The Universal Laws of Gravitation. Copyright 2012 Joseph A. Rybczyk

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1 The Universal Laws of Gravitation Copyright 2012 Joseph A. Rybczyk Abstract Close examination of Newton s universal law of gravitation and Galileo s discovery that all objects fall to Earth at the same rate reveals discrepancies in both theories that have gone unresolved to the present time. Subsequent resolution of these discrepancies leads directly to the new universal laws of gravitation presented here in complete form for the first time. 1

2 The Universal Laws of Gravitation Copyright 2012 Joseph A. Rybczyk 1. Introduction In a previous work 1 by this investigator it was shown that Newton s universal law of gravitation 2 along with Galileo s findings 3 involving the rate at which objects fall to Earth was only conditionally true. In this present work a simplified independent analysis not only corroborates the original findings but leads to additional findings that redefine our understanding of gravitational force. The final results are the new universal laws of gravitation that will be presented in as brief a manner as practical. 2. The Discrepancies in Newton s Universal Law of Gravitation We begin the analysis by demonstrating that Newton s universal law of gravitation does not actually give the total rate of acceleration between two masses as currently believed. Thus, we start with Newton s universal law of gravitation defined by 1 where G is the gravitational constant and F is the gravitational force between the primary mass M and the secondary mass m, and r is the center to center distance between the masses. It is understood that the primary mass is the greater of the two masses and subsequently the secondary mass is the lesser of the two. To determine the rate at which the secondary mass will accelerate toward the primary mass we then apply the mathematical relationship formally given as Newton s second law of mechanics. That is we take Ʃ Ʃ 2 where a is the rate of acceleration and ƩF is the vector sum of the forces acting on the mass m. Since it is only the gravitational force between the two masses that we are investigating we can discount the Ʃ term and apply the relationship of equation (2) to equation (1) in the form 3 where a is the rate of gravitational acceleration of mass m toward mass M. Since G, M, and m are all constants it therefore follows that for any distance r between the two masses there is a corresponding rate of acceleration a. In analyzing the right most expression in equation (3) it then follows that for any mass m, the rate of acceleration will be the same. Thus the false claim that all objects accelerate to the Earth at the same rate as will now be discussed. Still referring to formula (3) no one would argue that if primary mass M was the moon instead of the Earth and if mass m and distance r were unchanged from those used in the Earth analysis, then the rate of acceleration would be much less than that related to Earth. This then leads to a contradiction in the claim that all objects fall to Earth at the same rate assuming that 2

3 the secondary object came from the Earth. The point is that the Earth s mass is reduced by the mass of the object that is used in the experiment. Given that the Earth s mass is so great in comparison to the objects typically used in the experiments that demonstrate the principle that all objects fall at the same rate, it is understandable that the reduced mass of the Earth will cause an immeasurable effect on the outcome. But technically, the heavier the object is, the slower it will fall to Earth. Yes, according to Newton s law, slower. Since the mass of the object comes from the Earth, and the Earth s mass is reduced, even though the total force between the two masses is increased, the greater the secondary the mass becomes, the slower it will fall in the direction of the Earth. This is obvious from just looking at formula (3). If mass m increases to the point that it now has the mass originally contained in the Earth we have the condition given by formula (4). 4 Since G = x N m 2 /kg 2, and assuming 1 kg for mass m and given r = x 10 6 m (normally Earth s radius but now the radius of mass M) it should be rather obvious that the rate of acceleration is 0. As if this were not already enough to deal with, it turns out that Newton s universal law of gravitation is invalid in the example just given because it does not actually allow for the object to come from the Earth. That is, Newton s formula is only valid for objects that come from outer space since there is no provision for reducing the mass of the Earth by the amount of mass used in the experiment. For that matter, this even invalidates the falling apple example used with Newton s theory. That is, when the apple falls from the tree, the Earth s mass is reduced by the amount of mass contained in the apple, yet there is no provision in his formula to account for such change. This provision will be dealt with later in the paper, but for now let us simply revise Newton s formula to give the total rate of acceleration and not simply the rate of each mass to the center of mass as given by the present formula. 3. The Total Rate of Celestial Gravitational Acceleration To differentiate the outer space version of the total acceleration treatment we will refer to it as the celestial version of the universal laws of gravitation. The local version, to be developed afterwards, will then be referred to as the non-celestial version of the universal laws of gravitation. Thus, for the present case, where A is the rate of acceleration toward the center of mass by the faster moving less massive body and a is the rate of acceleration of the slower moving more massive body we have the relationship 5 were a s is the sum total of the two different acceleration rates. Then, given 6 and 7 3

4 in combination with equation (5) we derive 8 for the overall relationships involved in the total acceleration between the two masses. From these relationships we then derive 9 that can be solved for F to obtain and finally 12 for the gravitational force F based on the total acceleration between the two masses. Referring back to equation (8) we next derive 13 that simplifies to 14 for the final version of the sum total acceleration formula for celestial objects. We are now in a position to prove beyond doubt that the total acceleration formula correctly represents the acceleration between the two masses given in Newton s formula. 4. Universal Celestial Gravitational Force To re-derive Newton s universal law of gravitation formula that will now be correctly referred to as the universal law of celestial gravitation, we simply take the acceleration based force formula (12) and substitute the right side of the sum total acceleration formula (14) in place of a s, to obtain 4

5 15 that simplifies to 16 and 17 that as can be readily seen, is Newton s original universal law of gravitation formula. Now, however, we will refer to it as the universal law of celestial gravitation to distinguish it from the correct formula for non-celestial gravitation as will be developed next. 5. Universal Non-Celestial Gravitational Force Assuming that the secondary mass came from the primary mass we then have the relationships given by and where A is the higher rate of acceleration for the secondary mass and a is the lower rate of acceleration for the primary mass. Solving either of these two equations for force F gives 20 for the universal law of non-celestial gravitation. 6. The Total Rate of Non-Celestial Gravitational Acceleration If we now use the previously derived formula (5) for the sum total acceleration resulting from equations (18) and (19) we obtain 21 for the overall relationships involved in the total acceleration between the two masses. From these relationships we then derive 5

6 22 that can be solved for F to obtain and finally 26 for the gravitational force F based on the total acceleration between the two non-celestial masses. Referring back to equation (21) we next derive 27 that simplifies to 28 and finally 29 for the final version of the sum total acceleration for non-celestial objects. This should not be confused with equations (3) and (6) that give the individual rates of acceleration, a for the secondary mass under Newton s theory, and A for the secondary mass in the revised version of Newton s theory respectively. The importance of this finding cannot be overemphasized. If we now take equation (26) and substitute the right side of equation (29) into it in place of total rate of acceleration a s, we obtain 30 6

7 that simplifies into the universal law of non-celestial gravitation previously given as equation (20). 7. Radial Distances Considerations To avoid the possibility of arriving at false conclusions involving the nature of gravity it is important not to overlook the effects caused by the radial distance between the two masses. Since the radial distance r between the gravitational centers of the two masses can also be a reference only to the radius of the primary mass, such as the Earth, in experiments involving insignificant secondary masses in the form of feathers, steel balls, and other small objects, it is important to distinguish between the two cases. That is, since the radial distance r is the only other variable in addition to the two masses on the right side of the gravitational force formula and since r can be affected by a change in the either of the two masses if they are in contact with each other, such effect on r must be taken into consideration involving the formulation of conclusions associated with related theoretical analyses. Actually the radial distance r can have three different meanings. It can represent only the radius of the primary mass as already stated, or it can represent the distance between the centers of the primary mass and the secondary mass in the case where the size of the secondary mass is significant in comparison to the primary mass, or it can represent the distance between the gravitational centers of the two masses including a space between the surfaces of the two masses. To understand the significance of this more clearly let us now consider a theoretical example where the radial distance r is affected by an increase in the secondary mass. Assume that we are analyzing the effect of gravity on a small mass the size of a cannon ball on the surface of the primary mass such as the Earth. In this case the radius of the Earth is basically indistinguishable from the radial distance between the center of the Earth and the center of the secondary object. But then let us say we keep increasing the mass of the secondary object to the point where its size becomes significant relative to the size of the primary mass. In this case, regardless of whether the secondary mass came from outer space, or was taken from the primary mass, the radial distance r between its center and the primary mass center is affected when the two masses are in contact with each other. Taking the more misleading of the two examples where the secondary mass is taken from the primary mass and assuming that the two masses are adjusted to the form of spheres, it is readily shown that as the secondary mass increases in value toward the mass of the primary mass, the radial distance between the centers of the two masses will increase if the masses are in contact with each other. To do a complete analysis of this kind we can develop more advanced formulas than those already given for equations (20) and (29) as will be accomplished next. Where v is the volume of a sphere given by the formula we can solve for r to obtain

8 where r is the radius of the sphere. Then, given 33 where r is the center to center distance between two spheres and where r p is the radius of the primary sphere, r s is the radius of the secondary sphere and d is the surface to surface distance between them, we derive by substitution with appropriately modified versions of equation (32) in place of radii r p and r s where v p is the volume of the primary sphere and v s is the volume of the secondary sphere. Then given 35 where v o is the original volume of the primary sphere v p prior to the extraction of secondary sphere v s from it, we obtain by substitution into equation (34) as the final form of the equation for the center to center distance between the two spheres. This equation can then be substituted in place of r in equations (20) and (29) to obtain and for more complete forms of the non-celestial formulas for gravitational force and sum total acceleration. Then given

9 where v o is the original volume of the primary mass sphere based on its original radius r o we define the relationship 40 to be used along with 41 in conjunction with equations (37) and (38) where x can be any value required for the analysis. The value of x must be the same in both formulas, (40) and (41) for a given calculation. 8. Final Comments Except for the case just discussed, in order to keep the analysis as simple as possible in the appendixes provided at the end of the paper the investigator evaluated the gravitational force and acceleration behaviors at a distance equal to the Earth s radius. The intention was to enable comparison of the changes in force and acceleration using values that we are familiar with. To increase the center to center radial distance between the two masses enough to prevent surface to surface contact would result in different numerical results but would have no effect on the behavioral nature of the gravitational force and acceleration being investigated. In essence then, the secondary mass was treated as though it was an object of insignificant size regardless of its mass. Only in the latter case involving the analysis of Section 7, where the whole intention was to show the effect of the change in center to center distance between the two masses due to the reduction of the primary mass volume and corresponding increase in the secondary mass volume, was this practice not used. In the first case we are showing the behaviors associated with the center to center distance held constant and only the masses changing. In the latter case the center to center distance was also changed in order to accommodate the changes in mass volumes. 9. Conclusion When Newton s law of gravitation is properly understood it is apparent that it is not universal as claimed. In the case where the secondary object is taken from the primary body there is not only a decrease in the mass of the primary body but also a reduction in the size or volume of the primary body as well. Since the secondary object not only increases in mass but also in size and volume the center to center distance between the two masses is affected if the masses are in contact with each other. For such experiments conducted on the surface of the Earth the center to center distance between the two masses increases while the total mass is unchanged. The result is an extreme increase in gravitational force between the two masses accompanied by a decrease in the sum total acceleration rate as the mass of the secondary object approaches in value the reduced mass of the primary body. As the mass of the secondary object increases beyond the reduced mass of the primary body, the roles are reversed and if the exchange continues, the original force and acceleration rates are returned to what they were at the start of the experiment. This behavior contradicts the claim that all objects fall to Earth at the same rate. If the same exchange takes place where the distance between the two masses is just what is needed to be held constant, then the increase in gravitational force is the same as in the case just discussed but the sum total acceleration rate, though less than it was in the previous case, 9

10 remains constant. It is important to understand here, however, that the predicted constancy in the rate of acceleration is a result of the modified mathematical treatment and not a result of Newton s original law of gravitation. More specifically, the expression GM/r 2 in equations (3) and (6) of Newton s law gives only the individual rate of acceleration a, more correctly stated as, A of the secondary object relative to the center of mass between the two masses and not the sum total rate of acceleration a s of one mass toward the other as given by equation (29) of the revised laws. It is only in this latter case that the acceleration rate is unaffected by the change in mass of the secondary object. The overall behavior of unrelated masses, i.e. celestial bodies, is significantly different from the behaviors just discussed. In this case as the mass of the secondary body increases, while the distance between the two bodies is held constant, the gravitational force between the two bodies increases along with the sum total rate of acceleration at what appears to be a linear rate of increase. This change in behavior follows directly from the modified version of Newton s law clearly showing that the original form of Newton s law is not universal as claimed. In the investigator s opinion the findings presented in this work are beyond dispute and Newton s universal law of gravitation must be revised to the form herein presented. Appendixes The following Appendixes are provided at the end of the paper to aid the reader in comparing the behaviors predicted by the mathematical treatment presented in this paper: Appendix 1 Comparison of Celestial to Earth Related Non-Celestial Formulas Appendix 2 Graphs for Gravitational Force and Acceleration for m = 0 to M Appendix 3 Graphs for Gravitational Force and Acceleration for m = 0 to 10M Appendix 4 Advanced Analysis of Non-Celestial Gravitational Force and Acceleration Appendix 5 Graphs of Advanced Non-Celestial Mathematical Treatment for m = 0 to M REFERENCES 1 Joseph A. Rybczyk, Millennium Theory of Inertia and Gravity, (2004) 2 Isaac Newton s Three Laws of Mechanics, and his Universal Law of Gravitation, as presented in, Physics for Scientists and Engineers, second edition, (Ginn Press, MA, 1990), and Exploration of the Universe, third edition, (Holt, Rinehart and Winston, NY, 1975) 3 Galilei Galileo s principles of motion and gravitation, as presented in, Physics for Scientists and Engineers, second edition, (Ginn Press, MA, 1990), and Exploration of the Universe, third edition, (Holt, Rinehart and Winston, NY, 1975) 10

11 Appendix 1 Comparison of Celestial to Earth Related Non-Celestial Formulas 11

12 Appendix 2 Graphs for Gravitational Force and Acceleration for m = 0 to M 12

13 Appendix 3 Graphs for Gravitational Force and Acceleration for m = 0 to 10M 13

14 Appendix 4 Advanced Analysis of Non-Celestial Gravitational Force and Acceleration 14

15 Appendix 5 Graphs of Advanced Non-Celestial Mathematical Treatment for m = 0 to M 15

16 The Universal Laws of Gravitation Copyright 2012 Joseph A. Rybczyk All rights reserved including the right of reproduction in whole or in part in and form without permission. Note: If this document was accessed directly during a search, you can visit the Millennium Relativity web site by clicking on the Home link below: Home 16