There is something I don't quite undestand about the role the centrifugal term plays when describing motion in a non-inertial reference frame. In most Classical Mechanics books, you can find similar discussions on non-inertial reference frames. The "effective acceleration" felt by a body in one of those reference frames would be:
$$\vec{a}_{ef} = \vec{a}_{real} - \ddot{\vec{R}} - \dot{\vec\omega}\times\vec{a}-\vec\omega \times (\vec\omega \times \vec r) - 2\vec\omega\times v_{r}$$
Where $\vec{a}_{real}$ are the real forces acting upon the body, like grativational forces, and the rest of the terms correspond to fictitious forces. My doubt is the following:
For a mass moving in circles, like a child in a merry-go-round, there has to be a centripetal force that we have to account for. It is a real force, after all; making it move in circles. But, in that case, wouldn't the centripetal term and the centrifugal term cancel each other out every time we have a situation like this?
I think I am confused about something and would be thankful if someone could clarify this for me.