Reinterpreting Newton s Law of Gravitation Copyright 2012 Joseph A. Rybczyk Abstract A new approach in interpreting Newton s law of gravitation leads to an improved understanding of the manner in which such law applies to the bodies under its influence. 1
Reinterpreting Newton s Law of Gravitation Copyright 2012 Joseph A. Rybczyk 1. Introduction Previous works 1 by this investigator have revealed inconsistencies in Newton s law of gravitation 2 that when corrected give an improved understanding of the effects of gravity on the bodies under its influence. Of greatest importance is the difference in behavior involving what is now recognized as the single body problem, as opposed to the two body problem typically associated with gravitational behavior. Such difference in behavior is not at all intuitive and provides insights into gravitational effects that appear to have evaded detection until now. 2. Redefining the Second Law of Mechanics To derive the standard formula for gravitational acceleration, as typically given in textbooks, we begin with Newton s second law of mechanics 2 involving particle collisions. Unlike the typical textbook treatment, however, we will include an additional consideration that has a significant effect on how the laws of gravity are to be interpreted yet are not normally covered in the particle collision treatment. Accordingly then, the second law of mechanics can be defined mathematically starting with the formula 1 where m is the inertial mass of the less massive secondary particle, (assuming the particles have different masses), F is the force exerted on it during the collision with a primary particle of greater mass M, and A is the resulting acceleration (change in speed) of the secondary particle relative to the inertial frame that the motion is referenced to. The rate of acceleration of the primary particle relative to the same inertial frame is then given by the formula 2 where a is a lesser rate of acceleration and F is the same force as previously given but as exerted on the primary particle by the secondary particle. The Ʃ vector sum symbol normally applied to the force F has been omitted here on the assumption that all of the force is along the same path of motion. We complete the definition process by including the formula 3 that gives the sum total rate a s of the two different rates of acceleration. By the process of substitution with the acceleration rate versions of equations (1) and (2) we then derive 4 2
that reduces to 5 for the final form of the sum total acceleration formula that gives the total change in speed a s of one particle relative to the other. It is understood here that each particle only experiences the actual acceleration defined by its respective formula (1) or (2) and not the total rate of acceleration given by formulas (3) through (5). Nonetheless, the total change in speed between the two particles is given by acceleration rate a s and not the individual acceleration rates A or a that are referenced to the inertial frame that the motion takes place in. From a mathematical perspective the force relationships given in the second halves of equations (1) and (2) can be summarized in the form of 6 showing that F is a function of the product of the mass and acceleration of each particle. 3. The Two Body Problem of Gravitational Acceleration Once the relationships associated with Newton s second law of mechanics are properly understood we can apply those relationships to the behavior associated with the two body problem involving gravitational acceleration. That is, beginning with Newton s law of gravitation defined by 7 we apply the relationship given by equation (1) to derive 8 where F is the gravitational force between the two bodies, G is the gravitational constant, m is the mass of the secondary body, M is mass of the primary body, r is the center to center distance between the masses of the two bodies and A is the resulting acceleration of mass m relative to the system center of mass as illustrated in Figure 1. In preparation for the discussion involving Figure 1 let us first draw what would seem to be an obvious conclusion from what was discussed so far. As previously stated, with regard to particle collisions, each particle experiences its own acceleration during the collision depending on its mass in relation to the opposite mass. So it would seem reasonable to conclude that each mass involved in gravitational acceleration would also experience its own rate of acceleration in a similar manner. And that, according to the evidence, appears to be the case. Thus, returning to equation (7) we now solve it for the acceleration experienced by the primary body as previously defined by equation (2) of the second law of mechanics to derive 9 3
where a is the resulting acceleration of mass M relative to the same system center of mass previously referenced in relation to the acceleration of secondary mass m. We are now in a position to discuss the two body problem in detail. orbital path about system center of mass primary body M primary body center of mass r p orbital path about system center of mass r Figure 1 Center of Mass Relationships In referring to Figure 1, the relationships typically associated with the two body problem involving gravitational force and acceleration are clearly illustrated. As already discussed, where M is the mass of the more massive primary body and m is the mass of the less massive secondary body, r is the radial distance between their individual centers of mass. If these two bodies were not orbiting about the system center of mass as shown, they would accelerate toward that center at a rate of acceleration inversely proportional to their masses as given by r s system center of mass m secondary body secondary body center of mass 10 where, as previously stated, F is the gravitational force acting between the two masses and a is the rate of acceleration of primary mass M, and A is the rate of acceleration of secondary mass m. Since M is the more massive of the two bodies it will accelerate at a lower rate of acceleration a along radial path r p during the same period that the less massive body m accelerates at the greater rate of acceleration A along radial path r s. If the two bodies did not physically interfere with each other, as they normally would, they would simultaneously reach a position in which they would both be centered on the system center of mass at the same instant in time. For reference purposes the relationship between these three radial distances can be mathematically defined by 11 where r is the (mass) center to center distance between the two masses, r p is the distance that the primary mass travels and r s is the distance that the secondary mass travels. (It is understood that both masses cannot occupy the system center of mass at the same time unless the two masses physically merge.) (It is further understood that the rates of acceleration are dependent upon radial distance r as given in equations (8) and (9) and increase in accordance with the inverse square rule as distance r decreases during the acceleration.) 4
From what was just discussed involving the acceleration of both bodies toward each other along the same radial path r, it directly follows that the total rate of acceleration of the one body toward the other body is given by 12 where acceleration rate a s is the sum of acceleration rates A and a of masses m and M respectively. This is nothing more than a repeat of equation (3) previously given for particle acceleration but now applied to the two body problem of gravitational acceleration. By way of substitution using equations (8) and (9) we then derive 13 giving 14 and 15 for the total rate of acceleration a s of each body relative to the other. Although the relationships just discussed are fundamentally correct they are often misused, in the form of equation (8), to support Galileo s finding 3 that all bodies fall to Earth at the same rate regardless of their mass. Although such claim appears to be supported by the evidence, it is technically invalid as will be discussed later in the paper. 4. The One Body Problem of Gravitational Acceleration Now that the general relationships associated with gravitational force and acceleration have been defined we will turn our attention to the one body problem illustrated in Figure 2 that is the real focus of this paper. primary body M system center of mass coincident with primary body center of mass r p = 0 r s = r insignificant secondary body m on surface of primary body Figure 2 Coincident, System and Primary Body, Centers of Mass Although the two body relationships of Figure 1 are equally valid for the one body relationships to be discussed now, the relationships given in Figure 2 are more appropriate for examples involving an object that is actually or theoretically dropped on the surface of the Earth. In this 5
case the mass of the secondary object is typically insignificant in comparison to the mass of the primary body. More specifically, since the mass of the secondary object typically used in such examples is a feather, or metal ball weighing on the order of an oz. (0.028 kg) or so, the distance r p accelerated by the Earth in the direction of the secondary object is 0, resulting in all of the accelerated motion being accomplished by the secondary object. This in turn results in the distance r s between the secondary object and the system center of mass being equal to the (mass) center to center distance r between the two masses. And that, of course, gives a distance of r p = 0 for the primary mass M. In other words, for such insignificant secondary masses, the system center of mass is coincident with the center of mass of the primary body. Thus, equation (8) for the secondary mass of the two body problem and, for that matter, even equation (15) for the total rate of acceleration will appear to support the proposition that all objects fall to Earth at the same rate. More specifically, since equation (8) gives the rate of acceleration of the insignificant secondary object and since that rate is constant relative to the system center of mass that in such examples is coincident with the Earth s center of mass it follows that such objects will all appear to fall at the same rate as predicted by Galileo. In fact it would take an object with a mass of 1x10 8.8 kg to affect the rate of acceleration predicted by the correct total acceleration formula at the 15 th decimal place. And that formula is not the standard Newtonian formula (8) currently used for that application. The fact that the referenced formula (8) is not valid for the one body problem under discussion is obvious just by reviewing the formula as given. The problem is that there is no provision in it for reducing the mass M of the primary body (the Earth) to compensate for the secondary mass m that is removed from it. In actuality the above referenced formula is valid only for the case where the secondary object came from outer space and the mass of the primary body is unaffected. Moreover, there is another even more important consideration that is not accounted for in the two body problem treatment that affects its validity with regard to the one body problem. But prior to discussing it, let us first derive the correct formulas for the one body problem as will be accomplished now. As already stated, the first thing that must be taken into consideration with regard to the one body problem is to account for the reduction of the mass of the primary body equal to the mass removed from it in the form of the secondary body. Thus, we modify the previously given two body version of the gravitational law formula (7) to the form 16 where M is the original mass of the primary body and m is the mass of the secondary body extracted from it. Solving for the secondary body s rate of acceleration relative to the system center of mass using the second law of mechanics given by equation (1) then gives 17 where F is the gravitational force between the two bodies, G is the gravitational constant, m is the mass of the secondary body, (M m) is the mass of the primary body and A is the rate of acceleration of mass m relative to the system center of mass given in the previous illustrations. For the primary body acceleration formula we apply a modified form of the second law of mechanics formula (2) to equation (16) to derive 6
18 where a is the rate of acceleration of the primary body relative to the system center of mass. Then, as with the previous case involving the two body problem, by way of substitution of equations (17) and (18) into the previously given equation (12) we obtain 19 giving 20 and 21 for the total rate of acceleration a s of each body relative to the other body. We will now compare these one body formulas to the previously derived two body formulas and show that Newton s law of gravity is not universal as previously believed. 5. Redefining the Laws of Gravitation The reason that what is about to be shown was not discovered earlier is that it is very deceptive in the manner in which the mathematical terms are defined. Thus, to make the comparison easier, we will first restate the just derived one body formulas for gravitational force and acceleration in terms that have the same definitions as those used in the two body problem. In that regard we have M o = the original mass of the primary body and M = the actual mass being acted upon gravitationally. This then gives 22 23 24 for the first three main formulas and in so doing leads to an apparent contradiction of what we are trying to show because they are the same formulas used for the two body problem. Therefore by using equation (12) to sum the results of equations (23) and (24) for the total rate of acceleration, we obtain 7
25 that is the same formula as formula (15) used for the two body problem. Yet if we use equation (21) where M is actually the original mass M o of the primary body and not its actual mass, we have 26 that forces us to consider the original mass of the primary body. The point being that in the one body problem M and m are dependent variables whose sum is a constant whereas they are independent variables in the two body problem. In other words, it is not enough to show that the total mass is the sum of masses M and m as given in formula (25). We must additionally show that 27 resulting in a simultaneous equation. It is for this reason that equation (25) will always give the same constant result for the same distance r when used in the one body problem regardless of the values of M and m while giving a different variable result depending on the independent values of M and m when used in the two body problem. This difference in behaviors between the two different uses of the gravitational formulas was apparent in the analytical graphs included in the appendixes of the previous paper 4 by this investigator. To summarize what was just discussed, the expression (M + m) in the two body total acceleration formula (15) is a variable depending on the independent masses M and m of the primary and secondary bodies. That is not the case for the one body total acceleration formula (25) in which case the expression (M + m) is a constant and masses M and m are variables that are dependent on each other. This, of course, affects all of the one body formulas. 6. The Confusion Surrounding Galileo s Claim The most compelling example of the confusion involving the laws of gravitation is that involving Galileo s claim that all objects fall to Earth at the same rate. As a matter of fact the only two formulas that properly take into consideration the total rate of acceleration between the two bodies are the two body total acceleration formula (15) and the one body total acceleration formula (21) of which only formula (21) is valid for that purpose. But even in the case where the proper formula is used, if the object is dropped on the surface of the Earth as Galileo proposed, then, as the mass of the secondary object increases so too does its volume. Conversely, the volume of the Earth decreases along with its mass. If the two masses remain in surface to surface contact with each other, the radial distance r between their individual centers of mass will increase as the mass of the secondary object approaches in value the reduced mass of the Earth. Since the total mass M o of the two bodies remains constant, the increase in radial distance r will result in a decrease in the total rate of acceleration of the one body toward the other. Only in the case where the distance r was already set at this increased value and allowed to remain constant would the total rate of acceleration between the two bodies be unaffected by the change in their masses. In the final analysis, Galileo was only conditionally correct in his claim. 8
7. Conclusion It has been clearly shown in this present work by this investigator and the associated works that preceded it that the laws of gravitation are far more complicated than they appear to be. The importance of not being misguided by their deceptively simple appearance cannot be overemphasized. Hopefully this present work will be given the full acceptance of the scientific community that it rightfully deserves. Acknowledgement The author is indebted to Dr. Johannes C. Valks for alerting him to certain oversights involving Newton s laws of mechanics at the beginning of the paper. The necessary corrections have been incorporated into this updated version of the paper within one day of the initial publication of the paper on the Millennium Relativity website. REFERENCES 1 Joseph A. Rybczyk, The Universal Laws of Gravitation, (2012); The Four Principal Kinetic States of Material Bodies, (2005); The Special Case of Gravitational Acceleration, (2005); Millennium Theory of Inertia and Gravity, (2004); All available on the Millennium Relativity website at www.mrelativity.net 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) 4 Joseph A. Rybczyk, The Universal Laws of Gravitation, (2012) Reinterpreting Newton s Law 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 9