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“free fall is any motion of a body where gravity is the only acceleration acting upon it.” -Wikipedia

Question 1 - If an object is falling freely, will it follow the Earth’s rotation too? Or will the earth rotate whilst the object falls in a straight line (in its own reference frame). Or more simply put, does it have centripetal acceleration?

Question 2 - Assuming that the earth IS spherical AND has uniform density. Weight of the same mass is different at the Pole and at the equator; thus, the gravitational field strength, $g$ is different at those two places. Are these different values of $g$ also the same as acceleration of a body falling freely at those different places?

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Question 1. No. A projectile will not follow the Earth's rotation. This phenomenon results in very similar to the Coriolis effect. Let it freely fall, but launch it perpendicular to the equator toward one of the poles and you'll see this effect. There is, however, the effect of the wind in Earth's atmosphere, which complicates this answer.

Question 2. The weight is the same at the pole and the equator, i.e. if you consider weight as the gravitational force on a mass. True, the centrifugal 'force' will cause a scale to read a slightly lower result, but that is because part of the acceleration towards the center of the Earth now includes the object's centripetal acceleration. The centrifugal force is directed away from the Earth. The reading on a scale is the object's weight minus the centrifugal 'force' so that the net force is zero (assuming the object is not in freefall and resting on the surface of the Earth).

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