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Inertia forces in the rotating frame (then rotation rate depends on the radius)

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The problem: let us suppose we have some laboratory frame K, and frame K_rot which is rotating with the angular rate depending on R - distatnce to the 0.
Question: where I could find the way, how to get the expression for the inertia forces (like Centrifugal force and Coriolis one for the case of constant angulat rate).

Not sure what you mean by inertia forces. These are pseudoforces, only appearing because you are not in an inertial frame.

The Coriolis force is -2m w X v

Centrifugal/centripetal would be mw2r

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