Angular Momentum Calculator

Free Online Angular Momentum Calculator

This angular momentum calculator instantly solves for angular momentum, moment of inertia, or angular velocity using the formula L = I × ω — Angular Momentum equals Moment of Inertia times Angular Velocity. Enter any two known values, and the calculator converts everything automatically, works through the full step-by-step solution, and plots your numbers on a labeled Angular Momentum triangle diagram plus a conservation-of-angular-momentum illustration so the entire concept — not just the number — is visible at a glance.

Whether you're a physics student solving a rotational motion word problem, curious why a figure skater spins faster when they pull their arms in, or an engineer working with flywheels, turbines, or rotating machinery, this tool handles kg·m², g·cm², lb·ft², and slug·ft² for moment of inertia, rad/s, deg/s, rev/s, and RPM for angular velocity, and always shows its complete working, not just the final answer.

What Is Angular Momentum? (Definition and Formula)

Angular momentum is the rotational equivalent of linear momentum — it measures how much 'rotational motion' a spinning object has, based on both how its mass is distributed around the rotation axis (moment of inertia) and how fast it's spinning (angular velocity). A massive flywheel spinning slowly and a small wheel spinning fast can have similar angular momentum, since it depends on both quantities together. Angular momentum is a vector quantity, pointing along the rotation axis according to the right-hand rule.

The formula for angular momentum is one of the cornerstone equations of rotational mechanics:

• Angular Momentum (L) = Moment of Inertia (I) × Angular Velocity (ω)
• The SI unit of angular momentum is kilogram-meters squared per second (kg·m²/s).
• Example: a disk with moment of inertia 2 kg·m² spinning at 5 rad/s has angular momentum = 2 × 5 = 10 kg·m²/s

What Is Moment of Inertia?

Moment of inertia (I) is the rotational analog of mass — it measures how resistant an object is to changes in its rotational speed, and depends on both the object's mass and how far that mass is distributed from the axis of rotation. Unlike ordinary mass, moment of inertia isn't a single fixed property of an object; it changes depending on the axis of rotation and how the mass is arranged relative to that axis. For a simple point mass (or a good approximation for many real objects) at radius r from the axis, moment of inertia is I = m × r² — this calculator includes a built-in helper for exactly this calculation.

How to Find Moment of Inertia from Angular Momentum and Velocity

If you already know an object's angular momentum and angular velocity, you can rearrange the formula to solve for moment of inertia instead.

• Moment of Inertia (I) = Angular Momentum (L) ÷ Angular Velocity (ω)
• Example: an object has 15 kg·m²/s of angular momentum spinning at 5 rad/s → I = 15 ÷ 5 = 3 kg·m²
• This rearrangement comes directly from dividing both sides of L = I × ω by ω.

How to Find Angular Velocity from Angular Momentum and Inertia

If you know an object's angular momentum and moment of inertia, you can find its angular velocity by rearranging the formula once more.

• Angular Velocity (ω) = Angular Momentum (L) ÷ Moment of Inertia (I)
• Example: an object with 20 kg·m²/s of angular momentum and 4 kg·m² of moment of inertia → ω = 20 ÷ 4 = 5 rad/s
• This rearrangement comes from dividing both sides of L = I × ω by I.

The Angular Momentum Triangle

Just like the Force-Mass-Acceleration and Momentum-Mass-Velocity triangles, the angular momentum formula has its own 'magic triangle.' Angular Momentum sits at the top, with Moment of Inertia and Angular Velocity on the bottom row. Cover the quantity you're solving for with your finger, and what's left shows you the formula: covering L leaves I next to ω (I × ω); covering I leaves L over ω (L ÷ ω); covering ω leaves L over I (L ÷ I). This calculator's diagram builds exactly this triangle using your real numbers.

The Law of Conservation of Angular Momentum

One of the most powerful principles in physics is the law of conservation of angular momentum: in the absence of any external torque, the total angular momentum of a system stays exactly constant, even if the object's shape — and therefore its moment of inertia — changes. Since L = I × ω is conserved, if I decreases, ω must increase to compensate, and vice versa.

This is exactly why a figure skater spins faster when they pull their arms in and slower when they extend them back out: pulling the arms in reduces the moment of inertia (mass is now closer to the rotation axis), and since L stays constant, angular velocity must increase to compensate. This calculator's conservation diagram shows this directly, using your own L value split across an 'arms out' and 'arms in' state.

How to Use This Angular Momentum Calculator

Choose what you want to find — Angular Momentum, Moment of Inertia, or Angular Velocity — from the dropdown. Then fill in the two values you already know, picking whichever unit matches your problem. If you don't know the moment of inertia directly, use the built-in point-mass helper: enter a mass and radius, and the calculator computes I = m × r² for you, with a one-click button to drop that value straight into the main calculation.

Scroll down for the complete step-by-step solution, which mirrors exactly how you'd solve it by hand: the formula first, then your real values substituted in with unit conversion shown explicitly, then the arithmetic worked through to the final answer. The results panel also shows the equivalent RPM for your angular velocity, since RPM is often the more intuitive unit for real machinery.

Worked Examples

Spinning disk: moment of inertia 2 kg·m² spinning at 5 rad/s → L = 2 × 5 = 10 kg·m²/s.

Ice skater with arms out: moment of inertia 4 kg·m² spinning at 3 rad/s → L = 4 × 3 = 12 kg·m²/s. If the skater pulls their arms in and halves their moment of inertia to 2 kg·m², conservation of angular momentum means their angular velocity doubles to 6 rad/s, keeping L = 2 × 6 = 12 kg·m²/s unchanged.

Flywheel: moment of inertia 0.8 kg·m² spinning at 1200 RPM (≈ 125.7 rad/s) → L = 0.8 × 125.7 ≈ 100.5 kg·m²/s.

Finding moment of inertia: an object has 15 kg·m²/s of angular momentum at 5 rad/s → I = 15 ÷ 5 = 3 kg·m².

Finding angular velocity: an object with 20 kg·m²/s of angular momentum and 4 kg·m² of moment of inertia → ω = 20 ÷ 4 = 5 rad/s.

Real-Life Applications of Angular Momentum

Angular momentum and its conservation law explain a huge range of everyday and technical phenomena — understanding it makes many otherwise-mysterious behaviors make complete sense:

• Figure skating and diving — spinning athletes pull their arms and legs in to spin faster, and extend them back out to slow down, using conservation of angular momentum with zero extra effort.
• Gyroscopes and spacecraft — gyroscopes maintain a stable orientation because their large angular momentum resists changes from external torques, which is why they're used in navigation systems, drones, and satellite attitude control.
• Bicycle and motorcycle stability — a spinning wheel's angular momentum helps keep a moving bike upright, resisting tipping forces through the same physics that makes gyroscopes stable.
• Planetary and orbital motion — a planet's angular momentum around its orbit (and its own spin) is conserved, explaining why planets in elliptical orbits speed up near the sun and slow down farther away (Kepler's second law is a direct consequence).
• Rotating machinery and flywheels — flywheels store rotational energy and resist changes in speed because of their angular momentum, used in everything from engines to renewable-energy storage systems.
• Diving and gymnastics — divers and gymnasts control their rotation rate mid-air by tucking (reducing I, increasing ω) or extending (increasing I, decreasing ω), with zero angular momentum added after leaving the platform.
• Tornadoes and whirlpools — as air or water is drawn toward the center of a vortex, its moment of inertia decreases and its rotational speed increases dramatically, following the same L = I × ω relationship.
• Star and black hole formation — when a massive star collapses, its moment of inertia shrinks enormously, and conservation of angular momentum causes it to spin extremely fast — this is how pulsars (rapidly spinning neutron stars) form.

Angular Momentum vs. Linear Momentum

Angular momentum (L = I × ω) and linear momentum (p = m × v) are directly analogous concepts, describing rotational and straight-line motion respectively. Mass (m) in linear motion corresponds to moment of inertia (I) in rotational motion — both measure resistance to a change in motion (inertia). Velocity (v) corresponds to angular velocity (ω). Just as linear momentum is conserved in a closed system with no external force, angular momentum is conserved in a closed system with no external torque. This parallel makes it easier to build intuition for rotational mechanics by relating it back to the more familiar linear case.

Tips for Solving Angular Momentum Word Problems

A few habits make angular momentum calculations easier to get right the first time:

• Identify which two quantities are given and which one is being asked for — this tells you whether to use L = I × ω, I = L ÷ ω, or ω = L ÷ I.
• Convert RPM, degrees per second, or revolutions per second to rad/s before doing arithmetic by hand — this calculator does it automatically, but it's worth understanding for manual checks.
• For conservation-of-angular-momentum problems (like the ice skater), remember L stays exactly the same before and after — only I and ω change, and they change in opposite directions.
• If you only know an object's mass and how far it is from the rotation axis, use I = m × r² as a starting approximation — this calculator's built-in helper does this instantly.

Frequently Asked Questions

What is the formula for angular momentum? Angular Momentum = Moment of Inertia × Angular Velocity (L = I × ω). This is the fundamental equation of rotational momentum.

What is angular momentum, in simple terms? It's a measure of how much 'spinning motion' an object has — depending on both how fast it spins and how its mass is spread out around the axis of rotation.

Why does a figure skater spin faster with arms pulled in? Because angular momentum (L) is conserved — pulling the arms in reduces moment of inertia (I), so angular velocity (ω) must increase to keep L = I × ω constant.

How do you calculate moment of inertia from mass and radius? For a point mass or thin ring at radius r, use I = m × r². This calculator includes a built-in helper for this exact calculation.

What is the SI unit of angular momentum? The SI unit of angular momentum is kilogram-meters squared per second (kg·m²/s).

Is angular momentum always conserved? Total angular momentum is conserved in any closed system with no external torque acting on it — internal changes (like pulling arms in) redistribute I and ω but never change L.