rotatingframe

A rotating frame, or rotating reference frame, is a non-inertial frame of reference that rotates relative to an inertial frame. Such frames are useful for analyzing systems with rotation, curved motion on a rotating body, or dynamics tied to a spinning platform. In a rotating frame with angular velocity vector Ω, the position of a particle is r and its velocity measured in the rotating frame is v′.

The relationship between inertial and rotating quantities involves the velocity v = v′ + Ω × r and the acceleration a = a′ + 2 Ω × v′ + Ω × (Ω × r) + dΩ/dt × r. Because of the extra terms, Newton’s second law in the rotating frame takes the form m a′ = F − m [2 Ω × v′ + Ω × (Ω × r) + dΩ/dt × r]. The terms in brackets are fictitious forces; they do not arise from physical interactions but from using a non-inertial frame.

The Coriolis force corresponds to −2 m Ω × v′ and affects moving objects within the frame, the centrifugal force corresponds to −m Ω × (Ω × r) and acts outward from the axis of rotation, and the Euler force, −m dΩ/dt × r, appears if the rotation rate itself changes with time. If Ω is constant, the Euler term vanishes.

Applications of rotating frames abound. The Foucault pendulum demonstrates Earth's rotation as observed from a rotating frame. Meteorology and oceanography hinge on the Coriolis effect to explain large-scale circulations. In engineering and navigation, rotating frames simplify problems involving rotors, gyroscopes, and spacecraft attitude dynamics, provided the appropriate fictitious forces are included.