The “modern algebraic [equation for] Newton’s law of gravitation is
where F is the gravitational attraction [force] between two spherical bodies m1 and m2 in space, whose centers are separated by a distance R.” This equation was invented by French mathematician Pierre Laplace (1749 – 1827) during the late 18th century. The universal constant value for the gravitational force of attraction of any magnitudes of inertial mass can supposedly be determined from the value of “G,” which we will discuss in the next section.
Feynman interpreted the above equation as follows: “Two bodies exert a force upon each other which…varies directly as the product of their masses.” Gondhalekar interpreted such equation to mean that the total “force between two bodies is proportional to the product of their inertial masses.” Cohen’s interpretation is a bit more informative: “Each body attracts the other with a force of equal magnitude which is directly proportional to the product of the two masses.” But even Cohen does not tell us how or why.
One might be so bold as to ask the question: Is the above modern equation for gravitation and its interpretations completely correct? Is it even complete? Strangely enough, the answer to both of such questions is No.
First of all, the modern equation for gravitation only attempts to describe the total gravitational force (the product of the forces) emitted and received by two opposing masses [i.e. the Earth (82) and the Moon (1)] in any two-body gravitational system. The total gravitational force emitted in all possible directions by two opposing masses is a meaningless number, unless one is trying to compute the value of a gravitational field of force, which is also a meaningless concept. The total quantity of gravitational force received by each opposing body is also a meaningless number, unless one also knows the inertial resistance of each mass: in other words, the ratio of the masses. What can one learn from the modern equation for gravitation that is useful? The answer is: not much. At best, the modern equation is very incomplete and misleading.
The modern equation does not describe the intensity of the force emitted by each mass as a ratio. Nor does it describe the quantity of such intense force received by, and acting upon, each opposing inertial mass. Nor does it describe the different inertial resistances of each opposing inertial mass relative to such forces received and applied by each opposing mass.
In other words, it does not describe the critically important ratios of the opposing masses, the forces received and applied to the different inertial resistances, nor the different resulting accelerations and motions of action and reaction which are described in Newton’s three laws of motion. The mathematician who devised this modern equation must not have been considering Newton’s three laws of motion, and his law of gravitational attraction, conjointly.
In the Earth-Moon system, for example, the proportion of the relative mass of the Moon is 1, whereas the proportion of the relative mass of the Earth is approximately 82 (Figure 16, Column C). The product of these two masses, and the product of their gravitational attractive force emitted in all directions, is 1 x 82 = 82. By itself, 82 is a meaningless number.
In the previous Section P, we learned that the less intense force which is emitted by the Moon in all directions is proportional to the mass of the Moon (or 1), but that the quantity of such less intense force which is received by the Earth is proportional to the inertial mass of the Earth (or 82). We also learned that the more intense force which is emitted by the Earth in all directions is also proportional to the mass of the Earth, but that the quantity of such more intense force received by the less massive Moon is also proportional to the mass of the smaller body (the Moon), or 1 (Figure 17.1). In other words, as Cohen states: “Each body attracts the other with a force of equal magnitude…,” and our Section P will tell us how and why. These forces of equivalent magnitude are proportionally 1:1. This ratio does have some meaning. However, this is only part of the correct equation.
The correct modern equation should also tell us what happens with the gravitational accelerations of the masses involved, because of Newton’s three laws of motion. The best way to accomplish this feat is with ratios of the inertial masses and motions (accelerations) involved. As we learned in Sections I and J, when a force is applied to a body, its acceleration is inversely proportional to its inertial mass. Therefore, the ratio of the inertial masses (inertial resistances) of two gravitating bodies in space is the inverse of the ratio of their accelerations (their actions, reactions and motions):
It follows, that if an equivalent magnitude of force is applied to two different gravitating masses, each with a different inertial resistance, then the relative gravitational accelerations of each different mass (its inertial resistance) relative to the other body must be inversely proportional to its own mass, as required by Newton’s three laws of motion working in conjunction with Newton’s law of gravitational attraction. However, the modern equation for gravitation tells us nothing about these very natural results. Why? Because they would directly contradict Galileo’s Leaning Tower of Pisa results and Newton’s Proposition 6 in Book 3.
Let us again illustrate and explain these conclusions by means of a confirmed observational example. Since the relative attractive force received and acting on the Earth and the Moon is of equivalent magnitude, but the Moon’s inertial resistance is only 1/82 the magnitude of the Earth’s inertial resistance, the Moon accelerates toward the Earth about 82 times as fast and as far as the Earth accelerates toward the Moon. This is basically why the Moon is observed to orbit around the Earth, and not vice-versa (Figure 17.1 and Figure 28).
Thus, it is the ratio between different opposing masses (the relative intensity of one body’s attractive force which is received and acts upon the relative inertial resistance of the other mass, and vice-versa) that determines their relative gravitational actions and motions (accelerations) in any two-body gravitational system. We will discuss all of these concepts in much greater detail in later sections.
So what are the correct modern Newtonian gravitational equations and formulae? We shall describe these correct equations and ratios in detail later in Section JJ.
Another reason why the modern equation is flawed and incomplete is because it doesn’t even begin to tell us what happens in a system where more than two masses are opposing each other. For example, we need to know exactly what happens in the solar system of many planets orbiting around the Sun, or even in a three-body system like Galileo was dealing with at the Leaning Tower (two unequal cannonballs and the Earth). We will demonstrate the answers to these questions later in Section KK.