Magnus Force Calculator (sideways force on a spinning ball)

Understanding the Magnus force calculator — concept and principle

This calculator estimates the sideways force on a spinning ball, the Magnus effect, and how far it bends the ball's path over a given distance. It combines the ideal lift of a smooth spinning ball with a real-to-ideal factor, and is used for baseball, football, tennis, golf and physics classroom questions.

Spin pushes the ball sideways with a steady force; over 0.46 s that adds up to about 41 cm.

How it is calculated

Ideal lift (smooth ball): L = (4/3) × 4π² b³ s ρ V

Estimated force = factor × ideal lift

Acceleration a = F ÷ m; flight time t = distance ÷ V

Deflection = ½ a t²; spin ratio = b ω ÷ V

b is the ball radius in m, s the spin in revolutions per second, ρ the air density, V the speed.

The principle

A spinning ball drags a thin layer of air around with it. On one side this circulation adds to the oncoming flow, on the other it subtracts, so the air moves faster on one side and pressure is lower there. The ball is pushed towards the low-pressure side, perpendicular to its motion. The ideal formula comes from the Kutta–Joukowski lift theorem applied to a sphere; real balls slip relative to the air and the flow separates, so measured forces are much smaller. The default factor of 0.15 comes from NASA's discussion of big-league curveball data.

Worked example

Default inputs: diameter 73 mm, 1,800 rpm, V = 40 m/s, ρ = 1.225 kg/m³, factor 0.15, mass 145 g, distance 18.4 m.

b = 0.0365 m; s = 1,800 ÷ 60 = 30 rev/s; spin ratio = 0.0365 × 2π × 30 ÷ 40 = 0.172.
Ideal lift = (4/3) × 4π² × 0.0365³ × 30 × 1.225 × 40 = 3.763 N.
Estimated force = 0.15 × 3.763 = 0.564 N, which is 0.397 of the ball's weight.
a = 0.564 ÷ 0.145 = 3.89 m/s²; t = 18.4 ÷ 40 = 0.46 s.
Deflection = ½ × 3.89 × 0.46² = 41.2 cm.

Each point is the calculator re-run with only “Ball diameter (mm)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Ball diameter (mm)	Estimated Magnus force (N)
18.2	0.009
36.5	0.071
54.8	0.238
73	0.564
91.2	1.102
110	1.905
146	4.515

How to read the result

These inputs resemble a baseball pitched over the 18.4 m distance from mound to plate. A sideways break of about 40 cm is in the range of a curveball's visible bend. More spin, more speed or denser air increase the force.

Common mistakes

Using the ideal force as the real force.
Entering spin in rad/s instead of rpm.
Forgetting that gravity and drag also act, so the full path is more complex.

Keep in mind: The model ignores drag, gravity, changing spin and seams. Treat the deflection as an estimate.

Frequently asked questions

Why does a curveball curve?

Its spin creates a pressure difference across the ball, pushing it sideways.

Does topspin make a ball drop faster?

Yes. The Magnus force then points downward, adding to gravity.

Why is the real force so much smaller than ideal?

The air does not stick perfectly to the ball and the flow separates behind it.

Does altitude matter?

Yes. Thinner air reduces the force in proportion to density.

Sources: NASA Glenn Research Center, Beginner's Guide to Aeronautics, "Ideal Lift of a Spinning Ball" and "Lift of a Baseball"

Checked 9 Nov 2026. NASA Glenn Research Center, Beginner’s Guide to Aeronautics, “Ideal Lift of a Spinning Ball” (L = 4/3 · 4π²b³sρV) and “Lift of a Baseball” (real lift = experimental coefficient × ideal lift; 0.15 from big-league curveball data).

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