A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame.
An everyday example of a rotating reference frame is the surface of the Earth. But first, a few words on why fictitious forces are useful. We know that physical results are the same whether we use an inertial system or a non-inertial system (The Principle of equivalence ). In the inertial system, Newton’s laws hold without modification. In the non-inertial linearly accelerating system, we added a non-physical fictitious force
Including the fictitious force allowed us to treat the problem just like a problem in an inertial system. If we tried to treat motion in a rotating coordinate system from the standpoint of an inertial frame, we could easily get bogged down in geometry. We shall see in this section that by adding two fictitious forces, the centrifugal force and the Coriolis force, we can treat motion in a rotating coordinate system as if we were in an inertial system. The fictitious forces systematically account for the difference between the rotating non-inertial system and an inertial system.
The surface of the Earth provides an excellent example of a rotating coordinate system, and using fictitious forces, we will be able to explain observations on the Earth, for example the precession of the Foucault pendulum and the circular nature of weather systems. To analyze motion in a rotating coordinate system, we need an equation that relates motion in inertial and rotating systems. Our approach will be to find a general rule for calculating the time derivative of any vector in a coordinate system that is rotating with respect to an inertial system, and then to apply this to relate velocity and acceleration in the two systems.