the earth in that same unit of time. The moon, in other words, is always falling toward the earth, but it also "sidesteps." In a sense, this "sidestep" means that the earth's surface curves away from the moon just as fast as the moon approaches by falling, and the distance between earth and moon remains the same. This can be made plain if one supposes a projectile fired horizontally from a mountaintop on earth with greater and greater velocity. The greater the velocity, the farther the projectile travels before striking the ground. The farther it travels, the more the surface of the spherical earth curves away from it, thus adding to the distance the projectile covers. Finally, if the projectile is shot forward with
sufficient velocity, its rate of fall just matches the rate at which the earth's surface curves away, and the projectile "remains in orbit." It is in this fashion that satellites are placed in orbit, and it is in this fashion that the moon remains in orbit. In considering the moon's motion, therefore, we need only consider that component which is directed toward the earth, and we can ask ourselves whether that component is the result of the same force that attracts the apple. Let's first concentrate on the apple and see how to interpret the force between it and the earth in the light of the laws of motion. In the first place, all apples fall with the same acceleration regardless of how massive they are. But if one apple has twice the mass of a second apple, yet falls at the same acceleration, the first apple must be subjected to twice the force, according to the second law of motion. The force attracting the apple to the earth (often spoken of as the weight of the apple) must he proportional to the mass of the apple. But according to the third law of motion, whenever one body exerts a force on a second, the second is exerting an equal and opposite force on the first. This means that if the earth attracts the apple with a certain downward force, the apple attracts the earth with an equal upward force. That seems odd. How can a tiny apple exert a force equal to that exerted by the tremendous earth? If it did, one would expect the apple to attract other objects as the earth does, and the apple most certainly does not. The logical way to explain this is to suppose that the attractive force between apple and earth depends not only on the mass of the apple but on the mass of the earth as well. It cannot depend on the sum of the masses, for when the mass of the apple is doubled, the sum of the mass of the apple and the earth remains just about the same as before, and yet the force of attraction doubles. Instead it must depend upon the product of the masses. If we multiply the masses, the small mass has just as much effect on the final product as the large one. Thus, the minute quantity a is now multiplied by the tremendous quantity b yields the product ab . If a is now doubled, it becomes equal to 2 a . If that is multiplied by b , the product is . Thus doubling one of two factors in a multiplication, however small that factor may be, doubles the product. And doubling the mass of the apple doubles the size of the force between the apple
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