General Rotating Frame

General Rotating Frame in advanced mechanics. Theory, formulas, worked examples and applications.

Conceptual context

A passenger seated on a merry-go-round that turns feels something pushing them outward, and if they walk toward the centre, they also feel a lateral force that deflects them. Neither of those two forces exists for an observer standing outside the merry-go-round, looking from the ground: for that observer, the passenger simply moves in a straight line (or along whatever path the real forces dictate) while the merry-go-round turns beneath their feet.

This page gives the general recipe for going from the description of the outside (inertial) observer to that of the observer who turns with the frame: the absolute acceleration absolute acceleration decomposes into five terms measured or computed from inside the rotating frame, and of those five, four give rise to fictitious forces: the translation of the origin and the three that exist only for the rotating observer, namely Coriolis, centrifugal and Euler.

🟢 Essential level

The essential idea is that a general rotating frame can translate AND rotate at the same time, and both effects can be cleanly separated. The accelerated translation of the frame's origin (measured by acceleration of the origin) is the simplest effect: it is the same kind of fictitious force that is felt when a lift starts moving upward.

Rotation adds three new effects, distinct from one another. The centrifugal force (proportional to centrifugal force) pushes radially outward and acts even on a body at rest inside the frame, like the pull felt when sitting still on a merry-go-round. The Coriolis force (proportional to Coriolis force) appears only if the body moves within the rotating frame, and deflects it sideways with respect to its motion; it is why walking radially on a moving merry-go-round feels "twisted". The Euler force (proportional to Euler force) is the least intuitive: it appears only if the merry-go-round itself is speeding up or slowing down its rotation, not when it turns at constant speed.

An example that connects all three: on a merry-go-round that starts from rest, a passenger seated on the edge first feels the Euler force (while the merry-go-round speeds up its rotation), then, once cruising speed is reached, only the centrifugal force (constant), and if they also walk toward the centre, they will feel the Coriolis force while they move.

🔵 Formal level

Basic formulas

Before substituting values, it is advisable to identify which quantity each term represents, check that the units are compatible and predict the sign or trend of the result. This prior reading allows the relation to be used as a physical model and quickly detects an incoherent substitution, direction or hypothesis.

The formal level rests on four relations:

Acceleration in a general rotating frame

Real student questions

"Why is the Coriolis force not felt on a merry-go-round but the centrifugal one is?" Because the Coriolis force only appears if the body moves within the rotating frame; sitting still on the merry-go-round, its relative velocity is zero and that force vanishes, while the centrifugal force acts even at rest, as long as it is not exactly on the axis.

"Does the Euler force exist in real life?" Yes, although it is less well known: it is felt, for example, when a merry-go-round or a Ferris wheel is abruptly started or stopped; its absence from most textbook examples is because these usually assume uniform rotation.

Final synthesis

By the end of this page, the student should be able to decompose the absolute acceleration in a general rotating frame, identify which of the five terms are relevant in a specific problem, and calculate the three fictitious forces (Coriolis, centrifugal, Euler) in the common case of perpendicular vectors.

The relevant final check is one of consistency: verifying that, if the frame neither rotates (angular velocity of the frame zero) nor translates (acceleration of the origin zero), the master relation reduces exactly to absolute acceleration coinciding with relative acceleration, recovering the trivial case of an inertial observer.