 Equation of motion in a rotating frame

Suppose a rotating reference frame shares its origin with an inertial frame and rotates at constant angular velocity $\mathbf{\Omega }$. Let $\mathbf{x}$ be the relative position, with dots denoting derivatives of its rotating components. The two accelerations, expressed in the same axes, satisfyApply the derivative rule $(d\mathbf{A}/dt{)}_{\mathrm{inertial}}=(d\mathbf{A}/dt{)}_{\mathrm{rotating}}+\mathbf{\Omega }\times \mathbf{A}$ twice to the position vector to derive this relation. Newton's second law therefore gives the rotating-frame equationHere $\mathbf{F}$ contains the actual interaction forces; the extra terms are the Coriolis force and centrifugal force.^[ref]

For example, an object at rest on a rotating table still needs an inward interaction force to maintain its circular motion; in rotating coordinates it balances the outward centrifugal force. If the origin has acceleration ${\mathbf{a}}_O$, add $-m{\mathbf{a}}_O$ to the right side. If the angular velocity varies, also add the Euler force $-m\dot{\mathbf{\Omega }}\times \mathbf{x}$. These terms cannot be omitted merely because the object is instantaneously at rest in the rotating axes.