A translating accelerating frame already requires an inertial force because its origin accelerates. A rotating frame introduces additional effects because the coordinate axes themselves change direction.

Let a frame rotate with angular velocity $\boldsymbol\Omega(t)$. For any vector $\mathbf A$,

$$\left(\frac{d\mathbf A}{dt}\right){I} =\left(\frac{d\mathbf A}{dt}\right){R} +\boldsymbol\Omega\times\mathbf A,$$

where $I$ denotes an inertial frame and $R$ the rotating frame.

Applying this relation to position and velocity gives the acceleration transformation

$$\mathbf a_I=\mathbf A_O+\mathbf a_R +2\boldsymbol\Omega\times\mathbf v_R +\boldsymbol\Omega\times(\boldsymbol\Omega\times\mathbf r) +\dot{\boldsymbol\Omega}\times\mathbf r,$$

where $\mathbf A_O$ is the translational acceleration of the rotating frame's origin.

Newton's second law in the inertial frame is

$$m\mathbf a_I=\mathbf F_{\rm real}.$$

Therefore an observer using the rotating frame can write

$$m\mathbf a_R =\mathbf F_{\rm real} +\mathbf F_{\rm trans} +\mathbf F_{\rm Cor} +\mathbf F_{\rm cen} +\mathbf F_{\rm Euler},$$

with

$$\mathbf F_{\rm trans}=-m\mathbf A_O,$$

$$\boxed{\mathbf F_{\rm Cor}=-2m\boldsymbol\Omega\times\mathbf v_R},$$

$$\boxed{\mathbf F_{\rm cen}=-m\boldsymbol\Omega\times(\boldsymbol\Omega\times\mathbf r)},$$

$$\boxed{\mathbf F_{\rm Euler}=-m\dot{\boldsymbol\Omega}\times\mathbf r}.$$

The translational term is the same inertial force that appears in any nonrotating accelerating frame. The remaining three arise specifically from rotation.

Coriolis force

The Coriolis force acts only when the object moves relative to the rotating frame. Because

$$\mathbf F_{\rm Cor}\perp\mathbf v_R,$$

it changes the direction of the relative velocity but does no instantaneous work in the rotating frame:

$$\mathbf F_{\rm Cor}\cdot\mathbf v_R=0.$$

For a frame rotating with constant $\boldsymbol\Omega$, motion parallel to the rotation axis has no Coriolis deflection, while motion with a component perpendicular to the axis does.

Centrifugal force

For constant rotation, the centrifugal force points away from the rotation axis. If $r_\perp$ is the perpendicular distance to that axis, its magnitude is

$$\boxed{F_{\rm cen}=m\Omega^2r_\perp}.$$

An object stationary relative to a uniformly rotating platform can therefore be treated as being in equilibrium under real forces plus this outward inertial force.

For example, a mass held at radius $r$ by an inward tension $T$ satisfies in the rotating frame

$$T=m\Omega^2r$$

when it remains fixed relative to the platform.

In the inertial frame, the same relation is interpreted differently: the tension supplies the inward centripetal acceleration. The centrifugal force is not an additional interaction; it belongs only to the rotating-frame description.

Euler force

If the rotation rate changes with time, the Euler force

$$\mathbf F_{\rm Euler}=-m\dot{\boldsymbol\Omega}\times\mathbf r$$

appears. It is tangential to circles centered on the rotation axis and reflects the angular acceleration of the coordinate frame.

Example: Coriolis deflection on a rotating platform

Suppose a platform rotates with constant angular velocity

$$\boldsymbol\Omega=\Omega\hat{\mathbf z}$$

and an object moves radially outward with relative velocity

$$\mathbf v_R=v_r\hat{\mathbf r}.$$

Then

$$\boldsymbol\Omega\times\mathbf v_R =\Omega v_r\hat{\boldsymbol\phi},$$

so

$$\mathbf F_{\rm Cor} =-2m\Omega v_r\hat{\boldsymbol\phi}.$$

The object is therefore deflected opposite the local direction of platform rotation when moving outward. An inward-moving object has the opposite Coriolis deflection.

What these forces mean

Coriolis, centrifugal, Euler, and translational inertial forces are not new fundamental interactions. They are bookkeeping terms that allow Newton's second law to be used with coordinates whose origin accelerates or whose axes rotate.

They are nevertheless physically useful because many measurements are naturally made in non-inertial frames, including Earth's surface, rotating machinery, centrifuges, vehicles, and atmospheric or oceanic flows.