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I know that falling objects on the earth are deflected horizontally. But could anyone tell me how can we calculate the deflection of a mass m dropped from a tower of height h at the north pole?

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Assuming that air and wind resistance are not a factor: At the equator, an object above the Earth, that remains above the same spot on the surface, is moving a little faster (greater tangential speed) than the spot on the surface, This is because it is farther from the Earth's axis than the surface. So when it falls, moving faster than the surface, it lands slightly ahead of the spot it was above. As you move closer to the poles, this effect (Coriolis force) is less. Directly above the axial poles, the object will not be moving faster than the surface below it (no tangential speed), so it falls straight down.

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since the poles do not rotate, the mass goes straight down.

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    $\begingroup$ Yes and no. Yes the mass goes down, but the reason is that the rotation of the earth causes no movement along the spin axis. The pole does rotate, but the distance to the rotation axis is zero so no tangential effects are present. $\endgroup$ Nov 26, 2019 at 18:51

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