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Magnara

Magnus Effect Physics

\[\mathbf{F}_M = \alpha\,\boldsymbol{\omega}\times\mathbf{v}\]
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How does a curveball curve? Magnara is a spinning ball. As it moves through air, its spin drags the flow faster past one side than the other, and the air pushes it sideways: the Magnus effect. Backspin lifts the ball above its no-spin path, and sidespin bends it. Magnus studied it in 1852. 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

A spinning ball moving through air is pushed sideways, toward the side where its surface moves with the airflow. That Magnus force, proportional to spin crossed with velocity, makes spinning balls curve.

Magnara is a spinning ball that refuses to fly straight.

When a ball spins through air, the flow speeds up on one side and slows on the other. Bernoulli pressure then pushes sideways: the Magnus force F_M = α ω × v. Backspin (ω out of the page for a throw to the right) lifts the ball.

On the sticker the cyan curve is a numerical trajectory with backspin; the faint path is the same throw with ω = 0. Mid-flight the spinning ball sits metres higher. The cross product ω × v for ω = ẑ and v = x̂ points the lift along +ŷ.

Heinrich Gustav Magnus described the effect on cannonballs in 1852; footballs, tennis balls and curveballs live on it.

I spin, so I curve.

Equations

\[\mathbf{F}_M = \alpha\,\boldsymbol{\omega}\times\mathbf{v}\]
\[m\dot{\mathbf{v}} = m\mathbf{g} - b\mathbf{v} + \alpha\,\boldsymbol{\omega}\times\mathbf{v}\]
SymbolMeaningUnit
\(F_M\) Magnus force on the spinning ball \(\mathrm{N}\)
\(α\) empirical coefficient (depends on size, spin regime, fluid) \(\mathrm{kg}\)
\(ω\) spin angular-velocity vector \(\mathrm{rad/s}\)
\(v\) centre-of-mass velocity through the air \(\mathrm{m/s}\)
\(m\) mass of the ball \(\mathrm{kg}\)
\(b\) linear drag coefficient \(\mathrm{kg/s}\)
\(g\) gravitational acceleration \(\mathrm{m/s^2}\)

Sources

  1. Magnus (1852) Über die Abweichung der Geschosse, Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin
  2. Clancy, L. J., Aerodynamics (impact and spin chapters)
  3. NASA Glenn Research Center, Beginner’s Guide to Aeronautics: Magnus Effect

Source Magnus, G. (1853). Ueber die Abweichung der Geschosse, und: Ueber eine auffallende Erscheinung bei rotirenden Körpern. Annalen der Physik 164, 1–29.doi.org/10.1002/andp.18531640102

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