Rotating coordinate system

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. (This article considers only frames rotating about a fixed axis. For more general rotations, see Euler angles.)

All non-inertial reference frames exhibit fictitious forces. Rotating reference frames are characterized by three fictitious forces:

and, for non-uniformly rotating reference frames,

Scientists living in a rotating box can measure the speed and direction of their rotation by measuring these fictitious forces. For example, Léon Foucault was able to show the Coriolis force that results from the Earth's rotation using the Foucault pendulum. If the Earth were to rotate many times faster, these fictitious forces could be felt by humans, as they are when on a spinning carousel.

The following is a derivation of the formulas for accelerations as well as fictitious forces in a rotating frame. It begins with the relation between a particle's coordinates in a rotating frame and its coordinates in an inertial (stationary) frame. Then, by taking time derivatives, formulas are derived that relate the velocity of the particle as seen in the two frames, and the acceleration relative to each frame. Using these accelerations, the fictitious forces are identified by comparing Newton's second law as formulated in the two different frames.

To derive these fictitious forces, it's helpful to be able to convert between the coordinates ${\displaystyle \left(x',y',z'\right)}$ of the rotating reference frame and the coordinates ${\displaystyle \left(x,y,z\right)}$ of an inertial reference frame with the same origin. If the rotation is about the ${\displaystyle z}$ axis with a constant angular velocity ${\displaystyle \Omega }$, or ${\displaystyle \theta (t){=}\Omega t}$, and the two reference frames coincide at time ${\displaystyle t=0}$, the transformation from rotating coordinates to inertial coordinates can be written

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