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My doubt is the following: in a rotating frame of reference you have the centrifugal pseudo force.

When you analyze and object in circular motion (due to a rope) in the rotating frame of reference, the object will experience two forces, a real one due to the tension in the rope is centripetal, and the centrifugal pseudo force. Both forces compensate each other, thus the object, in the rotating frame, remains in equilibrium with no motion.

Now assume there is also another object not attached to a rope, and that the object is stationary in the nonrotating reference frame. But in the rotating reference frame this object will experience only the centrifugal force, so instead of seeing it rotating in the opposite direction than the reference system, we should see it accelerating away due the the centrifugal force.

What is it conceptually wrong with my reasoning?

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Actually, there are three pseudo forces in a rotating system of reference: the Coriolis force, the centrifugal force, and the the Euler force. You can see their expressions here Notice that, when there are no moving objects in the rotating reference frame, and in absence of angular acceleration, only the centrifugal force appears.

But in your example, the stationary object rotates with angular speed -$\Omega$, this you need to consider also the coriolis force. In this case, it reduces to: $F_{cor}=-2m\vec{\Omega}$ x $\vec{V}$ which points towards the center of motion.

Then, the forces equations becomes: $F_{total}=F_{cor}+F_{centr}=-2m \Omega V + mV^2/R=-mV^2/R$. Thus the result is actually a centripetal force with the magnitude you expected.

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