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Angular Momentum Calculator

Find the angular momentum of a particle.

How it is calculated

The tool multiplies three numbers: L = m × v × r. The mass m is in kilograms, the speed v in metres per second and the radius r in metres, so the answer is in kg·m²/s and is shown with four decimals. The speed must be the part of the velocity that is perpendicular to the radius, the tangential speed. For motion at an angle, use only the perpendicular component, v × sin θ. Mass must be above zero, and speed and radius cannot be negative; a speed or radius of zero gives 0.0000. The tool returns only the size of L, not the direction, and it treats the body as a single point mass.

Worked example

A 0.5 kg ball on a string moves in a circle of radius 0.8 m at a tangential speed of 12 m/s. Multiply the three values: 0.5 × 12 × 0.8. First 0.5 × 12 = 6, then 6 × 0.8 = 4.8. The tool shows 4.8000 kg·m²/s. If the string is pulled in to a radius of 0.4 m and no outside torque acts, L stays the same, so the speed must double to 24 m/s, since 0.5 × 24 × 0.4 is again 4.8.

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Common mistakes

  • Using weight in newtons or grams for the mass. The field is in kilograms, so a 500 g ball is entered as 0.5.
  • Entering the angular speed ω in rad/s instead of the tangential speed v. The relation is v = ω × r, so convert before using this tool.
  • Using the diameter as the radius. A circle of 1.6 m across has r = 0.8 m, and entering 1.6 doubles the result.
  • Applying it to a spinning solid body such as a flywheel as if it were one point. For an extended body use moment of inertia times angular speed instead.
  • Entering the full speed for motion that is not perpendicular to the radius. Only the perpendicular component counts toward L.

About this calculator

This calculator finds the angular momentum of a particle moving in a circular or curved path using L = m · v · r, where m is mass, v is the (tangential) velocity, and r is the radius from the axis of rotation — the rotational analog of linear momentum p = mv. Angular momentum is a vector quantity, but this formula gives its magnitude for motion perpendicular to the radius.

Because angular momentum is conserved in the absence of external torque, it explains phenomena from a spinning figure skater speeding up as they pull in their arms (reducing r increases the required v), to why planets sweep out equal areas in equal times as they orbit (Kepler's second law), to the stability of a spinning gyroscope or bicycle wheel.

Physics students studying rotational mechanics, engineers analyzing rotating machinery and flywheels, and astronomers modeling orbital dynamics all use this calculation to find angular momentum from mass, velocity, and radius.

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