# Centrifugal force and centripetal force in fixed coordinate frame

Hello I have a question related to centrifgual force. So in basic textbooks the movement of a particle in a rotating coordinate frame is derived. A starting point is to consider a fixed and a rotating coordinate frame. Then the velocity is expressed as $$$$\mathbf{\dot{r}} = \mathbf{\dot{r}}_0 + \sum\limits_{j=1}^{3} \dot{\overline{x}}_j \mathbf{\overline{e}}_j + \overline{x}_j \mathbf{\dot{\overline{e}}}_j$$$$ where $$\mathbf{\dot{r}}_0$$ denotes the relative velocity of coordinate systems origins, $$\dot{\overline{x}}_j \mathbf{\overline{e}}_j$$ represents the velocity in the rotating coordinate frame and $$\overline{x}_j \mathbf{\dot{\overline{e}}}_j$$ represents the velocity of a point ridigly connected with the rotating coordinate axes. The result of the latter term is the centrifugal force.

Now what I still do not understand is the meaning for the example of a simple pendulum. There the centrifugal force is compensated by the centripetal force. How is this related with the above formula, in particular with respect to the time dependence of the basis vectors. A pendulum is usually considered in a fixed coordinate frame. This would mean a vanishing of the term resulting in the centrifugal force. But then it would not be possible to compensate it with the centripetal force. Where is my error?