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Centre of Gravity:
Most people have at least an intuitive notion of the centre of gravity (CG) of an object: it is the point on which the object can be perfectly balanced. Grab a broom at one end and the other end tries to drop down; grab it at its centre of gravity, and it stays balanced, neither end tipping over. If you have learned to balance a chair or a broom on the palm of your hand, you know the trick is to place your hand right below the centre of gravity. Since your hand is not at the CG but below it, it must be constantly moved to keep that strategic position. There also exists a precise mathematical definition--it has nothing to do with gravity, which is why many scientists and engineers prefer the term centre of mass. However, it leads off the main subject and therefore we won\'t bother with it now. A lightweight stick with two balls of equal weight at its end obviously has its CG in the middle. When one ball is twice the weight of the other, the CG divides the distance between them by a ratio 1:2, in a way that makes it closer to the heavier mass (see figure). And similarly for other ratios
Balls that push each other
Now imagine that instead of a lightweight stick the above two heavy balls have a spring between them, held compressed by a string. Even though the balls are separate, one can speak of their common center of gravity, on the line connecting their centers, 1/3 of the distance from the center of the heavier ball. (The CG of the Earth-Moon system can be defined in the same manner. Since the ratio of masses of the two bodies is about 81:1, the CG is the point on the line between their centers dividing it by that ratio. It can be shown that--neglecting the pulls of the Sun and of other planets--the Moon does not orbit the center of Earth, but rather the common CG--and so does the Earth, reacting to the pull of the moon. Of course, since the Earth is much more massive, the CG is not very far from the center of the Earth--in fact, it is closer than the Earth\'s own surface.)Suppose next that a lit match is placed against the string, burning it through. As the spring expands, it pushes the balls apart; if it is sufficiently light, its own motion does not matter and we can assume that the balls push each other. By Mach\'s formulation of the equations of motion, if the heavy ball receives an acceleration a, then the light one gets 2a, twice as much. For each increment in the velocity of the heavy ball, the light one receives twice as much, and it follows that at any time, its total velocity, as well as the distance covered, are twice those of the heavy ball. If then the heavy ball is at a distance D from the initial position of the spring, the light one is at distance 2D--as in the earlier figure, reproduced here. No matter how much time passes, the center of gravity stays at the same spot.
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