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For uniform rotation, the Newtonian gravitational potential plus the centrifugal potential generates the non-Coriolis part of the equation of motion in a rotating frame. Stationary particles obey $\nabla {\Phi }_{\mathrm{eff}}=0$. The Coriolis force changes perturbation dynamics, so curvature of this potential alone is not a general rotating-frame stability criterion.

For a rigid frame with constant angular velocity $\mathbf{\Omega }$, add the Coriolis force and centrifugal force to the inviscid magnetohydrodynamic momentum equation. A constant magnetic permeability and solenoidal magnetic-field constraint split the Lorentz force density into magnetic pressure and magnetic tension. This nonrelativistic conducting-fluid approximation neglects the displacement current, net electric-charge force and viscosity. Differential rotation belongs to the fluid's velocity field, not to a spatially varying rigid-frame rotation vector. A time-dependent frame requires the additional Euler force.

A thermally driven fluid layer rotating about the vertical axis experiences a Coriolis force that couples vertical velocity to vertical vorticity. Its onset depends on the Rayleigh number, Prandtl number and Taylor number, as well as thermal and mechanical boundary conditions. The force does no mechanical work, but changes the linear mode balance and nonlinear convection roll competition.

A rotating reference frame uses spatial axes whose orientation changes relative to an inertial frame. Coordinates in those axes can simplify an apparatus rotating at a known angular velocity $\mathbf{\Omega }$, but the equations of motion acquire inertial terms because the basis vectors themselves change.

For a shared fixed origin and constant $\mathbf{\Omega }$, a particle at rotating-frame position $\mathbf{r}$ and velocity $\mathbf{u}$ satisfiesThe additional terms are the Coriolis force and centrifugal force. A time-dependent $\mathbf{\Omega }$ adds an Euler term $-m\dot{\mathbf{\Omega }}\times \mathbf{r}$; a translating origin can add another term. The rotating-axes theorem derives these contributions by differentiating the position twice, rather than treating them as new interactions with another body.^[ref]

The Technische Universität Darmstadt demonstration uses a steel ball launched outward from the centre of a rotating disc. A fixed lecture-room camera records a straight path, while a camera carried above the disc records a curved path. The instructions specify launching toward the fixed camera so that the laboratory motion is clearly visible. The rotating iPod transmits its view wirelessly through Apple TV; two vertical support rods of length $1\mathrm{m}$ are specified for the slow-motion arrangement. The paired video views are the readout: one physical motion is represented in two reference frames.^[ref]

A useful preparation control appears in the Universität Freiburg instructions: with the disc stationary, check that the ball travels radially toward its edge. A camera is mounted on supports attached to the disc; turn the disc by hand as the ball leaves its centre and observe the camera view.^[ref]

The inference is a dependence of the observed trajectory on the rotating axes; a curved camera trace alone does not isolate the Coriolis force. To compare with the constant-angular velocity free particle in rotating coordinates, measure the disc angular velocity, calibrate lengths in the camera image, and compare launches at rest and with opposite rotation directions. These are checks of the ideal model, not reported measurements from the manuals. A rolling ball can exchange forces with the surface through friction, and hand rotation need not have constant angular velocity; an Euler acceleration then enters too. The ideal free-ball drawing deliberately omits those effects. Neither manual supplies a numerical uncertainty or a precision measurement of the Coriolis acceleration.

The horizontal sea-ice drift momentum balance per unit area includes air and water drag forces, the Coriolis force, depth-integrated internal ice stress and the sea-surface pressure gradient. Here $m$ is ice-plus-snow mass per area and $\Sigma$ is a depth-integrated stress tensor.

// Target: geophysics.bigb

Sea-ice free drift neglects internal stress transmitted from neighboring ice. An isolated floe is governed by air and water drag forces, the Coriolis force and sea-surface tilt; steady drift also neglects acceleration.

// Target: geophysics.bigb

For steady flow of constant-density fluid in a uniformly rotating frame with conservative body forces, the displayed quantity is constant along each relative streamline. The centrifugal potential contributes the negative quadratic term, and the Coriolis force contributes no work because it is perpendicular to the relative velocity. The result follows by dotting the rotating Euler equations for an inviscid fluid with that velocity.

An incompressible Boussinesq fluid with uniform magnetic field, uniform vertical rotation and uniform temperature gradient supports waves combining magnetic tension, Coriolis force and buoyancy. Put $K=|\mathbf{k}|$, $h^2=K^2-k_z^2$, $a^2=(\mathbf{B}\cdot \mathbf{k}{)}^2/({\mu }_0\rho )$ and $H=g\alpha \beta$, with ${\partial }_z{T}_0=-\beta$. Eliminating pressure and the temperature and magnetic perturbations gives the displayed dispersion relation for nonzero frequencies. Stable thermal stratification has $H<0$. Clearing frequency denominators requires checking any resulting zero roots in the original equations; a stationary magnetostatic-buoyancy balance can exist independently of the propagating branches.

A surface gravito-inertial wave combines free surface gravitational restoration with the Coriolis force. For a uniform incompressible fluid occupying $z<0$, take gravity downward, rotation $\mathbf{\Omega }=\Omega {\mathbf{e}}_z$ normal to the surface and $f=2\Omega$. A mode proportional to $e^{i(kx-\omega t)+qz}$ must decay downward, so $q>0$.

The linear rotating momentum equations and incompressibility giveThe kinematic boundary condition is $w=-i\omega \eta$ at the surface. The dynamic condition is $p^{\prime }=\rho g\eta$, so vertical momentum gives ${\omega }^2=gq$. Eliminating $q$ yieldsFor $f=2\Omega$, this isFor nonzero horizontal wavenumber, this branch has ${\omega }^2>f^2$ and an evanescent vertical structure. With no rotation it reduces to the deep-water gravity wave relation ${\omega }^2=g|k|$.

The orientation and incompressible deep-fluid assumptions matter. A tilted rotation axis, finite depth or compressible polytropic atmosphere has a different eigenvalue problem; the displayed formula is the stated incompressible model.

For a layer rotating at angular speed $\Omega$, of depth $d$ and kinematic viscosity $\nu$, the Taylor number compares the Coriolis force to viscous damping through the squared ratio $(2\Omega d^2/\nu {)}^2$. Definitions in other geometries may include different geometric factors; this is the plane-layer convention.

A projectile thrown across a rotating disk's diameter at speed $v$ relative to the thrower inherits the thrower's tangential velocity. During its approximate flight time $2a/v$, a receiver at the opposite rim moves tangentially in the opposite direction. The relative sideways displacement is consequently $4\omega a^2/v$ to leading order in $a\omega /v$. Integrating the Coriolis force in the disk's frame gives the same result.

Wind is motion of atmospheric gas relative to the underlying surface. A pressure gradient accelerates air toward lower pressure, while the Coriolis force deflects moving air in a rotating frame and friction alters flow near the surface. Consequently, large-scale wind need not point directly from high to low pressure: approximate geostrophic balance can produce flow nearly along pressure contours.

An anemometer measures the local motion. A stated wind observation needs its height, averaging interval and directional convention, because gusts and surface obstacles produce variability. Vertical air motion is also physically important: an updraft transports heat and moisture and can support growing hail.

In a leading meridional-track approximation, an aircraft starts southward from the equator at constant speed $v$, with latitude $\lambda =-vt/R$. If lateral steering applies the Coriolis force rather than its negative, the two eastward effects add: ${\ddot{x}}_E=-4\Omega v\sin \lambda$. With zero initial cross-track velocity and displacement, double integration gives the displayed local-plane offset at southern latitude $-\Lambda$. For small $\Lambda$, it is $2\Omega R^2{\Lambda }^3/(3v)$. This assumes steering is a force transverse to the actual velocity while thrust and drag balance along it. It is not compatible with a literally fixed velocity vector. A spherical calculation also includes the transport of the local east and north basis, changing finite-latitude corrections.