### Newton’s Law of Universal Gravitation

Newton is famous for discovering that planets, moons, comets, and other celestial bodies all obey the very same laws of motion and gravitation that operate on terrestrial objects like apples falling from trees. Allegedly, he was pondering the latter phenomenon (apples falling from trees) when it first dawned on him that gravity might be the force that holds celestial bodies in orbit. Though he never mentioned this epiphany in his own writings, one of his assistants recounted the tale:

[W]hilst he was musing in a garden it came into his thought that the power of gravity (which brought an apple from the tree to the ground) was not limited to a certain distance from the earth but that this power must extend much farther than was usually thought. Why not as high as the moon, said he to himself, and if so that must influence her motion and perhaps retain her in her orbit, whereupon he fell a calculating what would be the effect of that supposition…Quoted in Richard Westfall, Never at Rest: A Biography of Isaac Newton (Cambridge: Cambridge University Press, 1983), 154

Eventually Newton realized that all objects that have mass exert a gravitational force on each other, no matter how far apart they are. Not only is the earth exerting a gravitational pull on the moon, and vice versa; any two objects in the universe are exerting a gravitational pull on each other. Your left shoe is exerting a (very weak) gravitational pull on your right shoe, for example; and your shoes are also pulling on the planets, and on the distant stars, and so on. Most of these forces are far too weak to be noticeable, but they are real nonetheless.

The force of gravity is described by Newton’s law of universal gravitation, which says that any two objects with mass exert an attractive force (gravity) on each other, and the strength of this gravitational force depends on their masses and the distance between them as follows:

F =
 G × m1 × m2 d2
Don’t confuse the gravitational constant (big “G”) with gravitational acceleration (little “g”). The gravitational acceleration g isn’t really constant; it’s only approximately constant, and only for relatively small objects located close to the surface of the earth. The gravitational acceleration on the moon is quite different from the gravitational acceleration on Earth, for example.

In the above equation, F is the magnitude (strength) of the gravitational force, m1 and m2 are the masses of the two objects, and d is the distance between them (or rather, the distance between their centers). G is a quantity called the gravitational constant and has the following value: G = 6.67 × 10-11 N m2/kg2

How did Newton figure out the value of G, you ask? He didn’t. The value of G was unknown until 1798 (more than 70 years after Newton’s death), when it was measured by Henry Cavendish. In his famous experiment, Cavendish used a sensitive torsion balance to measure the very weak gravitational pull between large lead spheres. Since the masses of the lead spheres and the distances between them were known, it was possible to calculate the value of G simply by filling in the other variables in Newton’s law of universal gravitation and solving for G. Cavendish is known as “the man who weighed the earth,” because knowing the value of G enabled him to calculate the mass of the earth from Newton’s laws. (The calculation is surprisingly simple. See appendix A for details.) The Cavendish Experiment Cavendish attached heavy lead spheres to the ends of a rod and hung the rod from a thin wire. Then he placed larger lead spheres near the hanging spheres. Although the gravitational pull between the spheres was very weak, it was enough to twist the wire a little. By measuring the angle of the rod’s rotation, Cavendish could determine the strength of the gravitational force pulling the smaller spheres toward the larger ones. Credit: Chris BurksImage adapted from this diagram by Chris Burks, who kindly released his work to the public domain.

Unlike gravitational acceleration (little “g”), the value of G is the same at all times and places throughout the universe. Quantities like this are called physical constants. (Another example of a physical constant is Coulomb’s constant, which will be introduced later in this chapter.)

The gravitational constant serves as a sort of calibration factor that relates the strength of gravity to the standard units of force, mass, and distance. Notice that G is a very small number: 0.0000000000667, in ordinary decimal notation. This means that the gravitational force is extremely weak, at least in terms of the standard units. For pairs of objects that have very small masses or are positioned very far apart, the gravitational force between them is negligible (though never actually equal to zero).

If we take two objects that are 1 kg each and place them 1 meter apart, the force they exert on each other will be only 0.0000000000667 N. According to Newton’s second law of motion, that force will cause each 1 kg object to accelerate at a slower-than-snails pace of only 0.0000000000667 m/s2, assuming there is absolutely no friction. At that rate, it would take more than 475 years for the object to reach a meager speed of just 1 m/s. That’s a pretty weak force indeed!