(4) Fgrav ~ m
where Fgrav represents the force of gravity on the object. But, by Newton’s third law, if the Earth is pulling on the anvil, or basketball, or whatever, then the anvil, or basketball, or whatever must be pulling on the Earth at the same time. The object falls down in response to the Earth’s gravity, but the Earth also “falls up” in response to the object’s gravity. The magnitude of the acceleration, and hence the fall, is inversely proportional to the mass (by Newton’s second law), and since the Earth is so much more massive than anvils and basketballs, its fall is totally unnoticeable. Nevertheless, fall it does; so the force of gravity must also be proportional to the mass of the Earth, which we’ll denote with a capital M:
(5) Fgrav ~ M
If a quantity is proportional to the first thing, and also proportional to the second thing, it must then be proportional to the product of the two things. We can therefore combine Equations 4 and 5 into:
(6) Fgrav ~ Mm
Since these masses help determine the strength of the gravity between the two objects, these are sometimes called the gravitational masses of the objects, although usually they are just referred to as mass, so long as there is no confusion about whether the mass is inertial or gravitational.
What else? The force of gravity might depend on the distance between the Earth and the object. In everyday experience, that doesn’t appear to be the case. If you drop an anvil a distance of 1 m on the top floor of a skyscraper, it doesn’t fall any differently than if you drop the anvil 1 m on a ground level sidewalk. However, gravity doesn’t emanate from the ground, it emanates from the center of the Earth, which is some 6,400 km below the sidewalk. So it might not be true that gravity is independent of the separation between two objects; it might just be that the difference in height between the top and bottom of a skyscraper is simply too small, in relation to 6,400 km, to be perceptible.
Fortunately, Newton had other data to work with. The German astronomer and astrologer Johannes Kepler (1571-1630) had earlier worked out three laws of planetary motion, the last of which said that the period T of a planet’s orbit—the time it takes to go around the Sun, in years—is related to its average distance r from the Sun. Specifically, after poring through detailed observations of the planets conducted by the Danish astronomer Tycho Brahe (1546-1601), Kepler discovered that:
(7) T2 ~ r3
How did this help Newton? Kepler’s law not only worked for planets orbiting the Sun, but also worked for objects orbiting the Earth, such as the Moon. If the Earth had two moons, they would also obey Kepler’s law. By combining Kepler’s law with what was known of orbital mechanics, he was able to derive another relationship for Fgrav:
(8) Fgrav ~ 1 / r2
The force of gravity is inversely proportional to the square of the distance between the two attracting objects. If you increase the distance by a factor of 2, the square of the distance goes up by a factor of 2 squared, or 4, and the force goes down by that same factor of 4.
Is there anything else? If the Earth and the falling object were ideal point masses, with no length, no width, and no depth, then there are no other properties to speak of. Of course, that’s not the case: the Earth is a big ball of rock and metal, and the other object could be any shape you want. Fortunately, Newton was able to show that these extra complications essentially didn’t matter, and therefore the force of gravity depends on the two masses and the distance separating them, and nothing else. By combining Equations 6 and 8, then, we can get (as Newton did):
(9) Fgrav ~ Mm / r2
As with Newton’s second law of motion, we can rewrite this by introducing a constant of proportionality. This time, however, the constant is not simply inherent in the falling object, or just inherent in the Earth—it is inherent and constant for any two attracting objects in the universe. This constant is denoted by the capital letter G, to indicate that it’s rather important. In addition, we write m1 and m2 to denote the masses of any two objects, rather than the Earth and some other object, and Newton’s formula can now be written as it usually is:
(10) Fgrav = Gm1m2 / r2
G is often called the gravitational constant, and is experimentally determined to be about 6.67×10-11 m3/kg s2. Newton first tried out this equation—without knowing the correct value of G—by comparing the Moon’s orbital motion with that of terrestrial ballistics. Because he started out with an incorrect value for the size of the Earth, the numbers didn’t work out right at first, and Newton disappointedly set aside the theory. Fortunately, some years later, the English astronomer Edmond Halley (1656-1742) convinced Newton to give it another try, with newer and better data, and this time it worked out so well that Newton wrote out his theory of gravitation in detail in his magnum opus, the Principia Mathematica, published in 1687.
A few comments about Newton’s law of universal gravitation. First of all, the constant G is a tiny number. If written out in ordinary figures, without scientific notation, it would be 0.0000000000667 m3/kg s2. That means that at least one of the objects has to be fairly massive in order for the force of gravity to be detectable using ordinary measures. The Earth is massive enough, at about 6×1024 kg, so objects do accelerate noticeably when you drop them here.
However, ordinary objects just aren’t massive enough. In principle, a pair of 10 kg anvils, set some distance apart in a room, will attract each other in accordance with Newton’s law, but in practice you can wait all you want, and the anvils will refuse to budge, because of friction. In order to get moving, the gravitational force has to be exceed the force of friction, and the size of the constant G insures that won’t happen, even if the anvils happen to rest on marbles.
All the same, the force is there, and in 1798 the English physicist Henry Cavendish (1731-1810) was able to measure the gravitational force between two lead balls, one 8 inches across, the other 2 inches across, using a delicate instrument called a torsion balance. In fact, it was his experiment that yielded the first accurate estimate of G.
Another observation is related to Galileo’s experiments with falling bodies. So long as we confine our experiments to the surface of the Earth, Equation 10 is highly constrained. The mass of the Earth M is, for all intents and purposes, constant, as is the distance r between the center of the Earth and the object. We can therefore rewrite Equation 10 as:
(11) Fgrav = m (GM / r2)
where everything in the parentheses is essentially constant near the surface of the Earth. The lone variable is the mass of the object, m. However, by comparing this with Newton’s second law as written in Equation 2a, we can derive a value for the acceleration due to gravity. We could write this as agrav, but conventionally, this quantity is known simply as g:
(12) g = GM / r2
This gives us, correctly, the acceleration experienced by falling objects near the surface of the Earth: 9.8 m/s2.
Now, observe something peculiar (and this is what that odd coincidence is all about). In order to derive this figure, we had to equate the gravitational mass of an object, in Equation 10 or 11, with the inertial mass of the object, in Equation 2a. The undeniable fact that both masses are denoted by the same letter m must not cloud the equally undeniable fact that there is no inherent reason why those two should be the same at all. For example, the electromagnetic attraction between two charged particles is given by Coulomb’s law:
(13) FEM = Kq1q2 / r2
You’ll notice that this formula looks very much like Newton’s formula for gravity; it states that the electromagnetic force is equal to a constant K, times the charge on both of the objects, divided by the square of their distance. The difference between Coulomb’s formula and Newton’s is that the property that generates the electromagnetic force is not mass but charge, while the property that determines the acceleration due to that electromagnetic force is still mass.
Therefore, if you have a basketball and an anvil with equal charges, and you let them move only in response to the attraction of another charged object, they won’t experience the same amount of acceleration—the anvil will move slower, because even though the attracting force is the same, the anvil is more massive and therefore harder to move. But if you let them move only in response to the gravitational attraction of another massive object, they always move with the same acceleration, to a very high level of precision.