Angular Momentum Calculator

Angular momentum is the rotational analog of linear momentum — it measures the quantity of rotation an object possesses. Just as linear momentum (p = mv) describes how hard it is to stop a moving object in a straight line, angular momentum (L) describes how hard it is to stop a spinning object. It is a conserved vector quantity, meaning that in an isolated system with no external torque, the total angular momentum remains constant. This conservation law explains why ice skaters spin faster when they pull their arms in, why planets move faster near the Sun, and how gyroscopes maintain orientation.

Angular momentum depends on two factors: moment of inertia (how mass is distributed relative to the rotation axis) and angular velocity (how fast the object spins). For a point mass moving in a circle, angular momentum can also be expressed in terms of linear velocity and radius — connecting rotational and linear descriptions of motion in a single formula.

This calculator supports three equivalent formulations depending on what information you have available:

One of the most powerful principles in physics is the conservation of angular momentum: in the absence of external torque, total angular momentum is constant. This means I₁ω₁ = I₂ω₂ — if moment of inertia decreases (pulling arms in), angular velocity must increase proportionally to conserve L. This principle governs everything from galaxy formation to quantum mechanical spin.

Input values into any of the three methods — fill just one method or combine them to cross-validate:

1. Method 1 — Moment of Inertia: Enter the moment of inertia I (kg·m²) and angular velocity ω (rad/s). This is the most general form and works for any rigid body regardless of shape. Look up the moment of inertia formula for your specific object shape.
2. Method 2 — Point Mass with Linear Velocity: Enter mass m (kg), linear velocity v (m/s), and radius r (m). Use this when you know how fast a point mass is moving along a circular path. The formula L = mvr is particularly useful for orbital mechanics problems.
3. Method 3 — Point Mass with Angular Velocity: Enter mass m, radius r, and angular velocity ω (rad/s). This uses L = mr²ω and is equivalent to Method 2 since v = ωr.
4. View Results: The calculator displays results from each method you filled in, all in units of kg·m²/s (joule-seconds). Cross-check results across methods to verify consistency.

Angular momentum is fundamental to aerospace engineering. Spacecraft use reaction wheels (spinning flywheels) to change orientation without expelling propellant — by changing the wheel's angular momentum, the spacecraft rotates in the opposite direction to conserve total angular momentum. The Hubble Space Telescope and the International Space Station both rely on this principle for precision pointing.

In astrophysics, angular momentum explains everything from the flattened disk shape of spiral galaxies to the formation of accretion disks around black holes. A collapsing gas cloud has some initial angular momentum; as it shrinks, conservation forces it to spin faster and flatten into a disk — the same physics that created our Solar System's ecliptic plane. Pulsars, the rapidly spinning remnants of supernovae, are extreme demonstrations of angular momentum conservation.