The formula
The whole subject is one rule applied twice. The rule is the transport theorem: the rate of change of any vector A as seen from the ground (the inertial frame) equals its rate of change as seen from the spinning platform, plus a rotation term.
(dA/dt)_inertial = (dA/dt)_body + ω × A
Read it: what the ground observer sees changing = what the rider sees changing + how much the frame itself carried the vector around. Here ω is the angular velocity vector of the rotating frame — direction along the spin axis by the right-hand rule, magnitude in rad/s.
Apply the rule once to the position vector r, and you get velocity:
v_inertial = v_body + ω × r
Read it: ground velocity = velocity relative to the platform + the platform sweeping the point around.
Apply the rule a second time — to that whole velocity expression — and the acceleration falls out with three extra terms:
a_inertial = a_body + 2(ω × v_body) + ω × (ω × r) + (dω/dt) × r
a_body— acceleration the rider actually measures on the platform.2(ω × v_body)— the Coriolis term. Only shows up when something moves relative to the frame.ω × (ω × r)— the centripetal term. Points inward toward the axis; its magnitude isω²·(distance to axis).(dω/dt) × r— the Euler term. Only shows up when the spin rate is changing (spin-up, spin-down).
Move everything but a_body to the other side and multiply by mass, and those three terms become the fictitious forces — centrifugal, Coriolis, Euler — that a rider on the platform feels as real.
Where you meet it
- Rate-table and centrifuge testing at Redstone. An IMU or accelerometer bolted to a spinning rate table doesn't just feel the commanded rate. Mount the unit off-center and it feels
ω²rcentripetal bias; spin the table up and it feels the Euler term. If you're checking a gyro's scale factor, you have to know which term is which or your calibration walks. - MEMS gyros — the ones inside every drone and phone in the building. A vibratory rate gyro is the Coriolis term turned into a sensor. A proof mass is driven back and forth; when the whole chip rotates, the
2(ω × v_body)term pushes the mass sideways, and the sideways pickoff reads angular rate. The formula on this page is the device's operating principle. - Long-range fire control and trajectory tools. A projectile in free flight over the ground is moving in a rotating frame (the Earth). A 155 mm round at ~800 m/s drifts on the order of ten-plus meters downrange from the Coriolis term alone — enough that gun tables and fire-control computers carry the correction.
- Any review board looking at a launch or reentry trajectory. Coriolis and centrifugal terms in the Earth-fixed frame are why the guidance equations have the shape they do. Skip them and your Earth-relative trajectory is wrong from the first integration step.
How it works
The cross products are not decoration — they encode direction, and the mistake people make is dropping the vector nature and treating everything as scalars.
Start with the centripetal term. ω × (ω × r) always points from the mass straight in toward the rotation axis, and its size is ω² times the perpendicular distance to that axis. A 0.5 m arm at 60 rpm (ω = 6.28 rad/s) pulls ω²r ≈ 19.7 m/s², about 2 g. That's the centrifuge's whole reason to exist. The outward "centrifugal force" a rider feels is just this term moved to the force side of the equation — it's real to the rider, fictitious to the ground.
The Coriolis term is the one that trips people up because it only exists when the object moves relative to the spinning frame. Sit still on the merry-go-round and there's no Coriolis force. Walk toward the center and you get shoved sideways, hard. A bug crawling outward at 0.1 m/s on a 33-rpm turntable (ω = 3.46 rad/s) feels 2ωv ≈ 0.69 m/s² of sideways push — seven percent of a g, from crawling. The factor of 2 is not a fudge; it comes out of the algebra when you differentiate twice, and half of it is the changing speed of the frame-carried motion while the other half is the changing direction. Anyone who writes ω × v without the 2 has a bug.
The Euler term, (dω/dt) × r, is the one people forget entirely because most bench setups run at constant rate. It's live during spin-up, spin-down, and any commanded rate change. On a rate table doing a rate reversal, the Euler term is what a radially-mounted accelerometer sees as a transient, and it's easy to misread as sensor noise.
Limits of validity: this is all rigid-body kinematics with ω describing a single rigid rotating frame. If your platform flexes, ω isn't well defined across it and you need more frames. The velocity and acceleration formulas as written also assume the rotating frame's origin stays fixed in the inertial frame — pure rotation, no translation [2]. If the frame origin itself moves — a rate table bolted to a moving vehicle, a ship motion platform, a shaker-mounted fixture — you must add the origin's own inertial motion: a_inertial = A₀ + a_body + 2(ω × v_body) + ω × (ω × r) + (dω/dt) × r, where A₀ is the frame origin's acceleration, and likewise add the origin's velocity V₀ in the velocity equation. Forget A₀ and every IMU calibration done on that carrier is quietly wrong. ω itself is a vector only in 3-D — it's the axis-angle rate — and while finite rotations don't commute (rotate a book about X then Z, then reverse the order — different result), the angular velocity ω does add as a vector because it's an infinitesimal rate. That distinction is the subtle one: you can add angular velocities of nested frames (ω_total = ω₁ + ω₂ + …), but you cannot add finite rotation angles like numbers.
The other classic trap is bookkeeping about which frame you're differentiating in. The transport theorem relates the two derivatives; it does not let you mix them in one line. Pick a frame, stay in it through the whole derivation, and only convert with the ω × term at the boundary.
History
The deflection came first as a gunner's problem, not a mathematician's. In 1651 the Jesuit astronomer Giovanni Battista Riccioli and his assistant Francesco Maria Grimaldi noted in the Almagestum Novum that a rotating Earth ought to bend the path of a cannonball, and in 1674 Claude François Milliet Dechales argued the same for both falling bodies and projectiles [1]. The effect was too small to measure with 17th-century instruments, so it sat as an argument rather than a formula.
The mathematics arrived before the name. Euler wrote down the acceleration terms for motion in a rotating frame in 1749, and Pierre-Simon Laplace carried the same terms in his tidal equations in 1778 [1][2] — decades before anyone attached a person's name to them.
The name comes from Gaspard-Gustave de Coriolis (1792–1843), a French engineer and mathematician who taught analysis and mechanics at the École Polytechnique [3][4]. He was already the man who had given engineering the modern words: his 1829 Du Calcul de l'effet des machines defined "work" and kinetic energy in the sense engineers still use [3]. Then in 1835 he published Sur les équations du mouvement relatif des systèmes de corps — "On the equations of relative motion of systems of bodies" — worked out while studying the energy of machines with rotating parts like waterwheels [1][3]. Coriolis wasn't chasing weather or cannonballs; he was doing machine dynamics, and the "compound centrifugal force" fell out of the rotating-frame bookkeeping. The meteorologists borrowed his term later, which is why the name lands on the engineer and not on Euler or Laplace who had the math first [1][2].
Related tools
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