What are the effects of centrifugal and Coriolis forces on an object moving in a rotating reference frame?

In a rotating reference frame, we usually come across the following two pseudo forces:

Centrifugal force: $$f_\mathscr{centrifugal}=-m\omega\times\omega\times r$$

Coriolis force: $$f_\mathscr{coriolis}=-2m\omega\times\dot{r}$$

I couldn't understand the effect of these two forces on a object in a rotating coordinate system. For e.g., if we consider a ball moving on a rotating disk, then what would be the effect of these two forces simultaneously on the motion of the ball. The discussion usually found in textbooks are quite confusing.

It would also be helpful if someone can provide any useful references to conceptually understand the effect of these two forces in a rotating reference frame.

• Do you want to build an intuitive sense for these forces, or do you want to know the way they are derived? – user243267 Dec 5 '19 at 16:53