formulas from mathematics and physics in this booklet.
Theorem-C becomes more understandable and meaningful
after seeing its application with many equations. Thus, the reader should reread the following
discussion and carefully examining the applications of the theorem in this
booklet.
Theorem-C states:
When there is a set
of equations that consist of similar variables and differ by a coefficient,
one general equation representing the entire set can be written, by
representing the value of the coefficients
with one variable, (one general term), as illustrated below:
If the following represent a set of
equations, and 
and 
represent coefficients with different
values.
Then this set of
equations can be represented by the following:
where K is a variable
that represents the values of any of the coefficients, and A, B,
C, and D represent the variables in the equations. All of the above is true by the postulate of
substitution.
The above concept can also be applied to exponents, when there is a set
of equations that differ by the value of their exponents, as follows: 
The entire set of
equations can be represented by
, where n represents
the different values of the exponents in the equations. (That is n equals A or C or E or B or D or
F.) W, X, Y, and Z represent the
respective variables in the equations. The
same basic idea can be applied to inequalities.
When Theorem-C is applied to a set of formulas, it is usually only
necessary to take one of the simpler formulas in the set and replace its
coefficient with a K. For example, the
simplest area formula is A=LW. This
formula (A=LW) is obviously for the area of a rectangle, and it has a
coefficient of 1. If the 1 is replaced
by 
as such 
then the equation that results is a general
equation that applies to the area of most, if not all, geometric forms, with a
different value of K for each form. (I
will discuss this in more detail in the chapter dealing with area.)
Theorem-C can also be thought of as a method of deriving an
infinite set of equations from a finite set of formulas. For example, there are a limited set of area
formulas, and when Theorem-C is applied to an area formula, an infinite set of
area equations result, which is represented by
This becomes
obvious, when you think about the number of geometric figures that can be
scribble on a piece of paper. The number represent an infinite set, and any figure that you
scribble will have a unique value of Ka. In addition, even if you increased or
decreased the size of the figure you scribbled, with a photocopy machine, its Ka
value will remain the same.
The General Area Equation
I will start this discussion with
the area of a rectangle, which is 
, were: 
, 
, and 
.
This simple formula, as already explained in the introduction, can be
generalized so it represents the area of all, or almost all, two-dimensional
figures. In addition, the resulting
formula will represent the surface area of most, if not all, three-dimensional
forms. The generalizing of the formula,
as implied above, simply involves replacing the coefficient, which is 1, with a
That is, the formula can be generalized by the application of
theorem-C. This means that the formula
is simply modified as follows: , were the value of relates to the pacific geometric form, and L
and W are linear measurement, such as length, width, length of a base, the
length of a radius or a diameter, etc.
Some examples of the value of are as follows: for a rectangle 1,
for a triangle 
, for a circle
, etc. This suggests a fundamental principle,
which is: Area is the product of two linear measurements, of a geometric
figure, multiplied by a constant.
(The two linear measurements are usually perpendicular to each other.) This concept of area will become apparent
with you look at the following examples.
The simplest cases involving the
general area formula is seen when it is applied to a rectangle
The most obvious case, as suggested above, is when the general area
formula () is applied to a rectangle. When this is done 
, 
, and 
The obvious proof for the value of is as follows:
When the value of is substituted back into the general area
formula, the equation for the area of a rectangle should result, if the above
proof is correct. This is done as
follows: 
.
The general area formula applied to
a square:
When the general area formula () is applied to a square, 
, 
side of the square, and 
other side of the square. That is, 
(a side of a square).
The formal proof of the value of is obvious, and is as follows: 
When the value of and the variable S is substituted back into
the general area formula, as a test of the above, the formula for the area of
the square results. This is done as
follows:,
.
The general area formula applied to
any type of parallelogram:
When the general area formula (
) is applied to any type of
parallelogram: , 
height of the parallelogram and length of its base. The proof of the value of is as follows:
When the value of and the variables b and h are substituted back
into the general area formula, as a test of the above, the conventional formula
for the area of a parallelogram (
) results, as follows:
The general area formula applied to
a triangle:
When the general area formula () is applied to a triangle 
, height, of the triangle, and base of the triangle. This was derived as follows:
When the value of is substituted back into the general area
formula, the equation for the area of a rectangle should result, if the above
proof is correct. This is done as
follows:
The general area formula applied to an
equilateral triangle:
When the general area formula () is applied to an equilateral
triangle, based on its three equal sides,
, L = the side of the triangle, and
W also = a side of the triangle, (or L = W = side of the triangle). The value of was determined with the conventional formula 
as follows:
When the value of is substituted back into the general area formula,
the equation for the area of a equilateral triangle
should result, if the above proof is correct.
This is done as follows:
When the value of is substituted back into the general area
formula, the equation for the area of a circle should result, if the above
proof is correct. This is done as
follows:
The general area formula applied to
an ellipse:
When the general area formula () is applied to an ellipse, 
, and 
longest radius, and 
shortest radius. This was derived with the conventional
formula for the area of an ellipse (
) as follows:
When the value of is substituted back into the general area formula, and L and
W are replaced by a and b, the equation
for the area of an ellipse should result, if the above proof is correct. This is done as follows:
Note that the general formula
derived in the above proof, 
is a general formula that can be used to
calculate the area of both a circle and an ellipse. In the case of the circle, any two radii of
the circle equal L and W. That is r = L
= W.
The general area formula applied to
an ellipse, with the length and width used as factors:
Another version of an area formula, for an ellipse, is based on the
actual length and width of the ellipse. With this formula 
and length, and width of the ellipse. This was derived with the conventional formula
for the area of an ellipse () as follows:
When the value of is substituted back into the general area
formula, and L and W are replaced by a and b,
the equation for the area of an ellipse should result, if the above proof is
correct. This is done as follows:
Note that the formula derived in the
above proof, 
is a general formula that can be used to
calculate the area of both a circle and an ellipse. In the case of the circle, any two diameters
of the circle equal L and W. That is d =
L = W.
Thus, the application of Theorem-Cin relation to area, should be
obvious, after examining the above examples.
Thus, the following four examples are presented without formal proof.
The general area formula applied the
lateral surface area of a right circular cylinder:
The general area formula () can be applied to the lateral
surface area of a right circular cylinder, if 
, height of the cylinder, and radius.
Thus, the formula become 
.
The conventional formula is usually written as 
, when 
, and 
.
The general area formula applied the
lateral surface area of a right circular cylinder, with the diameter used as a
factor: The above formula, for the lateral surface
area of a right circular cylinder, can also be basted on the diameter of the
cylinder. In this case:
, height of the cylinder, and diameter.
Thus, the formula becomes .
The general area formula applied to
the surface area of a sphere:
The conventional formula for the surface area of a sphere is usually
written as 
(
).
When the general area formula () is applied 
and 
.
The formula can be written as .
The general area formula applied to
the surface area of a sphere, based on the diameter:
A formula for the surface area of a sphere can be written as 
(
). When the general area formula () is applied to this equation and 
.
This can be written as .
The General Area Equation and Calculus
It is possible to apply the general
area equation to calculus, in relation to the area under the curve. This concept has theoretical value,
but it probably is not practical, because to find the value of the Ka requires calculus. (Keep in mind that the primary goal of this
booklet is to demonstrate a theoretical framework, as opposed to finding an
easier way of performing calculations. )
The general area equation can be
applied to calculus by first delineating an area that surrounds the curve. There are many ways that this can be done,
because the area can be of various sizes, and shapes, as long as it surrounds
the curve. The following formula is one
way of delineating an area surrounding the curve.
The area delineated by this formula can be converted to the area under
the curve by multiplying it by constant, 
which will differ for each mathematical
expression, depending on the exponents involved, the number of terms, etc. This results in the following equation for
area under the curve:
or 
The value of Ku can be
determined with the following formula:
The duration of this formula, should
be obvious.
In terms of the general area
formula, all of the above can be expressed as follows:
I will demonstrate the above with a few simple
examples in the following pages.
If 
and if we are interested in the area under the
curve from 
, which is the same as saying: with
the ordinance (0,0) and (2,4). I will
start by working out the value for Ku then proceed to work out the L
and W, as follows;
Thus, the area under the curve, for from X1=0 to X2=2,
is 
.
This can be checked by the conventional methods of calculus follows:
Another example is 
Thus, the area under the curve, 
X1=2 to X2=6, is 78.666666667 This can
be checked by the conventional methods of calculus as follows:
Thus, the area under the curve, 
X1=1 to X2=2, is 4.8333333333 This can
be checked by the conventional methods of calculus as follows:
Another example is 
Thus, the area under the curve, 
X1=1 to X2=2, is
10.83333333. This can be checked with
the conventional methods of calculus as follows:
The General Area Equation Applied to
Irregular Geometric Figures
As explained earlier, the general
area equation () can be applied to any regular
geometric figures, including the geometric figures randomly scribbled on paper,
or any unusual shape of a two-dimensional figure. The only requirement is to find the correct value for the geometric form that
you are working with. This can be done
in many ways. However, it may not be
practical to determine the value for irregular geometric forms that you
and/or others will not use repeatedly.
This will become apparent as follows.
To find the value of an irregular geometric
form, it is usually necessary to find the area, of the geometric figure, the
first time by applying difficult and time-consuming methods, such as, by
dividing the figure into rectangles of various sizes and calculating the area
for each rectangle, and adding results together. Once this is done, you can obviously
determined the appropriate value with the general area equation, by
solving for .
Calculating the area of irregular figures can probably be done
automatically with the appropriate software, such as by electronically counting
all the pixels that are contained within the geometric figure.
Now I will try to clarify the above
ideas with some examples. If we have
some irregular geometric figure that has an area of 28 
and measures L=7 Cm,
W=5 Cm, then the value can be determined with the
general area equation as follows:
A concept discussed above is applied
to surface area in the next chapter.
The Ka Value of the
General Area Equation, in Relation to Varying
Degree of Flatness or Roughness of a Surface
The Ka value of the
general area equation, in relation to a square, can be used to quantify the
degree of flatness or roughness of a surface.
The idea can have many theoretical and practical applications. The principle here is probably obvious, but I
will explain it as followers.
Any square or rectangle traced a
perfectly flat surfaces will have an area that is equal to its length
multiplied by its width. From the
perspective of the general area formula the value of is 1, or = 1.
However, if the surface area, consisting of a square or rectangle,
contains holes, bumps, pits, peaks, or valleys, its surface area is greater
then a perfectly flat area. The reason for this is bumps, holes, and other
imperfections have their own surface area.
For example, if you are on a perfectly flat surface that has a length of
L and the width of W, its surface area is LW.
However, if you dig a perfectly round hole that is h units deep,
and has a radius r, the surface area will increase by 
. Thus, the total surface area would
be +LW.
If you
now applied the general area formula to determine the value of as was done below, you would find
that the value has increased. Of course, the
values for L and W would remain same.
Noticed that Ka no longer
equals one, its value has increased by 
.
His concept can be applied to the irregular
surface areas that relate to specific geographic locations, especially as
assessed from an aircraft or satellite. In
such a case, a square section of land could be evaluated, with radar or other
methods, with the resulting data fed into a computer to determine the value. High values in wilderness areas, would
indicate very rough terrain, such as rocky mountainous areas. Low values would mean relatively smooth
terrain, such as farm land. This can
also be applied to the oceans, where a low value would obviously indicate the flat surface area
of calm seas, and high value would indicate rough seas. The total surface area of a city street, as
measured from high-altitude, would change according to the number of motor
vehicles, the higher the value the higher the concentration
of motor vehicles.
As roads and sidewalks age, their value tends to increase as a result of cracks,
potholes, and other damage. Thus, an
assessment of sidewalks, streets, roads, and highways can be made in terms of a
value, with low values relating to well
maintained surfaces, and vice versa.
Thus, the value can be used to express the
relative degree of smoothness or roughness in mathematical terms for any type
of surface.
Many things that age, including
roads, steels if it rusts, the exterior of buildings, and even human skin,
increase in area, with an increase indicate value, at a certain rate. Thus, it
might be feasible and practical to determine the rate that the value is increasing, or the rate
that a given square area is increasing.
When evaluating the degree of
roughness of smoothness of surface, it is important to understand that the
concepts of smoothness and roughness are relative to a predefined frame of
reference, in relation to the size of the surface variations, ( holes, crests, pits, peaks in valleys). This may not be an obvious
concept. This suggests that it is
necessary to delineate the size of the peaks and crests that you want to used to evaluate the surface. For example, if you evaluate the surface of a
typical glass window in terms of and crests and peaks that are larger
than one-tenth of a millimeter, the value would be 1, which indicates a perfectly
smooth surface. However, if the same
window was evaluated in terms of atomic dimensions, the value would be much greater than 1, indicating
a very rough surface. That is, from the
frame of reference of atomic dimensions the surface of the glass has many
surface variations, peaks and crests.
Some less extreme examples are, a piece of felt would be perfectly flat =1, to the naked eye, but at a
microscopic level it would be quite rough and the value of would be much greater than 1. Generally, it would be necessary to define is
a significant surface variation, under specific set of circumstances. Generally, it would be necessary to define
what is a significant surface variation is under given set of
circumstances. For example, a
significant surface variation for a piece of glass might be 100,000 of an inch,
especially if it was going to be used for optical equipment. A surface variation for a newly built
highway might be 1/16 of an inch might be considered in significant. However, for a country road, composed of
small rocks, a half-inch variations in surface might
be acceptable, and anything greater than a half-inch might be significant, and
less than desirable. For a mountain road
one inch variation might be quite acceptable.
For toa hiking trail, six inches might be
quite acceptable.
The General Volume Equation
The basic formula for cubic volume,
length, times width multiplied by height (), can be generalized with
theorem-K, which yields .
This formula can also be obtained by multiplying the general area
formula by height. This generalized
formula applies to the volume of most, if not all,
three-dimensional structure. This
formula, () suggests that volume is length
times width, times height, multiplied by a constant. The following examples will make these ideas
obvious.
The simplest cases:
A Three-dimensional box, and the
general volume formula:
The simplest example involving the general volume formula () is a three dimensional box, where 
, length, width, and 
height of the box. This results in the basic formula that we
started with, .
The general volume formula and a
three-dimensional box, with equal height, length and width:
Another very simple example, involving the general volume formula () is a box with equal length, width
and height. In this case and 
.
Thus the formula become S3=V.
The general volume formula applied
to a sphere:
When the general formula for volume is applied to a sphere, 
, and 
.
Thus, the formula, 
, or in the more traditional form 
.
The value of was derived with the conventional formula for
the volume of the sphere, as follows.
The general volume formula applied
to a sphere, using the diameter
It is possible to use the diameter to
calculate the volume of sphere. This is
essentially based on the length times width, times height concepts. For this version of the formula
, 
diameter, and the formula is
, or 
, if 
.
The general volume formula applied
to a right circular cone:
For the volume of a right circular cone 
, and L and W equal the radius of
the base of the cone. H equals the
height of the cone. Thus, the formula is 
.
This formula is usually written as 
.
The general volume formula applied
to a right circular cone, with the diameter as a factor:
The above formula, for the volume of a right circular cone, can be
basted on the diameter, if 
and if 
diameter. Thus the formula is 
or 
, when 
.
The general volume formula applied
to a cylinder:
The volume of a cylinder can be represented with the general volume
formula, if 
, and L and W both equal the radius
of the base. The height of the cylinder
is represented by H. Thus, the formula 
.
This formula is usually written as 
.
The general volume formula applied
to a cylinder, using the diameter:
The above formula, for the volume of a cylinder, can also be written in
a form that is basted on the diameter, if 
and if diameter.
Thus, the formula is 
.
This can also be written as 
, when 
diameter.
The general volume formula can be
applied to three-dimensional objects with regular dimensions, such as rocks, sand,
and even the volume delineated by various equations and calculus expressions
with three variables. However, as was
the case with the general area formula, you have to determine the value of.
It was quite difficult to do this with the area formula, for irregularly
shaped geometric figures, but determining the value for the general volume formula
is often quite easy, no matter how irregular the object is. Specifically, you can easily determine the
volume of almost any objects by placing it water, and measuring the increase in
volume. This simply involves measuring
cup partly filled with water. For
example, if you have 100 cc of water in the measuring cup, and you place a rock
in it, of water rises to 120cc, than the rock has a volume of 20cc. Measuring the height, length and width of the
rock will allow you to calculate value of .
This is of course all theoretical, because if you wanted to know the
volume of the rock, you obviously do not need the general volume
equation.
The General Dimension Equation
It is possible to write one equation
that represents distance, area, volume, and theoretical geometric forms with
more than three dimensions. This
equation is as follows: 
.
I am calling this formula the general dimension equation. The equation, in theory, can be used to
replace all, or almost all, formulas for distance, area, and volume.
Obviously, the value of Kg will
depend on the shaped and dimensions of the geometric forms.
When the general dimension equation is applied to distance, 
equals the total linear distance, as measured from two
points, 
the total distance, and 
This is the same as saying that 
is ignored.
For example a linear distance between town-X and town-Y might be 100
miles. However, an automobile might have
to travel 200 miles through winding roads to get from town-X to town-Y.
and 
equals the linear distance of 100
miles. Calculated as follows:
With a different value for Kg,
equals the total nonlinear dissidents travel
between two points, and the linear distance. For example, a linear distance between
town-A and town-B might be 10 miles.
However, an automobile might have to travel 20 miles through winding
roads to get from town-A to town-B.
and =20, the linear distance which is 10
miles. Calculated as follows:
When the general dimension equation is applied
to area, represents the area, 
represents linear measurements, such as length
and width, and 
.
This obviously is the same as saying that the irrelevant terms 
are ignored.
When the general dimension equation
is applied to volume represents the volume, 
represent linear measurements, such as length,
width and height, and the terms represented by 
,
This is the same as saying that the irrelevant term, represented by is ignored.
The expression, of course indicates that there are potentially
a limitless number of terms present in the formula, to deal with theoretical
geometric forms that have more than three dimensions. Thus, the equation has four terms, besides 
for a four-dimensional figure, such as 
and five terms for and five-dimensional
figure, such as 
etc.
Of the general dimension equation
probably does not have any practical utility, but in effect this equation
represents a general theory, dealing with spatial dimensions. That is distance, which is one dimension,
area that is two-dimensions, and volume that is three-dimensions. In addition, the equation deals with
theoretical and unknown forms that have more than three dimensions. The equation, deals with everything that is
known, as well as the unknown.
The General Distance Equation
All of the general equations
presented to this point, can be derived with the general distance equation 
.
However, the general distance equation is the simplest formula in this
booklet. Keep in mind that simplicity
can sometimes be confusing.
The general distance formula () apparently consists of only three
terms, which are as follows: 
distance as measured from two points. 
measured distance or
the total distance traveled. Keep in
mind that the measured distance or the total distance traveled is not
necessarily the same as the distance measured from two points. 
coefficient that
relates to the actual distance, as measured from two points. That is 
can be thought of as a correction factor that
when multiplied by X, will equal the distance as measured from two points. For example, if a car travels 10 miles in a
perfectly straight line, =1, X=10 and it is apparent that
there is very little to calculate, D=10 miles.
That is the actual distance is 10 miles.
However, if the car traveled in a zigzag manner through many winding
roads might be equal to 2.
Thus, in this case the distance D is obviously only five miles, but the
car travels 10 miles as a result of the nature of the roads.
The general distance equation can be
used to determine the efficiency of a travel route, by solving for as such: 
.
Thus, the
higher the value the greater the
efficiency of the route, but it cannot exceed 1.
The N-dimensional Equation
It is possible to derive a very simple general
equation that represents distances, areas, volumes, and theoretical geometric
forms. The derivation of this general
equation can be obtained by applying Theorem-Cto the
exponent on the term of the general distance equation ().
(The exponent of the X
term is of course 1.) This
becomes understandable, if we realize that 
The idea becomes
even more apparent if we realize that 
four-dimensional theoretical form, and 
five-dimensional theoretical form, etc. Thus it can be apparent, that the general equation
is simply a coefficient multiplied by 
.
I will write this equation as 
.
This equation apparently has less terms than the general dimension equation discussed
above. Thus, value of the exponent, n,
is determined by the type of geometric figure, as follows: n=1 for distance,
n=2 for area, n=3 for volume, n=4 for theoretical four-dimensional structure,
n=5 for a theoretical five-dimensional structure, etc.
I am calling this formula () the n-dimensional equation. This equation is much simpler than the
general dimension equation () presented above. However, the n-dimension equation is equal to
the general dimension equation, and the value of the X term can be determined
by setting both equation equal to each other as
follows:
When the n-dimensional equation is
used to calculate anything that has linear measurements that are not the same,
such as the length and with of a rectangle, there appears to be a problem,
because there is only the X term.
However, if the above relationship is used, the problem should be
solved. Of course, this equation is not
really meant to be practical, it is meant to illustrate an interesting
concept.
The General Equation for Moments of
Inertia
Theorem-Ccan
be applied to the moment of inertia, which results in
a general formula for moment of inertia, 
This idea will become clear with the following
examples.
The moment of inertia of a thin ring
or a hollow cylinder:
When the general moment of inertia formula () is applied to a thin ring, hoop, or
a hollow cylinder, with all of the mass assumed to be on the outer surface (see
note), with the axis in the center, 
.
This was derived with the conventional formula (
) as follows:
If the value of 
=1 is placed back into the general
moment of inertia equation (
), as a test of the proof, as such 
, the conventional formula is
obtained.
*Note, of course, it is not possible
to have a thin ring, or hoop, or a hollow cylinder with all of the mass on the
outer surface. However there are real
structures that are quite thin. With
such structure the thickness will not change the calculated result significantly. Thus, it makes sense to ignore the thickness
in such cases, and assume that all the mass is on the surface. When the thickness is significant the formula
that follows should be used.
The moment of inertia of a thick
ring or a thick hollow cylinder, and related ideas:
If the general inertia formula 
is applied to a ring,
hoop, or a hollow cylinder, that is relatively thick, with the axis in
the center, 
when 
, or when 
(See the explanatory note that follows this
paragraph.). 
=The smaller radius, which is the
radius of the hollow space. The larger
radius is 
, which includes the hollow space
and the solid structure surrounding it.
Note, both of the above, will result
in the same answer. However, one can
argue that from the point of view of physics that and is the correct choice. This is not important from the perspective of
theorem-K. The following proofs will
prove further insight.
Proof for when was derived with the conventional
formula 
as follows:
The moment of inertia of a solid disk
or cylinder:
If the general inertia formula, () is applied to a solid disk or
cylinder, with the axis in the center, .
This was derived with the standard formula (
) as follows:
If the value of is placed back into the general moment of
inertia equation () as a test of the proof, as such 
the conventional formula is obtained.
If the value of is placed back into the general moment of
inertia equation (), as a test of the above proof, the
conventional formula is obtained, as follows: 
.
The moment of inertia of a thin
uniform rod:
If the general inertia formula is applied to a rod that is thin, and uniform
in shape and mass, with a perpendicular axis in the center, the following
apples: 
if 
.
L is the length of the rod from one end to the other. This was derived with the conventional
formula (
) as follows:
If the value of is placed back into the general moment of
inertia equation (), as a test of the above proof, the
conventional formula is obtained, as follows:
.
The above, (the moment of inertia of
a thin uniform rod ) was basted on the length of the
rod from one end to the other, as indicated.
It is also passable to derive a value for basted on the radius of the rod, if
one-half of the length of the rod is considered to be the radius. This can be done with the conventional
formula () as follows:
The moment of inertia of a thin
uniform rod with a perpendicular axis at one end:
If general inertia formula () is applied to rod that is thin,
and uniform in shape and mass, with a perpendicular axis at one end of
the rod, the following will apples: 
and 
which is the length of the rod. This was derived with the conventional
formula (
) as follows:
Energy Formulas and Concepts, and
the General Energy Equation
A general energy equation can be
derived by applying Theorem-Cto the kinetic energy
formula (
), which yields 
.
This generalized formula () applies to the energy of all types
of moving particles, even photons. In
addition, it is possible to derive a number of other generalized equations with
the general energy formula ().
This will become apparent in the following paragraphs. However, before I present the examples, I
well discuss some related concepts.
I will use the term E for all types of
energy, including kinetic, in the formulas. The term V is not limited to the simple concept
of velocity. The general energy equation
() deals with all types of moving
particles, including randomly moving gas particles, circular motion, as well as
objects moving in a straight line. Thus,
V represents the more general concept of speed. The term m stands for mass, or mass equivalent 
when the equation is applied to photons.
Energy at some level relates to
movement, or potential movement, which takes place at a certain rate of
speed. Speed of moving particles is an
important concept of energy. There are
many different ways that particles or any object can move, such as swinging
back and forth, unpredictable movements from the wind, deliberately walking or
driving from one location to another, etc.
However, I am dividing the many types of movements into three general
categories, which are velocity, nonlinear
movements, and Repetitive motion. The three categories are explained as follows:
$
Velocity
involving change in location along a straight line. Examples are a car or jet plane moving under
ideal conditions.
$
Nonlinear movements, that involves change in location or position that is not
along a straight line, and the movements might be random. Examples are randomly moving gas particles,
the movements of molecule comprising a liquid, and the movements of caged
animals.
$
Repetitive motion, that involve movements that are not the
result of any significant change in location.
Examples are any type of rotational motion, such as from an electric
motor, the back and forth movements of a pendulum, the vibrating membrane of a
stereo speaker system, vibrating vocal cords, and the vibrations of atoms and
molecules comprising a solid.
$
Even
though the three types of speed velocity, nonlinear
movements and repetitive motion, are very different, they
all relate to energy. That is energy is
required to start the movements, and when the movements are stopped energy is
released or transferred to another object.
For example, if you stop a moving pendulum with your hand, the energy
from the pendulum will be transferred to your hand. If a swinging pendulum is permitted to
swinging independently, its energy will be released into the surrounding air.
The general energy equation can be used in calculations involving any of the velocity,
nonlinear movements, and repetitive motion.
The general energy equation (), and the formulas in this chapter,
deal with all of the movements mentioned above.
Many of the formulas in this chapter
deal with kinetic energy, and a brief discussion may help clarify the ideas I
am presenting in this chapter. Kinetic
energy is a relative concept from two separate perspectives. It is relative from the perspective
delineated by Einstein, just as mass, and time, are which are only relevant at
extremely high levels. The relativity I
am discussing in this section is more obvious, even at small velocities. Specifically, kinetic energy is relative,
just as velocity is relative. That is,
there is no absolute velocity or kinetic energy of a specific object. If you measure the velocity of an object from
different reference points, you will get different velocities. For example, if the speedometer of an
automobile reads 50 miles an hour, it is basted on defining the road as a
stationary frame of reference. If the
velocity of the car is measured from other reference points, it will generally
result velocities that are much higher or lower. If the car moving at 50 miles per hour, on
the surface of the earth, is measured from the frame of reference of another
planet, the car might be moving several thousand miles per hour. From the frame of reference of the people in
the car, the distant planet might be moving at several thousand miles per hour. The same principle applies to kinetic
energy. The kinetic energy of the car,
moving at 50 miles per hour, will be much greater if measured from the distant
planet that is move thousands of miles per hour.
Momentum, which will be discussed
latter in this paper, is also a relative concept from the two perspectives
presented above. This is suggested by
the fact that momentum is determined by the mass of the moving object
multiplied by its velocity.
Another important idea is the amount
of kinetic energy given off by a moving mass is actually related to the change
of velocity of the mass, as measured from the object that it strikes. The conventional kinesic
energy formula, (
) and the general energy equation () assumes the change in velocity is
from V to 0. This is not always
true. Thus, a modification of both
formulas is justified, as such:
The following will illustrate this
obvious point:
Kinetic energy remains within the system,
until there is an interference with the moving object from another object or
forces, such as gravity. For example, a
stone moving at 10,000 miles an hour away from point R, in space, will retain
its velocity and kinetic energy. The
kinetic energy will only be released if the stone collides with another object,
especially if it is moving at several thousand miles an hour toward point R,
which could result in disintegration of the stone, and a flash of light and
heat. However, if the stone collides
with another stone that is also moving slightly slower away from point R, at 9999.8 miles per
hour, very little energy would be released.
The rocks might bounce off each other in such case. The velocities would not change very much
either. If the rocks that collided had
identical masses, the change in velocity would be only 0.1 miles per hour. The amount energy that is released is
related to the change in velocity, as well as mass. If two entities collide and there is very
little change in velocity, little energy would be released.
The simplest case of kinetic energy
and the general energy equation ()) are presented below
The kinetic energy of an object
moving in a straight line: The most obvious example, is when an object is
moving in a straight line. In this case 
.
The value of can obviously be confirmed with the kinetic
energy formula () as follows:
(Incidentally, the above formula (
) can be considered another general
energy equation. When the value of is positive the equation relates to the amount
of energy obtained from a specific quantity of mass, in relation to a
specific type of exothermic reaction. (Exothermic means here a
releases of energy, such as in the form of heat, light, electricity, or
physical movement). The value of will relate to the type of reaction. For example, will have one value for nuclear reactions
involving uranium, and another for plutonium, and another value for the
combustion of coal, another value for the combustion of natural gas, another value
for the combustion of gasoline, etc.
When the value of is negative the equation relates to an endothermic
reaction. (Endothermic means here
a reaction that absorbs heat or other type of energy.)
Rotational Kinetic energy and
related concepts
The kinetic energy of a thin
rotating ring or a hollow cylinder, and related ideas:
If the general energy formula () is applied to a thin rotating ring
or a hollow cylinder, with all of its mass assumed to be on its outer surface,
with the axis through its center the value of .
The formal proof is as follows:
The formula 
(
velocity, 
wavelength, and 
frequency) is the starting
point. However, this equation does not apply to all situations where
frequency and wavelength are involved.
For example, a particle, such as a molecule or atom might be vibrating
at a very high frequency, and have a velocity of 0. This is the case in solid material, where the
atoms and molecules vibrate at high frequencies, but they essentially remain in
the same position. That is, the
particles making up solids vibrate, but they have a zero velocity. Thus, in such case, frequency multiplied by
the wavelength might not equal the actual velocity of the particle(s). However, the vibrations as previously
indicated repetitive motion, that relates to
energy. Even in solids, the vibrating
atoms can be thought of as moving objects that return to the same position, similar to the way a pendulum moves back and
forth.
If Theorem-Cis
applied to the resulting equation would be a more general
representation of the concept, as such 
.
This equation would apply to cases where the frequency
multiplied by the wavelength was greater than or less than the actual velocity. With this formula () and the general energy equation (), a formula for energy basted on
frequency and wavelength can be derived as follows:
Thus, the new equation is 
.
This general formula is the same as 
when 
and 
.
That is, when the wavelength is equal to the radius, and the revolutions
per unit of time is equal to the frequency, the equation () applies to a rotating object. The same general idea applies to 
when and 
angular displacement per unit of time. For 
and 
,
with appropriate K values.
All of the above can be summed up
with the following statement. In
relation to the general energy equation ()
or 
.
Thus, in the examples that follow I will simply indicate this
relationship instead of giving the K values for 
, ,, and .
The kinetic energy of a thick
rotating ring or a thick hollow cylinder:
If the general energy formula () is applied to a ring, hoop, or a hollow
cylinder, that is relatively thick, with the axis in the center 
, when 
.
is the is the
velocity of the inner surface of the ring or cylinder, which moves slower than
the outer surface, because of the shorter radius,
. 
.
is the velocity of
the outer surface of the ring or cylinder, which moves faster than the inner
surface, because of the longer radius, 
.
The formal proof is as follows:
If the value of is substituted back into the result is as follows:
.
The kinetic energy of a rotating
solid disk or cylinder:
If the general energy formula () is applied to a solid disk or
cylinder, with the axis in the center, , and the resulting formula is
This can be seen as follows:
The formula
in relation to the V term is for
linear velocity, which is 
multiplied by the number of revolutions per
unit of time. That is 
, when 
number of revolutions per unit of time. Thus, the above energy formula can also be
written as: 
Thus, the above, can be represented in terms of
the general kinetic energy equation for rotating objects 
When this general equation is applied to a
spinning solid disk or cylinder, 
The kinetic energy of a Solid
rotating sphere:
When the general energy formula () is apply to a Solid
rotating sphere 
.
The proof is as follows:
If the value of is substituted back into the result is as follows: 
. If 
is substituted back into to above, as a file
test of the proof, the following results:
.
The kinetic energy of a thin walled
hollow rotating sphere: When the general energy formula () is apply to a thin willed hollow
rotating sphere 
.
The proof is as follows:
If the value of is substituted back into the result is as follows: 
.
The kinetic energy of a rotating
thin uniform rod, with a perpendicular axis in its center:
When the general energy formula () is applied to a rotating thin
uniform rod, with a perpendicular axis in its center, 
.
The proof is as follows:
If the value of is substituted back into the result is as follows: 
.
If is substituted back into to above, as a file
test of the proof, the following results:
.
The kinetic energy of a rotating
thin uniform rod, with a perpendicular axis at one end:
When the general energy formula () is apply to the above, 
.
The proof is as follows : 
If the value of is substituted back into the result is as follows: 
. If is substituted back into the above, as a file
test of the proof, the following results:
.
The General Energy Equation, Applied
to Temperature (the General Temperature Equation )
The general energy equation, can also be applied to temperature. This should be apparent, because temperature
relates to the average kinetic energy of the molecules and atoms. The greater the kinetic energy of the
molecules and atoms in a specific mass, the higher the temperature, and vice
versa. Thus, it is theoretically
possible to determine the temperature of a specific substance, with the general
energy equation, if you know the appropriate value of for the specific substance, the mass of the
substance, and the average speed of the molecules comprising the mass. Of course, this is not a practical way
to determine temperature, but it can offer some interesting insights, which
will become apparent in the following paragraphs.
It should be obvious from the above,
that the general energy equation () can be slightly modified to
represent the temperature, as follows: 
I am
calling this equation (
) the general temperature
equation. With this equation T=
temperature. The m term
represents a specific mass, such as the mass of one mole of the substance. The V term represents the speed of the moving
molecules. However, particles in a
substance at a specific temperature generally do not move at the same
speed. Thus, the average speed, or
similar concepts, can be used. 
is a constant that
relates to a specific substance(such as hydrogen ), and the units of mass and
temperature that are used in the formula.
The general temperature equation can
be modified so it can be used to find the speed of molecules at the giving
temperature. To do this the general
temperature equation () can be transformed by solving for
the V term and replacing the 
as follows:
I am calling this formula 
the equation of a set of moving
particles. This equation can easily be
used to find the average speed of molecules in a gas. It can also be used to find the root mean
square speed, and the most probable speed of molecules in a gas, as well
as an infinite number of other undefined speeds. This is possible, because there is a
different value of for each type of
speed. These constants are available in
almost any physics or chemistry college textbook, and are used with
conventional formula to calculate theoretical velocities for gasses molecules.
m/s for the average speed
m/s for the root mean square speed
m/s for the most probable speed
The above values are for ideal
gases. Gases that do not behave similar
to ideal gas, will have values that are different from the above. The above values are based on the Kelvin
temperature and the answers are in meters per second. When using the above values
for the
temperature, T, is based on the above the Kelvin temperature of the gas. The mass, m is the molecular weight of the
gas. Of course, different units can be
used, if appropriate value for are calculated.
The conventional equations, dealing
with temperature of gases are of course simplifications of reality. In real gases, liquids and solids there is
likely to be some molecular vibration as well as molecular movement. In addition, the atoms, and bonds that make
up molecules also vibrate. This is
especially true of organic chemicals, especially if they have long protruding
molecular structures.
The equation can be modified to represent the vibrations of
molecules in a solid. This is done as
follows:
I am calling this equation 
and its two variations 
the general equation of thermal frequency and wavelength. The challenge associated with these equations
is to determine the correct valid for
, such that
when the mathematics is carried out the correct frequency, wavelength, is
obtained for a given temperature, for specific solid. Of course, each solid would most likely have
a somewhat different value for, partly as a result of its
molecular weight, density, and many other factors. The value for essentially has to be determined
experimentally. This can be done by
measuring the frequency, and wavelength of the vibrating molecules, or atoms at
various temperature, for the specific solid. Then the value of can be
easily calculated. That is to
say, if we know the temperature, the molecular weight of the solid, and the
frequency and wavelength, at the specific temperature, we can obtain the value
of for that substance.
The above equations would only apply
to materials that produced a measurable degree of molecular or atomic
vibrations at a specific temperature.
Most likely, this would work well with metals, and it probably would not
work as well with organic solids, such as plastics, partly because of the
relatively complex molecules involved.
Complex organic molecules can vibrate, and
move many different ways as result of a complex structure associated with
organic molecules, as indicated above.
Specific, sections of the molecule can vibrating or move in its own
unique way.
The above formulas, do not provide
practical answers, but they suggest a series of experiments to determine the
relationship of temperature and vibrations (frequency and wavelength), in
various types of solids. Some solids
might produce several frequencies that relate to different sections of its
molecular structure.
A new concept becomes apparent from
the general temperature equation, but we are no longer dealing with just
temperature.
The concept of temperature,
is very often delineated as the average kinetic energy of the atoms and
molecules in a system. This concept is
expressed mathematically by the general temperature equation, () as well as some conventional
formulas. The above delineation of
temperature, coupled with the general temperature equation suggests an
interesting concept. Why not apply the
concept of temperature to other particles and entities. That is to say, why not calculate the
average kinetic energy of the various particles, or objects, in a system,
which can be done with the general temperature equation, providing the correct
value foris to
determined mathematically or experimentally.
This can be done for electrons, for photons, for stars in a galaxy, as
well as for any object that moves. Of course, the reader must keep in mind that
we are no longer dealing with the conventional concept of temperature. However, we are dealing with a more general
principle that includes temperature, as well as movements involving objects
that are smaller or larger than atoms and molecules. Such calculations,
can be carried out for objects that move or change very slowly, such as the
surface of the earth, as well as for a flash of light. This mathematical concept can even be applied
to people, automobiles, and animals. In
such cases, if the value of T is too high, there is likely to be problems with
dense crowds, and collisions. A simple
practical way of using this concept, for automobile traffic, is to calculate
the value for T on highway, if it is too high, the speed limit can be lowered,
which reduce the value of T. The value
of T can also be reduced by reducing the number of vehicles on the Highway.
Thus, the general temperature equation, is simply a
mathematical concept that can be applied to a large number of situations,
besides just temperature. If this is
confusing, he should realize that most mathematical concepts can be applied to
the large number of objects and situations.
For example, adding and subtracting can be used to calculate the
quantity of money, people, atoms, available departments, automobiles, etc. Most likely, many of the equations in this
booklet, and the formulas in physics, can be used as general mathematical
concepts, and applied to problems that are not related to physics.
The above concept of temperature
raises some interesting questions, but only one has an easy answer:
$
What is the subatomic temperature of matter
under standard conditions?
$
What
would happen if the subatomic temperature was raised to very high levels? (The matter would disintegrate in a nuclear
reaction. This idea might be useful in
producing nuclear energy, such as from deuterium or tritium.)
$
How
can the subatomic temperature be lowered or raised?
The general temperature equation and
entities with internal energy sources
The general concept of temperature
can be applied to objects that have their own internal energy source, such as
the fish in a lake, people in a crowd, automobile traffic in the city,
etc. In this case, especially with fish
and people
, we are dealing with statistically
random motion, similar to the movement of particles in a gas. The faster the motion, and
the larger the number entities the
greater the value of the general concept of temperature. There are obviously a number of equations
that can be worked out based on this idea, and the general temperature
equation. The m can be used to represent the mass of
the finish, people, or automobiles, or the average mess of a single
entity. However, very often the mass is
not particularly relevant in practical situations. The number and velocities
(or speed ) might be the primary concern. Thus, m used to represent the number of
entities.
Equations of this type could be used
to measure potential problems, especially involving collisions, with airplane
traffic, automobiles, and crowds.
The General Equation for Force
The general energy equation can be
used to derive an equation for force.
However, I will first demonstrate this idea with the conventional
kinetic energy formula, followed by the general equation, because it is simpler
and more obvious. I will also assume,
for the sake of simplicity, that the change in velocity (speed) is from the
initial velocity (or speed) to 0 velocity (or
speed). This derivation begins in the
following two paragraphs.
Force times distance equals work, and
in the ideal case, when all the energy is converted to work, the energy
will equal force times distance. This
idea can be confusing to some, because in the real world it is difficult to
convert more than 30 percent of the energy to work. To get around his confusion, think of the
following calculations as a small percentage of kinetic energy that was
successfully converted to work.
The E term, representing kinetic energy, in
this formula
and also in the general energy
equation () can be replaced by force times
distance, which is done in the following derivations, and then solved for
force, which is designated as F.
The following six headings, deal
with six equations for force, involving various types of objects and
movements. The value for KF was
determined by the calculations carried out
above, which resulted in this relationship: 
The six equations,
and the calculated value for KF was not confirmed by conventional
equations, at this point in time.
This is apparent
when you read the material.
However, based on my experience with deriving equations, I believe that they are
most likely accurate. However, the only
way to be certain of accuracy is to confirm the result by established
mathematical formulas, as was done with the other equations in this booklet.
The force from the acceleration of a
thin rotating ring or a hollow cylinder:
If the general equation for force (
) is applied to a thin rotating ring
or a hollow cylinder, with all of its mass assumed to be on its outer surface,
with the axis through its center the value of .
This means the value of 
because an 
.
The force from the acceleration of a
rotating solid disk or cylinder:
If the general equation for force () is applied to a solid disk or cylinder,
with the axis in the center, .
This means the value of 
because .
The force from the acceleration of a
solid rotating sphere:
If the general equation for force () is applied to a solid rotating
sphere .
This means the value of 
, because .
The force from the acceleration of a
thin walled hollow rotating sphere:
If the general equation for force () is applied to a thin walled hollow
rotating sphere .
This means the value of 
, because .
The force from the acceleration of a
rotating thin uniform rod with a perpendicular axis in its center:
If the general equation for force () is applied to a rotating thin
uniform rod, with a perpendicular axis in its center, .
This means the value of 
, because .
The force from the acceleration of a rotating thin uniform rod,
with a perpendicular axis at one end:
If the general equation for force () is applied to a rotating thin
uniform rod, with a perpendicular axis, .
This means the value of 
, because .
The General Equation of Moving
Particles
An interesting equation can be
created by applying Theorem-Cto the exponent of the V
term in the general energy equation .
When this is done, and if E is replaced by 
and is replaced by 
the following equation results
.
This () general formula deals with moving
particles, which involve a moving mass, or mass equivalent in the case of photons. Thus, I am calling this formula () the general equation of moving
particles. This equation represents
mass, momentum, energy, and a limitless number of other physical concepts
involving moving particles. That is
whether the term stands for mass, momentum,
energy, or some other concept, is determined by the exponent on the V
term. The other physical concepts
represented by this () equation are obviously undefined,
and some might have practical or theoretical significance. All of the above will become clear after
examining the following examples.
When the equation of moving
particles represents mass:
The equation of moving particles () represents mass when 
and 
.
The proof of this is as follows:
The above can be restated in terms
of Einstein=s theory. Specifically, the equation of moving
particles represents rest mass when and 
When the mass is moving 
.
This becomes obvious when we examine Einstein=s equation for a moving mass, which
is presented below:
When the mass is not moving is equal to one.
When the mass is moving at
conventional velocities, the value of is slightly greater than one, as measured
from a stationary position, but it is usually too small to be considered
significant.
However, the value of becomes significant when the velocity of moving particles approaches a
significant fraction of the speed of light.
For example, an object moving at 1/10 of the speed of light would have
an increase in mass, as measured from a stationary position, as follows:
This means that a mass moving at
1/10 of the speed of light would result in a mass increase of over three times,
as measured from a stationary point.
The equation of moving particles and
kinetic energy:
The equation of moving particles () represents Einstein=s equation for the mass energy
relationship when 
and The proof of this
is as follows:
General Equation of Moving Particles Applied to
the Unknown???
The general equation of moving
particles, can be used to represent many of the other equations in physics that
have velocity and mass as factors, which usually involves the process of
substitution of the appropriate value for n and Km,
and by itself is not particularly interesting.
However, the general equation of moving particles suggests many possible
channels of exploration and experimentation.
When various values of Km and n are plugged
into the equation, the results represent physical quantitative relationships
and perhaps physical principles that may or may not have any
significance or meaning in the real world.
Keep in mind, that this single equation 
represents an infinite
set of equations, involving mass and velocity.
From the infinite set, it is very possible that there is one or more
equations that would have theoretical and/or practical application. Keep in mind that this set comprises most of
the conventional formulas of physics that have mass and velocity as factors,
including in most formulas that deal with energy and momentum. Thus, there may be more meaningful and useful
formulas to be discovered in this infinite set.
Of course, identifying a finite number of useful equations in an
infinite set is not necessarily easy, or even feasible. Attempting such a process by random trial
and error, in an infinite set, would not have any significant chances of
success, even if they were many billions of useful formulas in the set. However, using science, intuition, educated
guesses, experimentation, as well as trial and error, it is likely that useful
formulas could be found in this set. In
the following paragraphs, I briefly try my efforts in this regard, and I do not
know if I obtained any useful formulas in the process, but I know for certain
that I came up with many interesting questions, as follows:
What would we obtain if we set the
value of n to be less than 1or greater than 2, or when n involves
a fraction or decimal point, such as 2 , 2.5, 1.02, 0.3, etc, such as the following:
What happens when n approaches 0,
such as in the following equations
What happens when the value of n
increase as in the following equations ?
From the above, it is obvious that
as the value of n approaches zero, the equations increasingly become more
representative of mass, and at 0, the equation represents mass or a fraction of
a mass, when the value of the constant does not equal one. When the value of n increases
obviously, the relative degree of significance of the velocity increases. That is to say, a tiny increase in velocity,
will produced a very dramatic increase in the value of E?
The opposite also holds true, when
the value of n decreases the significance of the velocity
decreases. When the value of n is very low a
dramatic increase in velocity would have very little effect on the value of E?
and when n equal zero any
increase or decrease in velocity will not change the value of E?
in the slightest.
Thus, from the above there is a
useful concept. If we want to create an
equation to calculate momentum-like or kinetic-energy-like phenomenon that
places more emphasis on velocity we can increase the value of the exponent on
the V in the equations, to varying degrees, and vice versa. Of course, the resulting equation and
calculations would not be momentum or kinetic energy.
What happens when the value of n is
negative, such as the following
Keep in mind, that this single
equation represents an infinite set of equations,
involving mass and velocity. From the
infinite set, it is very possible that there is one or more equations that
would have theoretical and/or practical
application. Keep in mind that this set
comprises all the conventional formulas for mass, energy, momentum. Thus, there may be more useful formulas to be
discovered in this set. Of course,
identifying a finite number of useful equations in an infinite set is not
necessarily easy, or even feasible.
However, I am presenting a few interesting possibilities under the
following headings.
The equation of moving particles and
momentum: The equation of moving particles () represents momentum
when 
, and , as previously indicated.
This raises the question in my mind:
What happens if the value of n in the above equations is increased or
decreased, such as 1.1, 1.2, 1.5.
What would happen if the values are
reduced slightly, such as 0.90. With this had any meaning or utility? What would happen if negative numbers are
used, such as n= -1. Does this happening
meaning? Is there any practical or
theoretical utility?
Conclusion
There is a large amount
of additional material that I could have added to this booklet, but I did not
because of time limitations. There are
many thousands of additional equations that could be derived with the theorem
of multiple constants. However, my
goal with this booklet was limited to presenting the basic concept and utility
of theorem-C, by illustrating the principle with a few formulas from physics
and mathematics. If I find that others
are interested in this topic, I will write a more detailed book, covering
additional concepts from physics and mathematics.
References
The basic formulas, I used,
to derive the general equations in this booklet, are standard formulas from
mathematics and physics. These formulas are widely available in just about any
reference source. In this regard, I
found the following books especially useful for the basic formulas:
Physics Cliffs Quick
Review,
first edition, by Linda Huetinck, Ph.D. copyright
1993 by Cliffs Notes, Inc.
Calculus, second edition, by
Roland E. Larson, and Robert P. Hostetler, copyright 1982 by D. C. Heath and
Company
