Spin turns the ball: the Magnus force

Everyone has seen a ball bend in flight. The cause is not the shape of the kick but the spin. A spinning ball drags the thin layer of air next to its surface around with it. On one side the surface moves with the air flowing past it; on the other side it moves against the flow. The wake behind the ball is pushed toward one side, and by Newton’s third law the ball is pushed toward the other. The sideways force is the Magnus force.

Its direction is given by the cross product of the spin and the velocity:

F_M ∝ ω × v

That is awkward to picture, so here is a rule you can use on a pitch: the front of the ball turns toward the side it will curve to.

• Looking from above, a ball spinning counter-clockwise curves to the left (as seen by the player who kicked it).
• Backspin (the front of the ball turns upward) gives an upward force: lift.
• Topspin (the front turns downward) pushes the ball down and makes it dip.

The Magnus force is always at right angles to the velocity. It cannot speed the ball up or slow it down. It only turns the flight direction, like a hand pushing a runner’s shoulder sideways.

How big is it? We use the spin ratio

S = r ω / v,

the speed of the ball’s surface divided by the speed of the ball. A free kick with 5–10 revolutions per second (ω = 2π × that) at 25–30 m/s has S of roughly 0.1–0.3. Experiments on footballs are fitted reasonably well by a lift coefficient C_L = 1 / (2 + 1/S), which gives about 0.1–0.2 here, and then

F_M = ½ ρ C_L A v².

This is a fit to measurements, not an exact law, and different balls give somewhat different values.

Worked example. At 30 m/s with 8 revolutions per second: ω ≈ 50 rad/s, so S = 0.11 × 50 / 30 ≈ 0.18 and C_L ≈ 0.13. Then F_M ≈ ½ × 1.2 × 0.13 × 0.038 × 30² ≈ 2.8 N, about two-thirds of the ball’s weight, pushing it sideways. That is a lot: enough to bend a free kick by metres.