Centrifugal force in inertial systems

Question

My textbook gives an example:

There are two systems, $O$ is inertial and $O'$ non inertial. $O'$ is rotating whit $\omega=\mathrm{constant}$ and $O=O'$. We assume that a disc is rotating whit the same $\omega$ of $O'$, and we put an object on the disc. There isn't friction between the object and the disc, so the object is quiet in system $O$. In system $O'$ the object seems to rotating (in opposite verse than $O'$).

For the second part of the exemple we connect the object whit the system $O'$ whit a rope, so the object starts rotating with $O'$. Now for $O'$ the object is quiet and $O'$ sees that the rope is in tension, so $O'$ hypotizes that there is a force that tenses the rope (centrifugal).

For $O$ the object is rotating, but how $O$ explains the rope's tension? Centrifugal forces does not exist in inertial system, I think. Am I wrong in thinking that? Why? If I'm not wrong, how is the tension explained?

Thank you all, sorry for bad english

Answer 1

You are correct in suposing centrifugal forces are not the explanation because they do not appear in inertial frames of reference.

In $O$ the object is performing a circular motion, and this motion does not appear when there is no force on the object (it would perform a straight motion).

What the rope does in $O$ is to constrain the motion of the object. The constraint is that the object must remain at a constant distance of the other point the rope is attatched to. For the object to satisfy this constraint it must perform the circular motion, and by the laws of mechanics there is no other way this could happen but the appearence of the tension force.
This tension force is "the rope's way to force a circular motion on the object".