Please look at Figure 26 as we analyze and scrutinize each scientist’s rationale for Galileo’s concept of equal acceleration of unequal masses.
Feynman referred to the result of “Galileo’s old experiment from the leaning tower of Pisa” as a “very interesting consequence” of the fact that the “force of attraction on an object is exactly proportional to…its [such object’s] mass.”
“If the pull is exactly proportional to the mass, and the reaction to the force, the motions induced by the forces, changes in velocity [acceleration],…are inversely proportional to the mass. That means that two objects of different mass…will fall [accelerate] the same way to the Earth.”
What was wrong with this analysis? First of all, the Moon’s force of attraction acting on the Earth is proportional to the mass of the Moon, but it is not “exactly proportional to [the Earth’s] mass.” The fact that the Earth’s force of attraction acting on the Moon is proportional to the mass of the Moon does not correct the above mistake (Figure17.1). Secondly, these statements by Feynman are self-contradictory. If the accelerations of two different masses are inversely proportional to their own different masses, then such accelerations must also be different…rather than the same, as Feynman asserted. If Feynman’s attempted explanation was really true, then the very different masses of the Earth and the Moon would fall (accelerate) the same way toward each other (Figure 21), which obviously does not empirically happen (Figure 26). Feynman obviously did not know what he was talking about.
Similarly, Gamow attempted to rationalize the validity of Galileo’s equal gravitational acceleration result based on the ad hoc idea that the magnitude of the gravitational force received by and acting upon a pulled mass is always proportional to the inertial resistance of the pulled mass. As Gamow stated:
“…[G]ravitational force might also be proportional to the mass of another body.”
It is true that the equivalence of the Gravitational Force Received by two opposing objects (i.e. the Earth and the Moon) is always proportional to the inertial mass and relative inertial resistance (1) of the smaller mass (i.e. the Moon). But such equivalent force is always inversely proportional to the inertial mass and relative inertial resistance (82) of the larger mass (i.e. the Earth) (Figure 17.1). For this reason, the Earth’s much greater inertial mass resists changing its inertial state with 82 times the force of resistance as the Moon, and its resulting motion (action and acceleration) is 82 times less than the Moon’s (Newton’s three laws of motion). Thus, contrary to Gamow’s assertion, the gravitational force emitted by the smaller body (i.e. the Moon) which is received by and acting upon the larger body (i.e. the Earth) is always inversely proportional to the mass of the larger body (i.e. the Earth).
If Gamow’s rationale was really true, then a priori both the Earth and the Moon would somehow orbit equally around each other like binary stars of equal mass (Figure 24B), and both should somehow accelerate equally towards one another (Figure 21A), which observationally does not happen. The same results would occur with respect to the Sun and the Earth (Figure 21B), which obviously does not happen. Gamow also did not know what he was talking about.
Many others have attempted to justify Galileo’s equal gravitational acceleration result based on the equivalence of the Gravitational Forces Received by each opposing body. For example, in a similar but somewhat different attempt to justify Galileo’s mystifying equal acceleration result, Stephen Hawking asserted that:
“One can now see why all bodies fall at the same rate: a body of twice the weight will have twice the force of gravity pulling it down, but it will also have twice the mass [inertial resistance]. According to Newton’s second law, these two effects will exactly cancel each other, so the acceleration will be the same in all cases.”
But, of course, Newton’s second law makes no such assertion. Instead Newton’s second law states that “[T]he change of motion [acceleration] is proportional to the motive force impressed,” and it implicitly asserts that such acceleration will be inversely proportional to the mass (inertial resistance) of the accelerated object (Figure 9). Newton’s third law results in the same inversely proportional accelerations (action and reaction) as his second law (Figure 10). These relative phenomena are completely different than the ad hoc “cancelling and same acceleration” phenomena which Hawking was incorrectly asserting.
Consider the following empirical example based on Hawking’s assertions. If the Moon had twice its current mass (Figure 23), a priori it would receive twice the quantity of relative gravitational force from Earth, which would cause its new twice as large relative inertial resistance to accelerate somewhat slower toward the Earth, and at a new acceleration ratio of 41:1. Likewise, the Earth would receive twice the quantity of relative gravitational force from the Moon which would cause its new relative inertial resistance (41) to accelerate somewhat faster toward the Moon, and at a new acceleration ratio of 1:41.
The net result would be a different ratio of the masses, a different magnitude of relative forces received and applied, a different ratio of inertial resistances, and somewhat different relative accelerations between the two opposing bodies (compare Figure 17.1 to Figure 23). But the proportionally increased magnitude of force received by the Moon and its increased inertial resistance would certainly not “exactly cancel each other, so the acceleration will be the same in all cases.” This result is also reciprocally true with respect to the Earth. In other words, when Hawking doubled the mass of one opposing body he also changed the mass-ratio of the opposing gravitating bodies, and therefore changed the magnitude of their reciprocal gravitational forces received, and the ratio of their inertial resistances, accelerations, actions, and reactions, which he obviously failed to consider. Thus, Hawking did not know what he was talking about either.
Cohen claimed that “when an apple falls from a tree, the force pulling it down is its weight W.” Therefore, the equation
proves that the equal accelerations of all terrestrial objects toward the Earth are solely determined by the centripetal force exerted by the Earth, and that such equal accelerations do not in any way depend upon the mass of the falling body. This dubious claim is dismissed by our prior Section W. Cohen also claimed that Newton’s pendulum experiments proved that “all sorts of heavy bodies fall to the earth from equal heights in equal times.” But this claim is refuted by our prior Section R. Cohen also did not know what he was talking about.
Finally, D’Abro attempted to justify Galileo’s paradoxical result with the following fuzzy circular reasoning: if the gravitational mass and the inertial mass of all bodies were not equal, then two balls with unequal masses “would not reach the earth’s surface simultaneously,” because the inertial resistance of the larger ball “would oppose the acceleration of the fall more strenuously.” D’Abro obviously had no clue what he was talking about.
All of the above physicists and scholars were so mystified, so fooled, and so convinced by Galileo’s empirical illusion of equal gravitational acceleration of unequal masses, that they repeatedly contradicted fundamental laws of physics and obvious astronomical observations in futile and ad hoc attempts to explain, justify and confirm it. In the next few sections we will continue to empirically explain the reasons for Galileo’s absolute gravitational acceleration paradox, and why it does not exemplify or constitute a law of nature.