The Magnus force calculator above computes the sideways force that spin produces on a body moving through air — the effect that curves a football, hooks a golf ball, drops a topspin tennis shot and drives a Flettner rotor ship. It offers the ideal inviscid relation for a sphere and for a rotating cylinder, and a lift-coefficient form for when you have real measured data.
Arb Digital publishes free physics calculators. The boundary with the live lift coefficient calculator is worth stating: that page works the standard lift equation for a wing, from a lift coefficient, an airspeed and a wing area, and it has nothing to say about spin. This page starts from spin, computes the circulation that rotation produces, and turns that into a force. The two meet in the coefficient mode, where the same lift equation is applied to a spinning sphere's frontal area.
What This Magnus Force Calculator Does
A spinning body drags a thin layer of air round with it. On one side that dragged air adds to the oncoming flow and on the other it opposes it, so the flow speeds up on one side and slows on the other. The pressure difference that results pushes the body sideways, perpendicular both to its direction of travel and to its spin axis.
The tool evaluates that force three ways. The ideal sphere relation applies the Kutta–Joukowski theorem to a rotating ball, which is the textbook derivation. The rotating cylinder relation does the same for a two-dimensional cylinder and multiplies by its length. The coefficient form takes a lift coefficient you supply from measurement and applies the standard lift equation to the frontal area.
Every mode also reports the spin ratio, which is the surface speed divided by the airspeed, and the lift coefficient the computed force implies. Those two together are how the ideal result gets compared honestly with reality.
How to Use It
- Choose the body and method. Start with the ideal sphere relation to see the upper bound, then switch to the coefficient form once you have measured data.
- Enter the radius in millimetres, and the length as well if you are modelling a cylinder.
- Enter the spin rate in revolutions per second, revolutions per minute or radians per second.
- Enter the airspeed and the air density. Airspeed is relative to the air, not to the ground.
- Add a lift coefficient for the coefficient mode and a mass to see the force as a multiple of the body's own weight.
The Formula: How the Magnus Force Is Calculated
The underlying result is the Kutta–Joukowski theorem: a body with circulation Γ moving through a fluid of density ρ at speed V feels a lift per unit length of ρVΓ. Spin generates that circulation.
For a rotating cylinder of radius r spinning at angular rate ω, the ideal circulation is Γ = 2πr²ω, so the force on a cylinder of length L is
F = ρ V Γ L = 2π ρ V ω r² L
For a sphere spinning at s revolutions per second, NASA's Glenn Research Center guide to the ideal lift of a spinning ball derives the corresponding result:
L = (4/3) · 4π² r³ s ρ V
and the same page is explicit that this is an ideal-flow answer. It assumes inviscid flow with no boundary layer, whereas a real ball has a turbulent boundary layer, surface roughness, seams and dimples, all of which change the answer. Real predictions use an experimentally determined lift coefficient.
The coefficient form is the standard NASA lift equation, L = CL · ½ρV² · A, applied with A the frontal area πr² of the sphere.
The dimensionless group that ties them together is the spin ratio, S = ωr / V, the ratio of surface speed to airspeed. Rearranging the ideal sphere relation shows that it corresponds to a lift coefficient of exactly 16S/3, which is the cleanest way to see how far the ideal result sits from reality.
Work the defaults through by hand. A sphere of radius 36.6 mm spinning at 30 rev/s, moving at 40 m/s through air at 1.225 kg/m³. The cube of the radius is 4.902 × 10⁻⁵ m³, and 4/3 of 4π² is 52.64. Multiplying through: 52.64 × 4.902 × 10⁻⁵ × 30 × 1.225 × 40 = 3.793 N. The spin ratio is (2π × 30) × 0.0366 / 40 = 0.1725, and the implied lift coefficient is 16 × 0.1725 / 3 = 0.920. Against a 145 g mass, that force is 2.67 times the body's own weight.
Why the Ideal Answer Is Too Big, and by How Much
The implied lift coefficient of 0.92 in that worked example is the giveaway. Wind-tunnel and ball-tracking measurements on real spheres at a spin ratio near 0.17 typically give lift coefficients in the region of 0.2 to 0.3, not 0.9. The ideal relation overpredicts by roughly a factor of three, and NASA's own page says as much.
The reason is the boundary layer. The ideal derivation assumes the surface drags the surrounding fluid perfectly, so the circulation is the full 2πr²ω. A real surface only entrains a thin layer, and the effective circulation is a fraction of the ideal figure. Surface finish decides how large that fraction is, which is why a scuffed cricket ball, a seamed baseball and a dimpled golf ball all behave differently from a smooth sphere and from each other.
Use the ideal modes for the physics and for an upper bound on the effect. Use the coefficient mode with a measured coefficient when a number has to be right.
The Spin Ratio Matters More Than the Spin Rate
This is the most useful practical idea on the page. What governs the size of the effect is not revolutions per minute but the ratio of surface speed to airspeed. A ball spinning at 2,500 rpm has a much larger spin ratio at 20 m/s than at 60 m/s, and correspondingly more deflection.
Two consequences follow that surprise people. First, deflection is often at its most dramatic late in a flight, after drag has bled off speed while spin decays comparatively slowly. That is why a curving free kick or a hooked golf shot appears to bend most sharply near the end of its path. Second, at very high spin ratios the effect saturates and can even reverse: for a smooth sphere at low spin ratios, an inverse or negative Magnus effect has been observed, caused by the spin tripping the boundary layer to turbulence asymmetrically. That is a genuine reversal of the force direction and is one more reason to distrust an ideal formula outside its comfortable range.
Radius enters strongly too. The ideal sphere force goes as the cube of the radius while the mass goes as the cube as well, so the acceleration is roughly size-independent for a constant density; but real balls are not constant density, which is why a light plastic ball curves spectacularly and a shot put does not curve at all.
Flettner Rotors and Why a Cylinder Beats a Ball
A rotating cylinder is a far more effective Magnus device than a sphere, because a cylinder has circulation along its whole length and no three-dimensional relief at the ends. Lift coefficients well above what any conventional aerofoil achieves are attainable at high spin ratios, which is why rotor ships exist.
The Flettner rotor is a tall vertical cylinder spun by a small motor. In a crosswind it generates thrust along the ship's axis, and because the driving power scales with rotor friction rather than with the thrust produced, the net energy balance can be strongly favourable. Modern installations on cargo vessels are a working application of exactly the relation in the cylinder mode above.
The same physics limited the rotor aircraft experiments of the 1920s. Rotating cylinders generate lift but also very large drag and severe structural loads, and they need continuous power to spin, so they never competed with a wing for aircraft. On a ship, where the cylinder can be enormous and the drag penalty matters far less, the trade works out differently.
Where This Sits Next to the Other Aerodynamics Tools
The live lift coefficient calculator works the standard lift equation for a wing from a coefficient and a wing area, with no spin term at all. The drag force calculator gives the component along the flight path, which for a spinning ball is what removes the speed the Magnus force depends on. The air density calculator supplies the density input from temperature, pressure and humidity.
For flow regime and trajectory, the Reynolds number calculator tells you whether the boundary layer is laminar or turbulent, which is what decides the real lift coefficient, and the projectile motion calculator covers the ballistic path before any aerodynamic force is added. The angular velocity calculator converts between revolutions per minute and radians per second.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Treating the ideal result as a prediction — it assumes inviscid flow and typically overstates the force on a real ball by around a factor of three.
- Entering a diameter as a radius — the ideal sphere force goes as the cube of the radius, so this one error changes the answer by a factor of eight.
- Including spin about the velocity axis — only the component of spin perpendicular to the direction of travel produces a Magnus force. Bullet spin produces none.
- Using ground speed in a wind — the relation needs airspeed, and a headwind or tailwind changes it directly.
- Assuming a constant lift coefficient through the flight — the spin ratio changes as drag removes speed, so the coefficient changes with it.
Related Free Tools From Arb Digital
Pair this with the live lift coefficient calculator for wing lift and the drag force calculator for the along-path component. The air density calculator supplies the density, the Reynolds number calculator identifies the flow regime, the angular velocity calculator converts spin units, and the projectile motion calculator covers the underlying trajectory. The air viscosity calculator gives the viscosity a Reynolds number needs. Everything Arb Digital publishes sits on the free online tools hub.
Frequently Asked Questions
It is the sideways force a spinning body feels when moving through a fluid. The spinning surface drags a thin layer of air with it, speeding the flow on one side and slowing it on the other, and the resulting pressure difference pushes the body perpendicular to both its direction of travel and its spin axis. It is what curves a football and hooks a golf ball.
Because it assumes inviscid flow in which the surface drags the surrounding fluid perfectly, giving the full theoretical circulation. A real surface only entrains a thin boundary layer, so the effective circulation is a fraction of the ideal value. NASA's own page on the ideal lift of a spinning ball says explicitly that real balls require an experimentally determined lift coefficient.
It is the surface speed of the spinning body divided by its airspeed, and it is the dimensionless group that actually governs the effect. The same spin rate produces a much larger spin ratio, and much more deflection, at low airspeed than at high. It also explains why a ball appears to bend most sharply late in its flight, after drag has removed speed while spin persists.
No. Only the component of spin perpendicular to the velocity contributes. A rifle bullet spinning about its own flight axis experiences no Magnus force from that spin, which is precisely why gyroscopic stabilisation works without curving the trajectory. A ball with a tilted spin axis produces a force perpendicular to both, which is how a slice and a topspin combine.
Yes. A reverse or negative Magnus effect has been observed on smooth spheres at low spin ratios, where the rotation trips the boundary layer to turbulence on one side and not the other, moving the separation points the wrong way. It is another reason not to rely on an ideal formula outside the range where it has been checked against measurement.
It is a tall vertical cylinder spun by a small motor on a ship's deck. In a crosswind the Magnus force acts along the ship's axis and provides thrust. Because the power needed to overcome the rotor's own friction is much smaller than the propulsive force generated, the energy balance can be strongly favourable, and modern cargo vessels use them for fuel saving.
Because circulation acts along its whole length with no three-dimensional relief at the ends, so the lift per unit frontal area is much higher. Rotating cylinders can reach lift coefficients well beyond what a conventional aerofoil achieves. The penalties are large drag, heavy structural loads and continuous power to keep it spinning, which is why rotors suit ships rather than aircraft.
Yes, proportionally. The force is directly proportional to air density in every form of the relation, so at high altitude where density is perhaps 20 per cent lower, the Magnus deflection falls by the same 20 per cent. That is one of the reasons breaking pitches behave differently at elevation, alongside the reduced drag that lets the ball travel further.
This tool is provided for educational use only. The ideal relations assume inviscid flow and overstate the force on real balls, typically by a large factor; the lift-coefficient form is only as good as the coefficient you supply, which must come from measurement of your own object at your own spin ratio. Surface finish, seams, dimples, boundary-layer state and spin decay all change the real result, and this page models none of them. It is not suitable for engineering design.
