Gravity (Why doesn't the Moon fall into the Earth?)

Question: Why doesn’t the Earth fall into the Sun or the Moon fall into the Earth?

A similar examination question is “explain why the satellite does not move in the direction of the gravitational force.” Students are expected to explain why the satellite is moving circularly instead of falling inwardly. They should understand how the centripetal force acting on the satellites rotates the object instead of pulling it closer to the center of the earth.

This question was first answered by Sir Isaac Newton as follows: “[i]f a lead ball were projected with a given velocity along a horizontal line from the top of some mountain by the force of gunpowder and went in a curved line for a distance of two miles before falling to the earth, then the same ball projected with twice the velocity would go about twice as far and with ten times the velocity about ten times as far, provided that the resistance of the air were removed. And by increasing the velocity, the distance to which it would be projected could be increased at will and the curvature of the line that it would describe could be decreased, in such a way that it would finally fall at a distance of 10 or 30 or 90 degrees or even go around the whole earth or, lastly, go off into the heavens and continue indefinitely in this motion. And in the same way that a projectile could, by the force of gravity, be deflected into an orbit and go around the whole earth, so too the moon, whether by the force of gravity – if it has gravity – or by any other force by which it may be urged toward the earth, can always be drawn back toward the earth from a rectilinear course and deflected into its orbit; and without such a force the moon cannot be kept in its orbit. If this force were too small, it would not deflect the moon sufficiently from a rectilinear course; if it were too great, it would deflect the moon excessively and draw it down from its orbit toward the earth. In fact, it must be of just the right magnitude, and mathematicians have the task of finding the force by which a body can be kept exactly in any given orbit with a given velocity and, alternatively, to find the curvilinear path into which a body leaving any given place with a given velocity is deflected by a given force (Newton, 1687, p. 406).” These are not the exact words of Newton, but they are translated by Cohen and Whitman such that Newton’s Principia can be more accessible for today’s scientists and students.

How would Feynman answer?

The focus is on the question “why doesn’t the moon fall into the earth?” Feynman’s answer might include the following three points: the confusing notion of fall, how the moon is falling around the earth, and problems of defining gravity.

1. The confusing notion of fall: In The Feynman Lectures on Physics, Feynman has discussed the question “why doesn’t the moon fall into the earth?” Feynman would consider the possibility of using these two descriptions: the “moon does not fall at all” and “the moon does fall.” For instance, he initially mentions that “[w]e might say that the moon does not fall at all (Feynman et al., 1963, section 7-4 Newton’s law of gravitation).” This is based on a narrower notion of fall. That is, the moon moves circularly such that it maintains the same distance from the center of the earth.

However, it is possible to calculate the distance that the moon falls in one second. According to Feynman, “[w]e can calculate from the radius of the moon’s orbit (which is about 240,000 miles) and how long it takes to go around the earth (approximately 29 days), how far the moon moves in its orbit in 1 second, and can then calculate how far it falls in one second. This distance turns out to be roughly 1/20 of an inch in a second (Feynman et al., 1963, section 7-4 Newton’s law of gravitation).” In short, the moon does fall a short distance in every second.

Importantly, in Feynman’s word, “[t]his idea that the moon ‘falls’ is somewhat confusing, because, as you see, it does not come any closer. The idea is sufficiently interesting to merit further explanation: the moon falls in the sense that it falls away from the straight line that it would pursue if there were no forces (Feynman et al., 1963, section 7-4 Newton’s law of gravitation).” In essence, we can have a broader notion of “falling.” In a sense, the gravitational (attractive) force still pulls the object inward such that it does not move in a straight line. The moon does “fall away” from the straight line.

2. The moon falls around the earth: Feynman has provided a good analogy that explains how an object “falls around” the earth: “[a]n object like a bullet, shot horizontally, might go a long way in one second — perhaps 2000 feet — but it will still fall 16 feet if it is aimed horizontally. What happens if we shoot a bullet faster and faster? Do not forget that the earth’s surface is curved. If we shoot it fast enough, then when it falls 16 feet it may be at just the same height above the ground as it was before. How can that be? It still falls, but the earth curves away, so it falls ‘around’ the earth (Feynman et al., 1963, section 7-4 Newton’s law of gravitation).” Simply phrased, the moon does fall around the earth, but the spherical earth curves away such that the moon does not reach the ground. In addition, the moon may maintain at different heights depending on its velocity.

Interestingly, Feynman would explain the direction of force from a historical perspective. For example, in the Messenger Lectures, Feynman (1965) elaborates that “[t]he next question was - what makes planets go around the sun? At the time of Kepler, some people answered this problem by saying that there were angels behind them beating their wings and pushing the planets around an orbit. As you will see, the answer is not very far from the truth. The only difference is that the angels sit in a different direction and their wings push inwards (p. 18).” In other words, the elliptical motions of planets around the sun are due to forces that are acting on the planets toward the sun.

3. Problems of defining gravity: Feynman might explain a problem of defining gravity due to the incompleteness of knowledge. For instance, Feynman (1965) elaborates that “[a]ll we have done is to describe how the earth moves around the sun, but we have not said what makes it go. Newton made no hypotheses about this; he was satisfied to find what it did without getting into the machinery of it. No one has since given any machinery. It is characteristic of the physical laws that they have this abstract character. ... Why can we use mathematics to describe nature without a mechanism behind it? No one knows (Feynman et al., 1963, section 7-7 What is gravity).” However, there are alternative mathematical models that also explain the nature of gravity.

On the other hand, another problem of defining gravity could be related to the difficulty in detecting gravitational waves. In Feynman’s Lectures on Gravitation for advanced graduate students, Feynman (1995) explains that “the quantum aspect of gravitational waves is a million times further removed from detectability; there is apparently no hope of ever observing a graviton (p. 11).” However, Feynman also provides a thought experiment on gravitational wave detector during the Chapel Hill conference: “[i]t is simply two beads sliding freely (but with a small amount of friction) on a rigid rod. As the wave passes over the rod, atomic forces hold the length of the rod fixed, but the proper distance between the two beads oscillates. Thus, the beads rub against the rod, dissipating heat (Preskill & Thorne, 1995, pp. xxv–xxvi).” Currently, there are reports that gravitational waves have been detected due to the collision of two black holes.

More interestingly, in the Messenger Lectures, Feynman (1965) suggests that “the most impressive fact is that gravity is simple. It is simple to state the principles completely and not have left any vagueness for anybody to change the ideas of the law. It is simple, and therefore it is beautiful. It is simple in its pattern. I do not mean it is simple in its action - the motions of the various planets and the perturbations of one on the other can be quite complicated to work out, and to follow how all those stars in a globular cluster move is quite beyond our ability (pp. 33-34).” In short, the nature of gravity is simple, and thus, it is beautiful. 

       In summary, Feynman might explain that the moon does not fall into the earth, but it does fall around the earth. Simply phrased, the moon continues to fall, however, it could not reach the earth because the earth is spherical. Essentially, it is the same gravitational force that causes an apple to fall onto the earth and the moon that rotates around the earth. Nevertheless, Feynman would elaborate on problems of defining gravity, and at the same time, he would state that the nature of gravity is simple and beautiful.

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. Feynman, R. P., Morinigo, F. B., Wagner, W. G. (1995). Lectures on gravitation (B. Hatfield, ed.). Reading, MA: Addison-Wesley. 
4. Newton, I. (1687/1999). The Principia: Mathematical Principles of Natural Philosophy, A New Translation (Trans. by B. Cohen, & A. Whitman). California: University of California Press. 
5. Preskill, J., & Thorne, K. S. (1995). Foreword to Feynman Lectures on Gravitation. In R. P. Feynman, F. B. Morinigo, & W. G. Wagner. Lectures on gravitation (B. Hatfield, ed.). Reading, MA: Addison-Wesley.