Why in every mechanics/dynamics book I look the centrifugal force due to Earth's rotation is measured from "the center of Earth"? For example in Marion's dynamics, the fixed inertial axes are placed at the center of Earth, and the ones rotating with the Earth somewhere on the ground. Then, the centrifugal force on an object is introduced as [ω cross (ω cross (r+R))] term, where r is the position vector in non-inertial frame and R the vector connecting two origins. I was thinking what I measure on the Earth as the centrifugal force wouldn't have anything to do with where I place the origin of the inertial frame, is it correct? Then what if I place the origin of the inertial fixed system somewhere in the space? Will the centrifugal force I measure in the non-inertial frame still be the same?
This is from the book, where the highlighted term is introduced as the centrifugal force:
On the other hand, when discussing motion relative to Earth, it takes something out of one of the terms and add it to aforesaid, and introduces the whole thing as the centrifugal term :/