Magnus Effect Calculator
Estimate the sideways force on a spinning ball from its speed, spin rate, size and the air around it.
What it does
A ball that spins while it flies bends off line. Footballers call it a curl, baseball players call it a breaking pitch, and table tennis players know it as heavy topspin. The reason is the Magnus effect, which NASA describes on its aerodynamics pages: the spinning surface drags the air around it, so the flow on one side speeds up and the other side slows down, and the difference in pressure pushes the ball sideways.
The engineering version of that idea splits the force from the air into two pieces. Drag acts against the motion of the ball, and lift acts at right angles to it. Both use the same building blocks: half the air density, the reference area of the ball, a coefficient that depends on the ball and the flow, and the speed squared. The paper on World Cup balls by researchers at the University of Adelaide lays out both equations and the spin parameter that ties the coefficients to the spin rate.
The reference area is the circle the ball presents to the flow, so a radius of 11 cm gives an area of about 0.038 square meters. The spin parameter is the radius times the angular speed divided by the speed of travel, and it is the number that says how strongly the air is being dragged along.
The coefficients are the part every model struggles with. They depend on the surface of the ball, its seams, the speed, and how the flow changes from smooth to turbulent. That is why both coefficients are inputs here rather than hidden constants.
The size 5 football used as the default meets the rules of the game: the IFAB Laws of the Game require a circumference of 68 to 70 centimeters and a mass of 410 to 450 grams, which is about 11 cm in radius and 0.43 kg.
How to use
- Pick metric or imperial units.
- Enter the radius and the mass of the ball.
- Enter the speed of the ball through the air, not the speed of the throw.
- Enter the spin rate in revolutions per second.
- Enter the air density, and lower it if you are playing at altitude.
- Set the lift and drag coefficients, then read the forces and the sideways acceleration.
Formula
- rho
- Density of the air in kg per cubic meter.
- A
- Reference area of the ball, pi times the radius squared.
- C_L
- Lift coefficient, which already reflects the spin you entered.
- C_D
- Drag coefficient of the ball at that speed.
- v
- Speed of the ball relative to the air.
- S
- Spin parameter, r times omega divided by v.
The lift coefficient you enter is taken as given, so spin is not applied a second time. Spin enters the model through the coefficient, because the lift a real ball generates depends on the spin parameter, the Reynolds number and the surface. The ideal lift formula on the NASA page comes from the Kutta-Joukowski theorem and assumes inviscid flow, so it gives a rough upper case rather than a prediction of a real match.
Worked example
Example: a size 5 football at 108 km/h with 10 rev/s of spin
Defaults: metric units, 11 cm radius, 0.43 kg, 108 km/h, 10 revolutions per second, air at 1.225 kg per cubic meter, and 0.25 for both coefficients.
- Reference area: pi x 0.11 squared = 0.0380 m2.
- Speed: 108 km/h = 30 m/s.
- Angular speed: 2 x pi x 10 = 62.83 rad/s.
- Spin parameter: 0.11 x 62.83 / 30 = 0.2304.
- Lift: 0.5 x 1.225 x 0.0380 x 0.25 x 30 squared = 5.24 N.
- Drag: the same calculation with the drag coefficient gives 5.24 N.
- Sideways acceleration: 5.24 N / 0.43 kg = 12.18 m/s2, which is 1.242 g.
- The tool shows 5.24 N, 5.24 N, a spin parameter of 0.2304 and 12.18 m/s2 (1.242 g).
Reference table
| Speed | Lift | Drag | Spin parameter |
|---|---|---|---|
| 20 m/s | 2.33 N | 2.33 N | 0.3456 |
| 25 m/s | 3.64 N | 3.64 N | 0.2765 |
| 30 m/s | 5.24 N | 5.24 N | 0.2304 |
| 35 m/s | 7.13 N | 7.13 N | 0.1975 |
Limits
- The coefficients have to come from somewhere, and real values depend on the ball, the speed and the flow.
- Ignores the curvature of the flight path, so the forces are evaluated at one instant.
- Assumes steady flow and does not model the change of the spin while the ball is in the air.
- Uses the ball's speed relative to the air, so a headwind or tailwind changes the answer.
Common mistakes
- Multiplying the lift by the spin rate as well as using a lift coefficient, which counts the spin twice.
- Using the radius instead of the reference area, which makes the force far too small.
- Forgetting that the area scales with the square of the radius, so a bigger ball feels much more force.
- Leaving the air density at sea level for a match played in Mexico City, where the air is thinner.
- Expecting a number that matches a video analysis frame by frame, since the coefficients are only averages.
Try it yourself
Quick checks. Answers sit tucked under each question.
At 25 m/s with a lift coefficient of 0.20, what is the lift on the example football?
Lift = 0.5 x 1.225 x 0.0380 x 0.20 x 625 = 2.91 N, while the drag with a coefficient of 0.25 stays at 3.64 N.
The ball travels twice as fast. What happens to the lift?
Speed enters the formula squared, so doubling it multiplies the lift by four. Going from 15 m/s to 30 m/s takes the lift from 1.31 N to 5.24 N.
What is the spin parameter when the ball is not spinning at all?
It is zero, because the angular speed is zero. The lift then depends entirely on the coefficient you enter, which in reality would be close to zero for a non spinning ball.
Questions
Why does a spinning ball curve?
The spinning surface drags the air with it, so the air moving with the spin travels faster and the air against it slower. The pressure difference between the two sides pushes the ball toward the slower side.
What is the spin parameter?
The radius times the angular speed divided by the speed of travel. It measures how much the surface is dragging the air relative to how fast the ball is moving, and it is the number that links a spin rate to the lift coefficient.
How fast does the ball have to spin to curl?
For a football, a few revolutions per second is a gentle bend and ten revolutions per second gives a strong one. The table on this page shows how the spin parameter falls as the speed rises, because the same spin matters less on a faster ball.
Does the Magnus effect work in reverse?
Yes. Backspin lifts the ball and topspin pushes it down, which is why a topspin serve in tennis sits up and dives into the court, and why a backspin shot floats.
How does altitude change the answer?
Thinner air means a smaller density, so both forces drop. At a stadium 2,200 m above sea level the air density is roughly a quarter lower, and so are the forces on the ball.
Can I use this for a baseball or a tennis ball?
Yes, as long as you enter the radius, the mass and coefficients that fit that ball. The equations are the same, but the coefficients and the surface roughness change a lot between sports.
Sources
- NASA Glenn Research Center: Ideal Lift of a Spinning Ball
- Kiratidis and Leinweber, University of Adelaide: An Aerodynamic Analysis of Recent FIFA World Cup Balls
- The IFAB Laws of the Game: Law 2, the ball