The gravitational interaction of the Earth and Moon produces a number of effects. The most obvious of these is the orbital motion of the Moon around the Earth, but there is also a motion of the Earth around the center of mass of the Earth-Moon system (discussed in Gravitational Interactions of the Earth and Moon: Barycentric Motion), tidal effects on the Earth and Moon (to be discussed here), and precessional effects on the Earth's axis of rotation (to be discussed in Gravitational Interactions of the Earth and Moon: Precession.

Most tidal phenomena will be discussed in general terms on another page. This page will provide an introductory mathematical foundation for the discussion, for the benefit of those interested in such a foundation; but for the moment, it is merely a brief copy of an earlier incomplete discussion, provided as an introduction to the subject in response to a question from a student.

Tides and Tidal Forces
     Tides are caused by tidal forces raised on one body by another. These forces are a side-effect of the gravitational force between the two objects, caused by the apparent size of each object as seen from the other. If the two objects are very far apart, so each has negligible size in comparison to the distance between them, tidal effects are also negligible; but if the objects are close enough for one or both to have an appreciable size as seen from the other, then tidal forces are generated due to the difference in gravitational force acting on different parts of each object.

The Gravitational Force of the Moon on Objects on the Earth
      Just as the Earth pulls on the Moon, the Moon pulls on the Earth and everything else in space. The strength of the Moon's gravitational force is given by Newton's Law of Universal Gravitation

where G if Newton's gravitational constant, m is the mass of the object being pulled on by the Moon, M_Moon is the mass of the Moon, and r_Moon is the distance between the Moon and the object. A similar force acts between the Earth and various objects, except that we use the distance to the Earth, r_Earth, and the mass of the Earth, M_Earth, in place of the lunar values. Since the Earth's force on something is the object's weight, W, we can write

      M_Moon / M_Earth is about 1/80, and for an object at the surface of the Earth r_Earth is about 4000 miles, while r_Moon is about 240,000 miles, or 60 times greater; so at the surface of the Earth the pull of the Moon is 80 times smaller than the object's weight because of its lesser mass, and another 3600 (= 60 squared) times smaller because of its greater distance. Combining these two effects, the Moon's pull on objects near the Earth is only 1/300,000th of their Earth weight. So if something weighs 150 pounds due to the pull of the Earth, the pull of the Moon on that object would be about 150/300000, or 1/2000th of a pound. This is a very small force but it produces a number of interesting effects because it acts on every object near the Earth, including the Earth itself.

What does this force do?

(most of what follows will be considerably revised, being too incomplete to leave as-is)

(For a discussion of the motion of the Moon around the Earth see The Motion of the Moon, and in the future, other pages on changes in the lunar motion.)

(2) Tidal Effects
(incomplete first draft)
      Gravitational Interactions of the Earth and Moon: Barycentric Motion describes the effect of the Moon's gravity on the Earth as a whole. But there is a second effect, due to the fact that if you are on the "front" side of the Earth (the side where the Moon is up), you are closer to the Moon than if you are on the "back" side of the Earth (the side where the Moon is down), and given the inverse square nature of the Law of Gravity, this means that you are pulled on harder when the Moon is up than when it is down. This produces what are referred to as differential forces, or because of their observable effect, tidal forces.
      On the side of the Earth facing the Moon you are as much as 1/60th closer to the Moon than if you were in the middle of the Earth, producing a force which is 1/30th larger than average. On the side of the Earth facing away from the Moon, you are as much as 1/60th further away from the Moon than if you were in the middle of the Earth, producing a force which is 1/30th smaller than average.
      So on the near side, you are pulled by a force which is 1/300,000th of your weight, which moves you around the barycenter every month, AND by an additional force equal to 1/30th of this, or 1/10,000,000th of your weight, which tends to pull you away from the rest of the Earth. And on the far side, you are pulled on this much less than the rest of the Earth, which tends to pull the rest of the Earth away from you.
      KEEP IN MIND that at the Equator, where things seem to weigh 1/3% less than at the Poles because of the Coriolis effect of the Earth's rotation, the Earth bulges out by about 1/3%. In other words, it bulges out by a fraction of its radius approximately equal to the apparent reduction in weight. IF THE SAME THING happened with the differential force = tidal force of the Moon, the 1/10,000,000th difference in force on different parts of the Earth would make various parts of the Earth deviate from their normal position by 1/10,000,000th of the radius of the Earth, which is about half a meter, or 1 1/2 feet. And on the average, that is exactly what happens in some places; but the actual tides involve a number of other factors, so they can differ from this simple result by a considerable amount.

(Much more to come in final version of this discussion)