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If an object is suspended above the earth, as in an airplane, would it fall directly below the point it was dropped from, or would the rotation of the earth cause it to end up somewhere else?

If the object is suspended at a fixed position relative to the earth, then it is in circular motion around the earth (this is referred to as a geosynchronous orbit). In this case it has angular momentum, which must be conserved as the object falls. Since it is at a larger radius from the center of the earth than the surface is, its velocity must increase as the radius decreases to conserve the angular momentum. Thus, it will land at a spot ahead of the point directly below its starting position. If this seems counter- intuitive, think of a figure skater pulling in his/her arms while spinning. The angular velocity of the skater increases sharply to conserve conserve the angular momentum as the arms are brought in. By a similar argument it can be shown that an object initially at rest above a rotating object will fall behind the spot directly beneath it as it falls.

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