WHY THE MOON DOESN’T FALL TO THE EARTH 樊洁茹，1128402016,微电子 Although Newton at first didn’t name the force explicitly, he knew that something had to attract the moon if it was to remain in orbit .The law of inertia stated that the moon would tend to travel in a straight line unless some force acted on it. He coined the word centripetal force for any force that is directed inward, toward the center of an object’s motion (centripetal means center seeking).Gravity is the centripetal force that holds the moon in its orbit. Before he compared the force of gravity on a falling apple and on the moon, Newton realized that any satellite (the moon of example)is a projectile .He considered horizontally projecting a stone from a high mountain .If the stone is given a small initial speed, it doesn’t travel far horizontally before hitting the earth, its inertia keeps it moving horizontally with a constant speed while at the same time it is falling under the influence of gravity .Now if the stone is projected with a greater speed ,it travels farter before it is at last brought to the ground . The greater the initial speed of the projectile, the farter it goes before it falls to the earth. But we must also remember that as the projectile falls, the earth curves away from under it, In the absence of air resistance, the stone could be projected so fast that it would return to the mountaintop. The stone is continually falling, but because of the curvature of the earth it never reaches the ground. Instead, the stone orbits the earth. The moon, Newton realized, has just the right speed to orbit the earth as it does; it is always falling toward the earth, but never reaching it. This, too, is how satellites and space shuttles orbit the earth. Once Newton understood that the moon is falling, he could determine how far it should fall in 1 s in comparison to an apple falling near the surface of the earth. We already know that the distance an apple falls is described by sa 1 = gt 2 ……………………………………………….(1) sa 2 In 1 s this turns out to be 16 ft. And we can express g as

M E g  G 2 …………………………………………………(2) RE

And write 1 S  G(M / R2 )t 2 …………………………………………………(3) a 2 E E The moon is not close to the surface of the earth, and consequently it falls with an acceleration different than that given by Eq(2).What is its acceleration? We can find out by using the universal law of gravity as we did earlier to calculate the acceleration on the surface of the earth. The answer is

M E a=G 2 . .…………………………………………………(4) rm

where rm is the distance from the center of the earth to the moon. The moon, therefore, should fall according to 1 S  at 2 , m 2 Or 1 S  G(M / r 2 )t 2 ………………………………………………….(5) m 2 E m On the earth, an apple falls 16ft (4.9m) in 1 s. How far, Newton asked, does the moon fall? In other words, what is the ratio sm of to sa ? Using Eqs (3) and (4),we find

1 2 at 2 2 Sm 2 a GM E / rm RE    2  2 ………………………………………………(6) S 1 2 g GM / R r a gt E E m 2 The ratio of the distance the moon falls in 1 s to the distance an apple falls is equal to the square of the ratio of the radius of the earth to the distance to the moon. Ancient Greek mathematicians had figured out that the distance to the moon is about 60 times the radius of the earth. In his Principia, Newton cited values of this distance from Ptolemy, Kepler, Tycho, and Copernicus and used the value of 60 earth radii. Thus the ratio is

Sm 1 1  2  Sa 60 3600 …………………………………………………(7)

This means that if an apple falls 16 ft in 1s, the moon in 1 s falls 16 ft S   0.05in ……………………………………………………(8) m 3600 1 Each second the moon is falling in. 20 Now we have a prediction. How can we check it? In other words, how do we calculate how much the moon falls in 1 s? We know that the moon goes around the earth in a nearly circular orbit, taking about one month to complete a revolution. According to the law of inertia, the moon does not want to travel in a circle, but rather fly tangentially out and keep moving in a straight line. Gravity from the earth always makes it fall, thereby causing to move in a circle.

dsm

rm

The distance the moon falls in 1 s, sm is shown above. The distance d is the horizontal distance the moon moves in the same amount of time, From the theorem of Pythagoras we have

2 2 2 (r m  sm )  rm  d , …………………………………………………….(9)

So 2 2 2 2 rm  2rmsm  sm  rm  d …………………………………………………(10)

2 Canceling rm ,we get

2 2 2rmsm  sm  d …………………………………………………(11)

2 Now sm is a quantity so much smaller than rm and d that we can safely neglect it to get

2 2rmsm  d …………………………………………………(12)

Or d 2 sm  2rm If we know d, then we know how far the moon falls in 1 s. Because d is so much smaller than rm , we can approximately d as part of the circle constituting the moon’s orbit. Then the ratio of d

to the circumference of the circle ( 2rm )is equal to the ratio of time it takes the moon to the distance d, which is 1 s ,to the time for the moon to travel once around the circle ,which is its period –one month .This means that

d 1s 7   4.210 …………………………………………..(13) 2rm 1month

Therefore, we know that

d  2  240000mi  4.2107  0.63mi ………………………………………(14)

Substituting this value into our expression for sm we obtain

2 7 sm  (0.63mi) /(2 240000mi)  8.310 mi  0.05in …………………….(15)

Aristotelian physics had been crumbling for more than a century .But all through that time, no one imagined that an experiment done on the earth could reveal the laws of the heavens. That 1 moment arrived when Isaac Newton realized that the moon falls in. every second, just as his 20 theory of universal gravitation predicted. The physics of the heavens and physics on the earth became united in one coherent science.