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Coriolix

Coriolis Effect Physics

\[\mathbf{a}' = -2\,\boldsymbol{\Omega}\times\mathbf{v}' - \boldsymbol{\Omega}\times(\boldsymbol{\Omega}\times\mathbf{r}')\]
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Why does a straight throw curve on a merry-go-round? Coriolix throws a ball from the hub at a flag on the rim. Seen from the ground the ball flies straight, but the disc turns while it is in the air. Seen from the disc, it bends aside, the Coriolis effect, which Coriolis described in 1835. A 3" pixel-art sticker; equation, symbols and sources in the physics panel below.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

In a rotating frame, moving objects appear to be pushed sideways by the Coriolis term, −2Ω × v. Something moving in a straight line as seen from outside traces a curve as seen from the rotating platform.

Coriolix lives at the hub of a merry-go-round that turns anticlockwise. He throws a ball straight at the flag on the rim.

Seen from the ground, the ball flies in a straight line. But the flag moves on while the ball is in the air, so the ball reaches the rim behind it. Seen from the disc, the ball curves away to the right: x′ = vt cos Ωt, y′ = −vt sin Ωt.

On the disc this looks like a sideways push, the Coriolis acceleration −2Ω × v′, plus the outward centrifugal term. In the art the disc turns 1 radian while the ball crosses it, so it misses by 57.3°. The drawn curve is checked against a numerical solution of the turning-frame equation.

Gaspard-Gustave de Coriolis wrote down these terms in 1835. On the turning Earth the same effect bends winds and sets the spin direction of large storms.

I aimed. You spun.

Equations

\[\mathbf{a}' = -2\,\boldsymbol{\Omega}\times\mathbf{v}' - \boldsymbol{\Omega}\times(\boldsymbol{\Omega}\times\mathbf{r}')\]
\[x' = vt\cos\Omega t,\qquad y' = -vt\sin\Omega t\]
\[\Delta\varphi = \frac{\Omega R}{v}\]
SymbolMeaningUnit
\(\mathbf{a}', \mathbf{v}', \mathbf{r}'\) acceleration, velocity and position seen on the turning disc \(\mathrm{m/s^2, m/s, m}\)
\(\boldsymbol{\Omega}\) angular velocity of the disc (anticlockwise) \(\mathrm{rad/s}\)
\(v\) speed of the ball, thrown from the centre \(\mathrm{m/s}\)
\(t\) time since the throw \(\mathrm{s}\)
\(R\) radius of the disc \(\mathrm{m}\)
\(\Delta\varphi\) angle by which the ball misses the flag at the rim \(\mathrm{rad}\)

Sources

  1. Coriolis (1835) Sur les équations du mouvement relatif des systèmes de corps, Journal de l’École polytechnique 15 (cahier 24), 142
  2. Taylor, Classical Mechanics (2005), Ch. 9: Mechanics in Noninertial Frames
  3. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., Sec. 4.10

Source Coriolis, G.-G. (1835). Sur les équations du mouvement relatif des systèmes de corps. Journal de l’École polytechnique, 15 (cahier 24), 142–154.

Field notes

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