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Walk-through

How to Use This Calculator

3 steps

1

Choose your known values

On From I & ω, enter the object's moment of inertia and angular velocity. Don't know I? Use the Moment of Inertia Calculator first, or switch to Point Mass and enter mass, velocity, and radius instead.

2

Read the angular momentum

The result card shows L in kg·m²/s, the rotational equivalent of linear momentum. Both formulas — L = Iω and L = mvr — describe the same physical quantity from different starting points.

3

Solve for ω or I instead

Switch to the Solve tab, pick which variable you're missing, and enter the known angular momentum plus the other quantity — useful for conservation problems like a spinning skater or an orbiting satellite.

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Reference

Formula & Methodology

2 formulas

From moment of inertia and angular velocity

L = I ω

Angular momentum equals moment of inertia (I, in kg·m²) times angular velocity (ω, in rad/s). This is the rotational analog of linear momentum p = mv, with I standing in for mass and ω standing in for velocity.

For a point mass

L = m v r

For a single mass m moving with tangential speed v at distance r from the rotation axis, angular momentum equals mass times velocity times radius. Since I = m r² for a point mass, this is algebraically the same result as L = Iω.

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Glossary

Key Terms Explained

6 terms

Angular momentum (L) A measure of an object's rotational motion, in kg·m²/s. It plays the same role for spinning objects that ordinary momentum plays for objects moving in a straight line.

Moment of inertia (I) How much an object resists a change in its rotational speed, in kg·m². Depends on both mass and how far that mass sits from the rotation axis.

Angular velocity (ω) How fast an object is rotating, measured in radians per second (rad/s). One full rotation per second equals 2π rad/s.

Point mass A simplification that treats an object's entire mass as concentrated at a single point, useful for orbiting bodies or small objects far from the axis.

Conservation of angular momentum In the absence of any external torque, a system's total angular momentum stays constant. If I decreases, ω must increase (and vice versa) to keep L the same.

Spin Informal term for an object's rotation about its own axis, as opposed to orbital motion around an external point — both contribute to total angular momentum.

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Scenarios

Real-World Examples

3 worked examples

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Physics student

Spinning disk with known I and ω

Moment of inertia (I) 0.5 kg·m² Angular velocity (ω) 10 rad/s

L = Iω = (0.5)(10) = 5 kg·m²/s. This is the angular momentum of a disk with that moment of inertia spinning at 10 radians per second, roughly 1.6 revolutions per second.

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Astronomy hobbyist

A mass orbiting at a fixed radius

Mass (m) 2 kg Velocity (v) 3 m/s Radius (r) 0.5 m

L = mvr = (2)(3)(0.5) = 3 kg·m²/s. Treating the orbiting mass as a point at 0.5 m from the axis gives the same L you'd get by first computing I = mr² and then multiplying by ω = v/r.

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Ice skater

Pulling the arms in mid-spin

Angular momentum (L) constant — no external torque Moment of inertia (I) decreases as arms pull in

With no external torque, L stays constant. As the skater pulls her arms in, I drops, so ω must rise to keep L = Iω fixed — this is exactly why skaters spin faster with their arms tucked in.

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Deep Dive

Understanding the Angular Momentum Calculator

3 chapters · 4 min read

Angular momentum (L) is the rotational counterpart of ordinary momentum — it captures how much "rotational motion" a spinning or orbiting object carries. This calculator computes L from moment of inertia and angular velocity, from mass/velocity/radius for a point mass, or solves backward for ω or I when L is already known.

How the Angular Momentum Calculator works

The two formulas on this page describe the same physical quantity from different angles. L = Iω treats the object as a whole, using its moment of inertia (how mass is distributed relative to the axis) and how fast it's spinning. L = mvr treats a single mass as a point traveling in a circle at tangential speed v and radius r — since a point mass has I = mr² and ω = v/r, substituting into L = Iω gives L = mr²·(v/r) = mvr, the exact same result. Use whichever formula matches the numbers you already have. For an object with a more complex shape, first compute I with the Moment of Inertia Calculator, then bring that value here.

Inputs and what they mean

Moment of inertia (I) is in kg·m² and depends on both mass and how it's distributed relative to the rotation axis — a mass spread far from the axis has a larger I than the same mass concentrated near it. Angular velocity (ω) is in radians per second; one full revolution per second equals 2π ≈ 6.283 rad/s. Mass (m), velocity (v), and radius (r) on the Point Mass tab describe a single object moving in a circular path. On the Solve tab, angular momentum (L) is treated as the known, conserved quantity, and you solve for whichever of ω or I is missing.

Conservation and limits

Angular momentum is conserved whenever there's no net external torque acting on a system — this is why an ice skater spins faster pulling her arms in, and why a neutron star spins far faster than the star it collapsed from. But if any external torque acts (friction, an applied force, gravity from another body), L is no longer constant and this simple relationship won't capture the full motion. These formulas also assume a rigid body or a true point mass; for extended, non-rigid, or precessing systems, the full vector treatment of angular momentum is needed. See the Rotational Kinetic Energy Calculator for the related energy stored in a spinning object.

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Questions

Frequently Asked Questions

6 questions

What is the formula for angular momentum?+

Angular momentum is L = Iω (moment of inertia times angular velocity) or, equivalently for a point mass, L = mvr (mass times velocity times radius). Both give the same result in kg·m²/s — use whichever inputs you already know.

Is angular momentum conserved?+

Yes, whenever there's no net external torque acting on the system. If I changes but no torque is applied, ω must change to keep L = Iω constant — this conservation law is why spinning objects speed up or slow down when their mass redistributes.

Why does a spinning skater speed up when she pulls her arms in?+

Pulling her arms in decreases her moment of inertia I (mass moves closer to the spin axis). With no external torque, L = Iω must stay constant, so a smaller I forces a larger ω — she spins faster.

What units does this calculator use?+

Moment of inertia in kg·m², angular velocity in radians per second (rad/s), mass in kilograms, velocity in m/s, radius in meters, and the resulting angular momentum in kg·m²/s (equivalently, kg·m²·s⁻¹ or J·s).

What's the point-mass formula for angular momentum?+

For a single mass m moving at tangential speed v around a circle of radius r, angular momentum is L = mvr. This is the same as L = Iω once you substitute a point mass's moment of inertia I = mr² and its angular velocity ω = v/r.

Can this calculator solve for angular velocity or moment of inertia?+

Yes — the Solve tab lets you enter a known angular momentum L plus either I or ω, and it algebraically solves for the missing one (ω = L/I or I = L/ω). This is useful for conservation problems where L stays fixed but I and ω change.

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Angular Momentum Calculator

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