Rotating reference frame

For uniform angular velocity $\omega$, potential $\varphi$, and additional forces doing no work in the rotating frame,The Coriolis force does no work; the centrifugal potential accounts for the remaining apparent force. This energy is conserved precisely when the potential has no explicit time dependence in the rotating coordinates. An inertial time-independent conservative force need not meet that condition unless its potential is also invariant under the frame's rotation. The balance prevents applying a rotating-frame conservation law to a nonaxisymmetric stationary inertial potential.

For a smooth plane at inclination $\theta$ rotating about a vertical axis at constant angular velocity $\omega$, choose rotating coordinates $\xi$ horizontally and $\eta$ uphill, with their cross product the upward normal vector. A sliding particle under gravity satisfiesThe normal reaction is $N=m(g\cos \theta -2\omega \dot{\xi }\sin \theta +{\omega }^2\eta \sin \theta \cos \theta )$ while contact is maintained. The Coriolis force does no work in the rotating frame, giving the conserved specific energyThe last term is the centrifugal potential; this quantity need not equal the inertial mechanical energy.

For a smooth horizontal circular hoop of radius $a$, rotated at constant angular velocity $\omega$ about a vertical axis through a rim point, measure $\theta$ at that pivot from the diameter toward the bead. In rotating Cartesian coordinate system coordinates, $\mathbf{r}=a(1+\cos 2\theta ,\sin 2\theta ,0)$. The tangential component of the centrifugal acceleration gives the displayed equation; the Coriolis acceleration is normal to the hoop. The farthest point $\theta =0$ is stable for $\omega \ne 0$, with small oscillation angular frequency $|\omega |$. The pivot itself is unstable. This geometry differs from a vertical rotating hoop and has no tangential gravitational term.

If a frame rotates with angular velocity $\mathbf{\Omega }$ relative to an inertial frame, the derivatives of any vector $\mathbf{a}(t)$ satisfyApplying this formula twice to a position gives the Coriolis acceleration, Euler acceleration, and centrifugal acceleration terms.

In a frame rotating with angular velocity $\mathbf{\Omega }$, a moving particle of mass $m$ has apparent Coriolis force $-2m\mathbf{\Omega }\times \mathbf{v}$.

In a frame rotating with angular velocity $\mathbf{\Omega }$, a moving particle has apparent Coriolis acceleration $-2\mathbf{\Omega }\times \mathbf{v}$.

When a reference frame has time-dependent angular velocity $\mathbf{\Omega }(t)$, a particle at position $\mathbf{r}$ has apparent Euler acceleration $-\dot{\mathbf{\Omega }}\times \mathbf{r}$.

In a uniformly rotating frame, centrifugal acceleration points away from the rotation axis and equals $-\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{r})$.

For uniform rotation about an axis, centrifugal acceleration is the negative gradient of the potential $-{\Omega }^2{s}^2/2$, where $s$ is distance from the rotation axis.

A pendulum of length $a$ whose pivot rotates at radius $L$ can remain at angle $\varphi$ from the downward vertical whenIf the connector is a spring of extension $x$, replace $a$ in the radius by $a+x$ and use vertical balance to determine $x$.

For a smooth circular hoop of radius $R$ rotating about its vertical diameter at angular speed $\omega$, a bead at angle $\theta$ from the downward vertical obeys

At ${\omega }^2=g/R$, the stable downward equilibrium of a rotating-hoop bead loses stability and two stable equilibria withemerge symmetrically.