Barycenter (page 2)
In step 7, you should have found that the balls balance when each is the same distance from the sling. After doubling the mass of one of the balls, the lighter ball must be twice as far from the sling as the heavier ball to balance. After tripling the mass, the lighter ball must be three times as far from the sling as the heavier one.
Two objects balance when the ratio of the distances from sling is equal to the inverse of their mass ratio. Put another way, if one ball is twice as heavy as another, the heavier ball has to be twice as far away from the balancing point. If one ball is five times as heavy, it must be five times as far away from the sling. This can be summed up in a simple equation: m1d1 = m2d2, where m1 and m2 are the masses and d1 and d2 are the distances from the sling.
The balls can be thought of as a planet and a moon, or a planet and a sun. The dowel is like the gravitational force that holds the two near each other. Because more mass means more gravitational force, when a moon and planet orbit each other, they orbit around their center of mass—the place where their masses balance each other. The barycenter is the same as the center of mass!
So if a planet is three times heavier than its moon, where is the barycenter? Based on your ball-and-dowel model, it would be three times closer to the planet than the moon. The Earth has nearly 80 times more mass than the Moon does. Can you guess how much closer the Earth-Moon barycenter is to the Earth’s center? What about the barycenter of the Earth and Sun, where the Sun is 333,000 times more massive than Earth?
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