Angular Momentum Calculator: L = I × ω
Angular momentum (L) is one of the most important conserved quantities in physics, governing everything from the spin of figure skaters to the rotation of galaxies. For a rigid body rotating about a fixed axis, it is calculated with the compact formula L = I × ω, where I is the moment of inertia (in kg·m²) — a measure of how mass is distributed relative to the rotation axis — and ω is the angular velocity (in radians per second). The result has units of kg·m²/s, equivalent to the joule-second (J·s) used in quantum mechanics.
This calculator is designed for physics and engineering students, mechanical designers, robotics engineers, and educators who need quick numerical answers. Enter the moment of inertia and angular velocity; get the angular momentum and a brief physical interpretation.
The single most powerful consequence of the formula is the Conservation of Angular Momentum: in any isolated system with no external torque, L remains constant. This is why a figure skater spins faster when pulling in their arms (smaller I forces larger ω to keep L unchanged), why neutron stars rotate hundreds of times per second after collapse (collapse from R ≈ 10⁶ km to R ≈ 10 km reduces I by ~10¹⁰), and why the gyroscope on your phone can detect rotation without external reference.
Angular momentum (L) is calculated with L = I × ω, where I is the moment of inertia in kg·m² and ω is the angular velocity in rad/s. For example, a flywheel with I = 0.5 kg·m² spinning at 10 rad/s has L = 5 kg·m²/s. The unit kg·m²/s equals J·s — the same as Planck's constant. When no external torque acts, L is conserved: if I decreases (skater pulls arms in), ω increases proportionally.
When to use this calculator
- Mechanical engineer designing a flywheel energy storage unit: I = 50 kg·m² at ω = 314 rad/s (3,000 rpm) → L = 15,700 kg·m²/s, stored rotational energy ≈ 2.46 MJ.
- Physics student solving textbook problem: solid disk of mass 2 kg and radius 0.3 m (I = ½mr² = 0.09 kg·m²) spinning at 20 rad/s → L = 1.8 kg·m²/s.
- Aerospace analyst calculating reaction wheel torque budget for a small satellite: wheel I = 0.001 kg·m² at ω = 6,000 rad/s → L = 6 kg·m²/s, enough for fine pointing.
- Sports scientist evaluating figure skater spin: pre-tuck I ≈ 4 kg·m² at ω = 5 rad/s → L = 20 kg·m²/s; after pulling arms in (I drops to 1 kg·m²), ω jumps to 20 rad/s to conserve L.
- Automotive engineer modeling wheel angular momentum: 18-inch tire + rim assembly (I ≈ 1.2 kg·m²) at 100 km/h (ω ≈ 80 rad/s) → L = 96 kg·m²/s per wheel.
- Robotics designer sizing servo motors for a robotic arm joint: payload I = 0.05 kg·m², required acceleration to ω = 10 rad/s in 0.5 s gives τ = ΔL/Δt = 1 N·m.
- Wind turbine engineer estimating rotor angular momentum: utility-scale 1.5 MW turbine rotor (I ≈ 5 × 10⁶ kg·m²) at ω = 1.8 rad/s (17 rpm) → L ≈ 9 × 10⁶ kg·m²/s — explains why braking requires massive systems.
Worked Example: Spinning Bicycle Wheel
- Wheel moment of inertia: I = 0.1 kg·m²
- Angular velocity at ~30 mph: ω = 30 rad/s
- L = I × ω = 0.1 × 30 = 3 kg·m²/s
How it works
2 min readHow the Formula Works
Angular momentum L measures how much rotational motion an object carries. The formula L = I × ω has two inputs:
The product gives L in kg·m²/s (= J·s), the same unit as Planck's constant in quantum mechanics.
Reference Table: Common Angular Momentum Values
| Object | I (kg·m²) | ω (rad/s) | L (kg·m²/s) |
|---|---|---|---|
| Spinning top (small) | 0.001 | 100 | 0.1 |
| Bicycle wheel | 0.1 | 30 | 3 |
| Figure skater (arms in) | 1 | 20 | 20 |
| Figure skater (arms out) | 4 | 5 | 20 |
| Car flywheel (idle) | 0.5 | 100 | 50 |
| Car flywheel (highway) | 0.5 | 300 | 150 |
| Wind turbine rotor | 5,000,000 | 1.8 | 9,000,000 |
| Earth (rotation) | 8.04×10³⁷ | 7.27×10⁻⁵ | 7.07×10³³ |
Note how the figure skater rows have the same L — that is conservation of angular momentum in action.
Moment of Inertia Quick Reference
| Shape | Formula for I |
|---|---|
| Thin hoop / ring (axis through center) | I = mr² |
| Solid disk / cylinder | I = ½mr² |
| Solid sphere | I = ⅖mr² |
| Hollow sphere | I = ⅔mr² |
| Thin rod about its center | I = (1/12)mL² |
| Thin rod about one end | I = (1/3)mL² |
Key Physical Insight
Conservation of Angular Momentum: In the absence of external torque, L remains constant. This means I and ω trade off: pull mass inward (↓ I) and rotation speeds up (↑ ω). This is the 'spinning skater effect' and also explains why neutron stars (collapsed from stellar cores) spin hundreds of times per second.
This calculation is for classical mechanics — fixed-axis rotation. For asymmetric bodies, precession, or 3D rotation, the full vector form L⃗ = I̅·ω⃗ (with I as a tensor) is required.
Frequently asked questions
What is angular momentum and how do you calculate it?
Angular momentum (L) measures the rotational motion of an object. For a rigid body rotating about a fixed axis, L = I × ω, where I is the moment of inertia in kg·m² and ω is the angular velocity in rad/s. Example: a flywheel with I = 2 kg·m² spinning at 50 rad/s has L = 100 kg·m²/s. The unit kg·m²/s is equivalent to J·s.
What does the conservation of angular momentum mean in practice?
When no external torque acts on a system, L = I × ω stays constant. This means if moment of inertia I decreases (e.g., a figure skater pulls arms inward), angular velocity ω must increase proportionally. If I drops by half, ω doubles. This principle also explains why neutron stars spin so fast (stellar collapse reduces I by 10¹⁰, so ω increases by 10¹⁰) and why planets maintain their orbits.
What is the difference between angular momentum and linear momentum?
Linear momentum p = mv (kg·m/s) describes straight-line motion. Angular momentum L = I·ω (kg·m²/s) describes rotational motion. They are independent — a spinning top has angular momentum but may have zero linear momentum if its center of mass is stationary. Both follow conservation laws: p is conserved with no external force; L is conserved with no external torque.
How do I find the moment of inertia I for my object?
For common shapes: solid disk = ½mr²; solid sphere = ⅖mr²; thin hoop = mr²; thin rod (center) = (1/12)mL². For composite bodies, sum the contributions from each part. If the rotation axis is not through the center of mass, add md² (parallel axis theorem), where d is the distance between the two parallel axes. Reference: Halliday & Resnick Physics, Chapter 11.
How do I convert rpm to rad/s for the angular velocity input?
Multiply rpm by 2π/60: ω (rad/s) = rpm × 0.10472. Examples: 100 rpm = 10.47 rad/s; 1,000 rpm = 104.7 rad/s; 3,000 rpm = 314.2 rad/s; 6,000 rpm = 628.3 rad/s. Alternatively, if you know the frequency f in Hz (rotations per second), then ω = 2πf.
Do planets and stars follow L = I × ω?
Yes. Earth's rotation gives L ≈ 7.07 × 10³³ kg·m²/s; its orbital angular momentum around the Sun is ~2.66 × 10⁴⁰ kg·m²/s. Jupiter holds ~60% of the solar system's total angular momentum despite being only 0.1% of its mass. For a point mass in orbit, L = m·v·r (simpler form), while for a rotating extended body the full L = I·ω applies.
Why are angular momentum units the same as Planck's constant (J·s)?
kg·m²/s is dimensionally identical to J·s. This equivalence is fundamental in quantum mechanics, where angular momentum is quantized in units of ℏ = 1.054 × 10⁻³⁴ J·s. Electron orbital angular momentum takes values 0, ℏ, 2ℏ…; spin is ℏ/2. The classical formula L = I·ω is the large-scale limit where these discrete steps become imperceptible.
How are reaction wheels and gyroscopes in satellites related to angular momentum?
Reaction wheels store angular momentum: spinning a wheel up inside a spacecraft applies an equal-and-opposite torque to the spacecraft body, rotating it without expelling propellant. Typical I = 0.001–0.1 kg·m², ω up to 6,000+ rad/s, giving L = 6–600 N·m·s per wheel. The International Space Station's four Control Moment Gyroscopes each store L ≈ 4,760 N·m·s for precision attitude control. When wheels saturate, thrusters dump angular momentum back to space.
When does L = I × ω not apply?
This scalar formula applies to a rigid body rotating about a single fixed, symmetric axis. It fails for: (1) asymmetric rotation — use the full tensor form L⃗ = I̅·ω⃗; (2) deformable bodies (gymnast in mid-air changes I continuously); (3) systems at quantum scales, where L is quantized; (4) relativistic speeds, where special-relativistic corrections apply. For a single point particle, the alternative form is L = r × p = m·v·r·sin(θ).
What causes angular momentum to change (not be conserved)?
Angular momentum changes only when an external torque τ acts: τ = dL/dt. Real-world examples: friction at a bearing slows a flywheel; atmospheric drag and tidal forces gradually transfer Earth's rotational L to the Moon's orbital L (the Moon recedes ~3.8 cm/year as Earth's day lengthens ~2 ms/century); a brake pad applies torque to stop a wheel; a propeller torques both the aircraft and the air mass it accelerates.
How accurate is this calculator?
The formula L = I × ω is exact within classical mechanics — no approximations are introduced beyond your input precision. Results are accurate to the significant figures you provide. For engineering applications (machine design, aerospace), cross-check with primary sources: NIST physical constants (physics.nist.gov), machine design handbooks (Shigley's, Roark's), or vendor-supplied inertia data for specific components.
Is this calculator free to use?
Yes — completely free, no sign-up, no ads. The calculation runs entirely in your browser; no data is sent to any server. Works on desktop and mobile.