Our Magnus force calculator lets you calculate the total force produced by a spinning cylinder moving in a fluid medium. The Magnus effect is the phenomenon that produces this force — it is caused by the interaction between the moving fluid and the spinning object.
Are you curious to learn more about the Magnus effect? Read on to find answers to these questions and more:
- What is the Magnus effect, and how is the Magnus force generated?
- How do you calculate the Magnus force generated by a spinning cylinder?
- How do you use our Magnus force calculator to calculate the Magnus force?
Finally, you will find a numerical example demonstrating how to calculate the Magnus force using our calculator.
What is the Magnus effect?
If you are a fan of ball-based games such as baseball, table tennis, or football, you might have noticed that spinning the ball creates a "magic" phenomenon that deflects the ball from its original path. We call this "magic" phenomenon the Magnus effect. The word "Magnus" in the "Magnus effect" was named after the German physicist Gustav Magnus, who studied the mysterious effect produced by spinning objects in the 19th century.
We define the Magnus effect as the phenomenon that produces a force on a spinning object moving in a fluid medium due to the generation of a pressure gradient across it. On the side where the surface of the cylinder moves in the same direction as the flow, the fluid speeds up and the pressure drops; on the opposite side the fluid slows down and the pressure rises. This pressure difference results in a net sideways force — the Magnus force.
In this particular case, our Magnus force calculator deals with the Magnus force produced by a spinning cylinder. Consider a cylinder of length ℓ and radius r, spinning at a rate s (revolutions per second) in a fluid medium of density ρ that flows past it at the free stream velocity v.
How to calculate the Magnus force generated by a spinning cylinder
The Magnus force on a rotating cylinder follows from the Kutta–Joukowski theorem, which relates the lift force to the circulation of the flow around the body:
F = 4 × π² × r² × s × ρ × v × ℓ
Where:
- F — Magnus force (N in SI, lbf in US units)
- r — radius of the cylinder (m or ft)
- s — spin rate of the cylinder in revolutions per second (rev/s)
- ρ — density of the fluid medium (kg/m³ or lb/ft³)
- v — free stream velocity of the fluid relative to the cylinder (m/s or ft/s)
- ℓ — length of the cylinder (m or ft)
This expression comes from combining the lift per unit length, F/ℓ = ρ · v · Γ, with the circulation around a spinning cylinder, Γ = 2π · r² · ω, where the angular velocity is ω = 2π · s. Substituting gives the compact form above.
How to use the Magnus force calculator
- Choose your unit system — metric (SI) or American/Imperial (US).
- Enter the radius of the spinning cylinder.
- Enter the length of the cylinder.
- Enter the spin rate in revolutions per second (rev/s).
- Enter the free stream velocity of the fluid relative to the cylinder.
- Enter the fluid density (use 1.225 kg/m³ for air at sea level, or about 1000 kg/m³ for water).
- The calculator instantly returns the Magnus force (in both newtons and pound-force), along with the angular velocity, surface speed, and circulation.
Numerical example
Suppose a cylinder of radius r = 0.1 m and length ℓ = 1 m is spinning at s = 10 rev/s in air (ρ = 1.225 kg/m³) while moving at v = 20 m/s. Plugging these values into the formula:
F = 4 × π² × (0.1)² × 10 × 1.225 × 20 × 1 F = 4 × 9.8696 × 0.01 × 10 × 1.225 × 20 F ≈ 96.7 N
So the spinning cylinder experiences a Magnus force of roughly 96.7 newtons (≈ 21.7 lbf) perpendicular to its direction of motion.
Reference fluid densities
| Fluid | Density (kg/m³) | Density (lb/ft³) |
|---|---|---|
| Air (sea level, 15 °C) | 1.225 | 0.0765 |
| Air (hot, 40 °C) | 1.127 | 0.0703 |
| Fresh water | 1000 | 62.43 |
| Sea water | 1025 | 63.99 |
| Olive oil | 920 | 57.43 |
Frequently Asked Questions
- What direction does the Magnus force act?
- It acts perpendicular to both the spin axis and the direction of motion, pushing the object toward the side where the surface spin and the airflow move in the same direction.
- Why does a spinning ball curve?
- The spin drags a thin layer of air around the ball, speeding up the flow on one side and slowing it on the other. The resulting pressure difference produces the Magnus force, which curves the ball's path.
- Does a faster spin produce a larger Magnus force?
- Yes. The Magnus force is directly proportional to the spin rate s, so doubling the spin (with everything else fixed) doubles the force.
- What is a real-world application of the Magnus effect?
- Rotor (Flettner) ships use large spinning cylinders as sails to harness wind and reduce fuel consumption — a direct application of the Magnus force on a rotating cylinder.