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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 barycenter of the Earth-Moon system, tidal effects, and precessional effects.
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
F = G m MMoon / r2Moon ,
where G the gravitational constant, m is the mass of the object being pulled on by the Moon, MMoon is the mass of the Moon, and rMoon 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, rEarth, and the mass of the Earth, MEarth, in place of the lunar values. Since the Earth's force on something is the object's weight, W, we can write
W = F = G m MEarth / r2Earth .
MMoon / MEarth is about 1/80; and for an object at the surface of the Earth, rEarth is about 4000 miles, and rMoon 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?
(motion of the Moon around the Earth; link to The Motion of the Moon, and future pages on lunar orbital changes)
The Earth's Barycentric Motion
Just as the Moon moves around the Earth once every 27 1/3 days, as a result of the Earth's pull on the Moon, the Earth moves "around the Moon" once every 27 1/3 days, as a result of the Moon's pull on the Earth. More accurately, each moves around a point in between them, which would be the balance point between them, if they were on a seesaw, called the center of mass or barycenter of the Earth-Moon system.
At any given time, the bodies are on opposite sides of the center of mass, moving in opposite directions. As shown in the diagram (below), each exerts a force on the other which, according to Newton's Third Law of Motion (the Law of Action and Reaction), is equal and opposite to the force that the other is exerting on it; but although the forces are equal, their effects are not, because the more massive Earth is accelerated less by the same force, than the less massive Moon.
*diagram to be inserted here*
The Earth and Moon are each moving around the center of mass of the Earth-Moon system. Since the Earth is much more massive, it is much closer to the center of mass, which is actually inside the Earth. |
As the Earth rotates to the east each day, the Moon appears to move to the west, and the place where the Moon is overhead moves west as well. This has no physical effect, because the center of mass is not a real, physical object, but an imaginary point, defined so the complex motion of the Earth and Moon are easier to visualize and understand. Each body moves around the Sun in a path which is almost identical to the elliptical orbit of the Earth, but weaves back and forth relative to that path every 27 1/3 days, due to their mutual interaction. This complex motion can be simplified by breaking their motion into two parts: the 27 1/3 day motion around the center of mass of the Earth-Moon system, and an annual motion of that point around the Sun.
Note that whether the Moon exerted any force on you or not, you would be compelled to move around the center of mass every month, because the Earth's gravity holds you on the surface while it goes around the center of mass; but you aren't just along for the ride, as the Moon is pulling on you with a force equal to 1/300,000th of your weight, in a way that would cause you to follow the path that the Earth makes around the barycenter, even if you weren't firmly held to the Earth by its gravity.
(2) Tidal Effects
(working on first draft)
That is the effect of the overall pull 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 effect, tidal forces.
On the side 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 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.
(More to come)
(3) Precession
(to be added later)
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