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I got a naive question about the relation between the free-fall acceleration vs gravitational acceleration.

When you consider the rotation of the earth around the axis which runs through the north and south poles, the free-fall acceleration "g" of an object with mass m is different with its gravitational acceleration "a_g".

For example, when the object is at the equator of the earth, then "m*g=m*a_g-m*w^2*R", where "w" is the angular velocity of the earth rotation.

This exmaple is a simple case. What I do not understand is the case that the object is located in a place which is between the equator and poles.

For this case, the component of the gravitational force which is perpendicular to rotational axis acts as the centripetal force for the rotation. How about the component of the gravitational force which is parallel to the rotational axis? Is the magnitude of this component equal to the weight of the object m*g? Or how to relate the free-fall acceleration "g" with the gravitational acceleration "a_g" for this case?

Best wishes

W.