Magnus Force Calculator
How hard does a spinning ball get pushed sideways? Enter the ball speed, spin rate, radius, and air density — get the Magnus force F = ½·ρ·v²·A·CL, its direction, the spin ratio S = ωr/v, and how it compares to the ball's own weight. Presets for baseball, tennis, golf, and soccer use the middle of each sport's official regulation range. 100% client-side, no data leaves your device.
Magnus Force Solver
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- Force direction (ball moving forward)
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- Spin ratio S = ωr/v
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- Magnus lift coefficient CL
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- Ball cross-section A = πr²
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- Ball weight W = m·g
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- F / W (ball-weights of force)
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- Magnus acceleration
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Magnus CL vs Spin Ratio
Empirical smooth-sphere fit used here: CL = S/(2+S), valid to S = 2.5 (shaded zone beyond is clamped). The dot marks your current operating point.
How it works
F = ½ · ρ · v² · A · CL · S = ωr / v · CL = S / (2 + S)
Spin drags a boundary layer of air around the ball, making airflow faster on one side and slower on the other; the resulting pressure difference pushes the ball sideways — always perpendicular to both the flight path and the spin axis. The lift coefficient CL here is not a wing property: it is a function of the spin ratio S, fit to smooth-sphere wind-tunnel data. Rough balls (seamed baseballs, dimpled golf balls, felt tennis balls) produce slightly more lift than this fit, so treat the result as a baseline.
Frequently Asked Questions
What is the Magnus force equation?
The Magnus force on a spinning sphere is F = one half times rho times v squared times A times CL, where rho is air density, v is the ball speed, A = pi times r squared is the ball cross-section, and CL is the Magnus lift coefficient. Unlike a wing, CL here is not fixed by geometry: it depends on the spin ratio S = omega times r divided by v, via an empirical curve. This calculator uses CL = S divided by (2 + S), a smooth-sphere fit valid up to S = 2.5.
How do you calculate the spin ratio S?
S = omega times r divided by v. Use omega in radians per second (rpm times 2 pi divided by 60), r in metres, and v in metres per second so S is dimensionless. S = 1 means the ball surface moves at exactly the flight speed. Typical sports values are S between 0.1 and 0.6: for example a 28 m/s tennis shot at 3000 rpm with a 33.5 mm radius gives S about 0.31.
Why does the Magnus force point perpendicular to the ball's motion?
Spin drags a thin layer of air around the ball, so airflow is faster on one side and slower on the other. Bernoulli's principle gives lower pressure on the fast side, deflecting the ball toward it. The force vector equals the cross product of the spin axis and the velocity, so it is always perpendicular to both. That is why topspin dives, backspin lifts, and sidespin curves sideways.
What is the difference between topspin, backspin, and sidespin?
Topspin rotates the top of the ball forward: the pressure drop is below the ball, so the Magnus force points downward and the shot dives. Backspin is the reverse: force points upward, carrying the ball farther. Sidespin rotates about a vertical axis, so the force points sideways and the ball curves left or right. Choose the spin type in the calculator to see the direction for your case.
How large is the Magnus force compared to gravity?
The calculator shows the ratio F divided by W, where W = m times g is the ball weight. A ratio above 1 means the Magnus force exceeds gravity. For example, a tennis topspin shot typically gives a ratio around 0.4 to 0.8, comparable to gravity and enough to visibly bend the trajectory. A golf drive with high backspin can approach a ratio near 1, which is why drives stay airborne so long.
What is the difference between Magnus force and airfoil lift?
They are different physics that happen to share the equation F = one half times rho times v squared times A times CL. Airfoil lift comes from the fixed shape of a wing at an angle of attack; its CL is a property of the geometry and angle. The Magnus force comes from the rotation of a symmetric body; its CL is a property of the spin ratio. Use the lift-coefficient calculator for steady wings and this calculator for spinning balls. This page covers spinning spheres only, with no angle-of-attack modelling.
Does altitude or air density affect the Magnus force?
Yes, proportionally: F scales linearly with air density rho. At 2000 m altitude the density is about 1.0 kg per cubic metre versus 1.225 at sea level, so the Magnus force is roughly 18 percent weaker there for the same ball and spin. Enter your local density, or estimate it with our air-density calculator linked below.
Why do golf balls have dimples if spin already creates lift?
Dimples and spin are partners, not alternatives. Dimples trip the boundary layer to turbulent flow, which delays separation and cuts drag roughly in half versus a smooth ball, while also raising the Magnus CL above the smooth-sphere curve used in this calculator. That is why this tool is conservative for dimpled or seamed balls: a real golf drive gets slightly more lift than the smooth-sphere fit predicts.
What are the calculator's preset ball specs based on?
Each preset uses the middle of the official regulation range: baseball 145 g and 73.8 mm diameter (MLB range 141.75 to 148.83 g), tennis 57.7 g and 67.0 mm (ITF 56.0 to 59.4 g, 6.54 to 6.86 cm), golf 45.93 g and 42.7 mm (USGA maximum weight, minimum diameter), soccer 430 g and 220 mm (FIFA 410 to 450 g, 68 to 70 cm circumference). Custom mode lets you override radius and mass for any ball.
What are the limitations of this Magnus model?
The CL = S divided by (2 + S) curve is an empirical smooth-sphere fit, valid for spin ratios up to 2.5; higher spin is clamped and flagged. It assumes steady flight in still air, constant spin, and no ground interaction. Real balls lose spin over time, rough balls like seamed baseballs and dimpled golf balls get extra lift, and drag always acts along the flight path opposite to motion, which this force balance does not include.
Related calculators: Lift Coefficient Calculator (steady airfoil lift — geometry-based CL, the sibling of this spinning-ball tool) · Air Density Calculator (estimate ρ for your altitude and temperature).
Free tool by NRTechStudio