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 byThis 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.

 


In this booklet Theorem-C will also be applied to the set of formulas that relate to: volume, moment of inertia, energy, momentum, force, and other concepts from both mathematics and physics.  Some of the general formulas that will be derived include the following: for volume , for moment of inertia , for energy , for force .  These equations suggest general principles in mathematics and physics.  Revealing relationships of this type is a primary utility of theorem-K.  That is by applying the theorem to specific sets of equations fundamental principles are often revealed, in precise mathematical terms.  This will become apparent in the following pages.


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:

 

 

 


The general area formula applied to a circle:  When the general area formula () is applied to a circle,  and both L and W equal the radius of the circle.  That is L = W = r.  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 circle should result, if the above proof is correct.  This is done as follows:

 

The general area formula applied to a circle, in relation to the diameter:  Another version of the above formula, for the area of a circle, is based on the diameter of the circle.  Thus, W and L of the general area equation () are both equal to the diameter, and .  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 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:

 


 

 

Another example is 

 

 

 

 

 

 

 

 


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:


If the values of and r are placed back into the general moment of inertia equation (), as a test of the proof, the conventional formula is obtained, as follows:         The proof for  when was also derived with the conventional formula, as follows:


If the values of and r are placed back into the general moment of inertia equation () as a test of the proof, the conventional formula is obtained, as follows:     

Incidentally, the equation is the formula that is present in most physics books.  This formula can be derived with the simpler equation .  To do this, assume that all the mass is on the outer surface of the ring or cylinder, and make the calculations.  Then assumed that all the mass is on the inner surface of the ring or cylinder, and make the calculations.  Then take the average of the results.  The proof derivation is 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.  


 

The moment of inertia of a solid sphere:  When the general inertia formula () is applied to a solid sphere, with the axis in the center, .  The value of  was derived with the common 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 moment of inertia of a thin wall hollow sphere:  When the general inertia formula () is applied to a thin wall hollow sphere, with the axis in the center, .  The value of  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 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:


The above can be checked by substituting the value of into the general energy equation ().  This will result in the kinetic energy formula that we started with, as follows:

 

.

 

 

 

Einstein=s energy equation, and related ides:  The general energy equation () applies to Einstein=s equation () for the mass energy relationship, when  and when speed of light in a vacuum.  This can be worked out as follows:

If the value of  can be substituted back into the general energy equation, to check the above, as follows:

.

Note, in the above equation () the  is usually considered to be a constant, but from the perspective of the general energy equation 1 is the constant and the  term is the square of the velocity.  However, if the term is considered a constant, it is possible to simplify the formula () by setting .  This would result in the following formula .  With this formula the value of  can easily be modified for any type of energy or mass units.


(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:


If the value of  is substituted back into to  the result is as follows:

.  If  is substituted back into to above, as a file test of the proof, the following results:

. 

The formula that was derived from the general energy equation (), in relation to the V term is speed in terms of a linear measurements.  The relationship of the two speeds is , when  number of revolutions per unit of time.  Thus, the above energy formula can also be written as:


 This version of the formula suggests another general formula.  That is, Theorem-Ccan be applied to  which will yield the following general equation   This formula is based on revolutions per unit of time.  When this general equation is applied to a thin rotating ring or a hollow cylinder,  

It is also possible to create a, formula similar to the, above based on the diameter ().  This can obviously be accomplished by using the two to one relationship between the radius and the diameter (  ).  The derivation of this formula is as follows: 

When this formula () is applied to the kinetic energy of a thin rotating ring or a hollow cylinder .  This can be derived from the above relationship of  as follows:

.   


Another general formula can be derived from the above.  This will become apparent if the left side of the first line of the formal poof is examined, which will show the following formula .  This formula is a kinetic energy equation for a thin rotating ring or a hollow cylinder, basted on angular displacement per unit of time, such as in radians per second.  If Theorem-Cis applied to  the result  is obviously,  when the formula is applied to a thin rotating ring or a hollow cylinder.

The above formula () can be written in terms of the diameter as follows:

When this formula () is applied to the kinetic energy of a thin rotating ring or a hollow cylinder .  This can be derived from the above relationship of  as follows:   

The general formulas presented above, , , and  can be represented by an even more general equation that relates to energy, frequency, and wavelength.  This new equation can be derived from the general energy equation () and the generally accepted assumption that frequency multiplied by wavelength equals velocity.  This will be done in the following paragraph.


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 formulain 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.


This formula  can be used to derive Newton=s equation for force, with the conventional formula for acceleration  by substitution, when , which results in .  If we examine the formula for force that was derived above it is apparent that  can be substituted into the formula, which results in  

 

The above idea can be used with the general energy equation to derive two equations for force, as follows. 

This formula  can be used to derive an equation that is similar to Newton=s formula, but it applies to a large number of cases.  This is done with the conventional formula for acceleration as follows:


 

This equation () can also be derived by applying Theorem-C to Newman=s formula for force ().  This involves replacing the coefficient of 1, with which of course results in   This is obviously much simpler procedure, but the more complicated derivation revealed the relationship between which is or .  This is a useful relationship, because it will allow the derivation of the values from the values, which was already used in this paper for the general energy equation.  This will become apparent in the following paragraphs.

 

 

A force produced by linear acceleration or deceleration   value is , which means the  value is one, that is .  If this value is substituted into the general equation for force () we obtain Newton=s equation will force ().

 


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