Reference frame

A reference frame assigns spatial coordinates and time coordinates to physical events.

An inertial frame is a reference frame in which a force-free particle moves with constant velocity.

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 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.