Section 7–4 Newton’s law of gravitation

(Universal force / Moon’s fall around the Earth / Moon’s pull on the Earth)

In this section, the three interesting concepts discussed are a universal force, the Moon’s fall around the Earth, and the Moon’s pull on the Earth.

1. Universe force:

“He already knew of the force holding us on the earth, so he proposed that this was a universal force—that everything pulls everything else (Feynman et al., 1963, section 7.4 Newton’s law of gravitation).”

Feynman mentions that Kepler’s second law is a direct consequence of the radial forces toward the Sun. In addition, an analysis of Kepler’s third law shows that the radial forces are inversely proportional to the squares of the orbital distances. Importantly, Newton supposed that the law of gravitation is applicable to the Sun holding the planets as well as to the planet Jupiter that has moons going around it. As Newton had knowledge of the force holding objects on the Earth, he proposed that this is a universal force—that everything pulls everything else. However, Newton did not claim that he understands the nature of gravitational force.

Kepler’s three laws of planetary motion are fundamentally superseded by Newton’s law of universal gravitation and Newton’s laws of motion. In one of his Messenger Lectures, Feynman (1965) explains that “[s]o far Newton has not said anything because he has only stated two things which Kepler said in a different language. One is exactly equivalent to the statement that the force is towards the Sun, and the other is exactly equivalent to the statement that the force is inversely as the square of the distance (p. 20).” Feynman elaborates Kepler’s second law in another Messenger lecture titled, “The Relation of Mathematics to Physics.”

Note: In Principia, Newton (1687) explains that Jupiter’s perturbation of Saturn as evidence of universal gravity.

2. Moon’s fall:

“This idea that the moon ‘falls’ is somewhat confusing, because, as you see, it does not come any closer (Feynman et al., 1963, section 7.4 Newton’s law of gravitation).”

Feynman identifies the problem whether the Earth’s pull on its people is the “same” as its pull on the Moon. Interestingly, he uses the following two contrasting descriptions: “the Moon does not fall at all” and “the Moon does fall.” Feynman initially mentions that the Moon does not fall at all based on a narrower notion of fall. This is because the moon moves circularly such that it maintains the same height from the center of the Earth. Alternatively, it is possible to calculate the distance that the Moon falls in one second which turns out to be approximately 16 feet in a second. In other words, the moon does fall a short distance (from a hypothetical straight line) in every second. Strictly speaking, the orbital path of the Moon is also elliptical.

Feynman has provided a good analogy that explains how an object “falls around” the Earth: an object like a bullet shot horizontally, might go a longer way in one second if we shoot a bullet faster and faster. Then he asks what happens? Importantly, the Earth’s surface is approximately spherical. Thus, if the bullet moves fast enough, it may fall 16 feet such that it maintains the same height above the ground as it was before. Essentially, the bullet still falls, but the Earth curves away such that it “falls around” the Earth. Similarly, the Moon also “falls around” the Earth because the spherical Earth “curves away” such that the Moon does not reach the ground.

3. Moon’s pull:

“If the moon pulls the whole earth toward it, why doesn’t the earth fall right “up” to the moon? (Feynman et al., 1963, section 7.4 Newton’s law of gravitation).”

Feynman provides a simple picture of two tidal bulges on the Earth: the water which is nearer to the Moon is pulled more than the average and the water which is farther away from it is pulled less than the average. Then he emphasizes that the water in the Earth can flow while the rigid Earth cannot. Furthermore, Feynman explains that the water on the farther side is “unbalanced” because the moon’s gravitational force at this location is weaker than it is at the center of the earth, where it just balances the “centrifugal force.” On the nearer side, the gravitational attraction from the Moon on the water is stronger than the centrifugal force, and the imbalance is in the opposite direction in space. Importantly, the Earth and the Moon both go “circularly” around a central position (center of mass of the Earth and the Moon) such that the forces on the Earth is “balanced” and thus, the Earth does not fall toward the Moon.

Feynman’s explanation of the tides may be summarized as three points: (1) An interplay between gravitational forces and centrifugal forces at different locations; (2) The water can flow while the rigid Earth cannot; (3) the Earth and the Moon both rotate around a central position such that the forces are balanced. In short, some may simply explain that the combinational effects of gravity and inertia create two tidal bulges on the Earth. However, Newton (1687) explains the tides as a combined effect of the gravitational forces of the Moon and the Sun.

Questions for discussion:

1. Why is the gravitational force considered to be a universal force?

2. Does the Moon fall around the Earth?

3. How would you explain the formation of tides?

The moral of the lesson: the gravitational force that keeps the moon in orbit is the same gravitational force that pulls objects towards the earth.

References:

1. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Newton, I. (1687/1995). The Principia (translated by A. Motte). New York: Prometheus.