What this calculator does
Angular momentum is the rotational counterpart of ordinary momentum. For a point mass moving in a circle it is simply mass times speed times radius, and like linear momentum it is conserved when nothing twists the system from outside.
That conservation is why a spinning skater speeds up when they pull their arms in. Nothing pushes them faster; reducing the radius must be matched by an increase in rotation rate to keep the total unchanged.
The formula
Angular momentum is mass times tangential speed times radius. The moment of inertia of a point mass is mass times radius squared, and angular velocity is tangential speed divided by radius, so the same quantity can be written as moment of inertia times angular velocity.
| Term | Meaning |
|---|---|
| Angular momentum (L) | Rotational momentum, in kilogram metres squared per second. |
| Moment of inertia (I) | Rotational equivalent of mass: how hard something is to spin up. For a point mass it is mr². |
| Angular velocity (ω) | Rate of rotation in radians per second. |
The inputs explained
| Field | What to enter |
|---|---|
| Mass (kg) | The mass of the object, in kilograms. |
| Radius (m) | The radius of the circular path, in metres. |
| Tangential speed (m/s) | The tangential speed along the circle, in metres per second. |
When to use it
Understanding why a skater speeds up
Pulling the arms in reduces the radius, and conservation of angular momentum forces the rotation rate up to compensate.
Analysing a rotating system
Angular momentum is the quantity that stays fixed through a change in configuration, which makes it the right tool for before-and-after problems.
Comparing rotational and linear motion
Every linear quantity has a rotational counterpart, and seeing them side by side makes the analogy concrete.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does angular momentum scale with tangential speed?
The same mass and radius at a range of tangential speeds.
| Tangential speed | Angular momentum | Angular velocity |
|---|---|---|
| 2 m/s | 2.000 kg·m²/s | 4.000 rad/s |
| 4 m/s | 4.000 kg·m²/s | 8.000 rad/s |
| 6 m/s | 6.000 kg·m²/s | 12.000 rad/s |
| 8 m/s | 8.000 kg·m²/s | 16.000 rad/s |
Questions
Why does pulling your arms in speed up a spin?
Because angular momentum is conserved. Bringing mass closer to the axis reduces the moment of inertia, and since the product of inertia and rotation rate must stay constant, the rotation rate rises to compensate.
Is angular momentum always conserved?
Whenever no external torque acts on the system. That condition is met far more often than people expect, which is what makes it such a powerful shortcut in problems that would otherwise be intractable.
How does this differ for an extended object?
The moment of inertia is no longer simply mr², because different parts sit at different radii. Each shape has its own formula, though the relationship between angular momentum, inertia and rotation rate is unchanged.
Does angular momentum have a direction?
Yes, along the axis of rotation, given by the right-hand rule. That directionality is what makes gyroscopes resist being tilted and keeps a spinning top upright.
For the linear equivalent, see the momentum and impulse calculator. For the torque that changes it, see the torque and rotational power calculator.