Let's say we have a rotating frame of reference, rotating with constant angular velocity $\omega$ and we observe a stationary object placed away from the origin of this frame. There are no forces acting upon this object, except the fictitious forces arising from our rotating reference frame, which in this case should only be the centrifugal force moving the object away from the origin. However, what we observe from this frame is the object rotating around the origin with angular velocity $-\omega$, corresponding to some fictitious centripetal force. So what's happening here? Why dont we observe centrifugal force, but centripetal force in a rotating frame of reference?
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except the fictitious forces arising from our rotating reference frame, which in this case should only be the centrifugal force
It is the centrifugal force and the Coriolis force. In this case the Coriolis force is twice the magnitude of the centrifugal force and it is pointed inwards. This produces the centripetal force you describe.
The Coriolis force is $\vec F=-2m\ \vec \omega \times \vec v$. The velocity in the rotating frame is $\vec v=-\vec \omega \times \vec r$. So $$\vec F=-2m \ \vec \omega \times (- \vec \omega \times \vec r) $$ Which is $-2$ times the centrifugal force.
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