Angular Momentum Calculator
Solve the rotational-motion relation L = I · ω for any of its three unknowns — angular momentum, moment of inertia, or angular velocity — with step-by-step math, a live rotating-body schematic, and companion kinetic energy metrics.
It is conserved when torque is zero
In the absence of external torque, angular momentum is conserved — the rotational analog of linear momentum. This is why a figure skater spins faster when pulling her arms in: I drops, so ω must rise to keep L constant. It is one of the deepest conservation laws in physics, tied by Noether's theorem to the rotational symmetry of space.
I depends on how mass is distributed
Unlike mass in linear motion, the moment of inertia is not a fixed property of the body — it depends on the axis of rotation. A rod spun about its centre has I = mL²/12; the same rod spun about one end has I = mL²/3. This is why the axis must always be stated when quoting I, and why engineers can tune rotational behaviour by moving mass in or out from the axis.
Rotational KE goes as ω²
The energy stored in a spinning body is KE = ½Iω². Like linear kinetic energy, it grows with the square of the rate — so doubling the spin quadruples the energy. This is why high-speed flywheels are so dangerous when they fail: a 100 kg disc at 30,000 RPM carries more energy than a small car at highway speed.
Torque is the rate of change of L
Newton's second law for rotation is τ = dL/dt — the same relation as F = dp/dt for linear motion, just in rotational form. Applying a torque changes L, and the change is in the direction of the torque vector. This is how gyroscopes, reaction wheels on spacecraft, and bike wheels all maintain orientation: they trade angular momentum with their surroundings.
How to Use This Calculator
The angular momentum relation links three quantities: the moment of inertia I, the angular velocity ω, and the resulting angular momentum L. Pick which one you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick L (angular momentum from I and ω), I (moment of inertia from L and ω), or ω (angular velocity from L and I) — the tool rearranges the formula automatically.
2. Use realistic presets
Presets span the practical range: L from 0.1 kg·m²/s (small rotor) to 10,000 (industrial flywheel); I from 0.01 kg·m² (small disc) to 100 (large flywheel); ω from 1 rad/s (slow rotation) to 10,000 (high-speed turbine).
3. Match units carefully
L is in kg·m²/s, I in kg·m², and ω in rad/s. If you have RPM instead of rad/s, multiply by 2π/60 = 0.10472 to convert. The hint under each field shows the value in alternative units.
4. Read the result and verdict
The result card shows the solved value plus the equivalent lb·ft²/s and rotational KE. A colour-coded verdict classifies the rotation from small instrument through consumer device, industrial machinery, high-speed rotor, and turbo machinery.
5. Check the schematic and charts
The schematic shows a rotating disc with the angular velocity arrow around it and the angular momentum vector along the axis. The chart plots L against ω for three moments of inertia.
6. Switch to Advanced for depth
Advanced mode adds an L-vs-I chart, a 1% sensitivity check on each input, and companion metrics including rotational KE, RPM, the radius of gyration, and the equivalent linear momentum at the rim.
What This Tool Offers
Built around the classic rotational relation L = I · ω, the fundamental equation behind every flywheel design, gyroscope, and spacecraft attitude control system.
Three-way solving
Solve for angular momentum, moment of inertia, or angular velocity. The calculator rearranges the formula for whichever variable you choose.
Live rotating-body schematic
A rotating disc with the angular velocity arrow traced around it and the angular momentum vector drawn along the axis, both scaled and labelled with current values.
Rotational scale verdict
Every result is classified from small instrument through consumer device, industrial machinery, high-speed rotor, turbo-machinery, up to extreme-scale flywheel.
Multi-unit hints
Each field shows its value in related units — lb·ft²/s for angular momentum, kg·cm² for inertia, and RPM for angular velocity.
Dual comparison charts
L vs ω for three moments of inertia, plus an advanced L vs I chart for three angular velocities.
Reference tables
Moment of inertia formulas for common shapes, typical values for real objects, unit conversions, and a full variable reference table.
Reference Tables & Conversions
A quick rotational estimator, unit converter, and reference tables for common shapes and real objects.
Quick angular momentum estimator
Unit Conversions
| Quantity | Unit | Equivalent in SI |
|---|---|---|
| Angular velocity | 1 RPM | 0.10472 rad/s |
| Angular velocity | 1 rad/s | 9.5493 RPM |
| Angular momentum | 1 lb·ft²/s | 0.04214 kg·m²/s |
| Angular momentum | 1 kg·m²/s | 23.730 lb·ft²/s |
| Moment of inertia | 1 kg·cm² | 0.0001 kg·m² |
| Moment of inertia | 1 lb·ft² | 0.04214 kg·m² |
| Torque | 1 N·m | 1 kg·m²/s² |
Moment of Inertia Formulas for Common Shapes
I depends on both the shape and the axis of rotation. Below, m is the total mass and R is the relevant radius or length.
| Shape (axis) | Formula | Notes |
|---|---|---|
| Point mass at radius R | I = mR² | The defining case — all mass concentrated at one radius |
| Hoop / thin ring (central axis) | I = mR² | All mass at the rim; maximum I for a given m and R |
| Solid disc / cylinder (central axis) | I = ½mR² | Uniform density; the classic flywheel geometry |
| Hollow cylinder (thin wall) | I = mR² | Same as hoop for a thin shell |
| Solid sphere (central axis) | I = ⅖mR² | Uniform density; used for planets and ball bearings |
| Hollow sphere (thin shell) | I = ⅔mR² | Same geometry as a soap bubble or thin metal ball |
| Thin rod (centre) | I = mL²/12 | Axis perpendicular to the rod, through the midpoint |
| Thin rod (end) | I = mL²/3 | Axis perpendicular to the rod, through one end |
| Rectangular plate (centre, perpendicular) | I = m(a²+b²)/12 | a and b are the side lengths |
Typical Angular Momentum Values for Real Objects
Approximate values. Actual values depend on geometry, mass distribution, and rotational speed.
| Object | I (kg·m²) | ω (rad/s) | L (kg·m²/s) | Notes |
|---|---|---|---|---|
| Hard drive platter (spinning) | ≈ 1×10⁻⁵ | ≈ 540 | ≈ 5.4 × 10⁻³ | 7,200 RPM, 3.5" drive |
| Bicycle wheel (upright) | ≈ 0.12 | ≈ 15 | ≈ 1.8 | At 25 km/h |
| Car wheel (tyre + rim) | ≈ 1.2 | ≈ 30 | ≈ 36 | At 100 km/h |
| Figure skater (spin) | ≈ 4 | ≈ 15 | ≈ 60 | Arms out; drops when arms pulled in |
| Industrial flywheel | ≈ 100 | ≈ 50 | ≈ 5,000 | Energy storage flywheel |
| Car engine crankshaft | ≈ 0.15 | ≈ 600 | ≈ 90 | At 6,000 RPM |
| Jet engine turbine | ≈ 15 | ≈ 1,500 | ≈ 22,500 | High-pressure spool at cruise |
| Earth (daily rotation) | ≈ 8 × 10³⁷ | ≈ 7.3 × 10⁻⁵ | ≈ 5.9 × 10³³ | About its polar axis |
| Earth (orbital) | ≈ 1.3 × 10⁴⁷ | ≈ 2 × 10⁻⁷ | ≈ 2.7 × 10⁴⁰ | About the Sun |
Rotational Scale Bands
| L (kg·m²/s) | Scale | Typical context |
|---|---|---|
| < 1 | Small instrument | Hard drives, precision gyroscopes, small rotors |
| 1 – 100 | Consumer device | Bicycle wheels, fans, small motors, figure skaters |
| 100 – 10,000 | Industrial machinery | Car wheels, flywheels, machine tools, turbines |
| 10,000 – 1,000,000 | High-speed rotor | Jet turbines, industrial compressors, energy-storage flywheels |
| > 1,000,000 | Astronomical | Planets, stars, galaxies, black hole accretion discs |
What Does L = I · ω Measure?
Angular momentum is the rotational analogue of linear momentum. Just as a moving object carries momentum p = mv, a spinning body carries angular momentum L = Iω. It quantifies how hard it is to stop the rotation, and it obeys its own conservation law: in the absence of external torque, the total angular momentum of a system stays constant. This is what makes a spinning top stand up, why a bicycle stays upright when moving, and why a reaction wheel can turn a satellite in orbit.
The Angular Momentum Formula
The defining relationship is:
Rearranged to solve for the other two variables:
ω = L / I
In full vector form, angular momentum is L = I · ω (vectors), and Newton's second law for rotation becomes τ = dL/dt. The magnitude relation L = Iω is what this calculator solves.
Angular Momentum Variables
| Symbol | Meaning | SI unit |
|---|---|---|
| L | Angular momentum | kg·m²/s (or J·s) |
| I | Moment of inertia | kg·m² |
| ω | Angular velocity | rad/s |
| KErot | Rotational kinetic energy = ½Iω² | J (joule) |
| τ | Torque = dL/dt | N·m |
| kg | Radius of gyration = √(I/m) | m |
Why Moment of Inertia Is Not Just Mass
In linear motion, mass alone determines how hard it is to accelerate a body: F = ma. In rotational motion, mass is not enough — how the mass is arranged around the axis matters enormously. Two discs of identical mass behave very differently if one is a solid plate and the other is a thin ring: the ring has I = mR² while the plate has I = ½mR², so the ring is twice as hard to spin up. This is why flywheels put most of their mass at the rim, and why a figure skater pulls her arms in to spin faster — moving mass closer to the axis drops I, which raises ω to keep L constant.
How Each Variable Affects the Result
Angular momentum responds directly and proportionally to both inputs. Doubling the moment of inertia doubles L at the same spin rate; doubling the angular velocity doubles L at the same geometry. Neither effect interacts with the other — the relationship is perfectly linear, exactly like F = ma or F = kx. This is why rotational problems are often simpler than they first appear: the algebra is the same, only the symbols and units change.
Worked Examples
L = 0.400 × 30.0
L = 12.0 kg·m²/s
I = 60 ÷ 15
I = 4.0 kg·m²
ω = 12 ÷ 0.4
ω = 30 rad/s = 286.5 RPM
I = ½mR² = 0.5 × 10 × 0.30² = 0.45 kg·m²
L = Iω = 0.45 × 50 = 22.5 kg·m²/s
Frequently Asked Questions
It is the rotational analog of linear momentum. Just as p = mv tells you how much translational momentum a body carries, L = Iω tells you how much rotational momentum it carries. The units are kg·m²/s (equivalently J·s). The larger L is, the harder it is to stop the rotation.
Because space has rotational symmetry — physics looks the same regardless of which direction you face. By Noether's theorem, every continuous symmetry of nature corresponds to a conserved quantity, and rotational symmetry gives conservation of angular momentum. In practical terms: if no external torque acts on a system, L cannot change. A figure skater pulling in her arms, a collapsing star, and a spinning bicycle wheel all obey this rule.
Mass is a scalar property of a body and does not depend on where you measure it from. Moment of inertia is a tensor property — it depends on the axis of rotation and on how the mass is distributed around that axis. A rod spun about its centre has I = mL²/12, but spun about one end it has I = mL²/3 — four times larger, from the same mass and length. This is why flywheels put their mass at the rim, and why a tightrope walker uses a long pole for balance.
Torque is the rate of change of angular momentum: τ = dL/dt. This is exactly analogous to F = dp/dt for linear motion. A torque applied for a duration Δt changes L by τ·Δt — the angular impulse. This is how reaction wheels on spacecraft turn a satellite: they spin up, changing their own L, and the spacecraft responds by rotating in the opposite direction to keep the total L of the system conserved.
Multiply RPM by 2π/60 = 0.10472. So 3,000 RPM = 314 rad/s, and 1 rad/s = 9.549 RPM. Angular velocity is almost always quoted in rad/s in physics and engineering calculations because that makes the formula L = Iω dimensionally consistent without any conversion factors — the radian is a dimensionless unit by design.
The radius of gyration k is defined by I = mk², so k = √(I/m). It is the distance from the axis at which all the mass could be concentrated (as a point mass or thin hoop) to give the same moment of inertia as the real body. It gives an intuitive feel for how far the mass sits from the axis, and it is used in structural engineering to characterise column buckling as well as in rotational dynamics.
The magnitude is never negative, but the vector can point either way along the axis — conventionally one direction is "positive" and the opposite is "negative". This is why two wheels spinning in opposite directions on the same shaft have L vectors that partially cancel. Sign matters in conservation problems: total L is conserved as a vector sum, so opposite spins reduce the total.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| L | Angular momentum | kg.m2/s |
| I | Mass moment of inertia about the rotation axis | kg.m2 |
| omega | Angular velocity | rad/s |
What it means
Angular momentum (L) is a measure of the rotational motion of a body. For a rigid body rotating about a fixed axis, L = I ω, where I is the moment of inertia about that axis and ω is the angular velocity. Angular momentum is a vector, with direction given by the right‑hand rule. The law of conservation of angular momentum states that if no external torque acts on a system, its total angular momentum remains constant. This principle explains phenomena like the spinning of figure skaters (they spin faster when pulling in arms) and the stability of gyroscopes. In orbital mechanics, it governs the motion of planets. In engineering, angular momentum is important in designing flywheels, gyroscopes, and propulsion systems. The unit is kg·m²/s. The concept extends to quantum mechanics, where it is quantised. Understanding angular momentum is crucial for analyzing rotating machinery and for control systems in aerospace.
Worked example
Angular Momentum – Two Examples
Real‑World| Parameter | Value |
|---|---|
| I | 5 kg·m² |
| ω | 10 rad/s |
| Parameter | Value |
|---|---|
| I | 2 kg·m² |
| ω | 20 rad/s |
Common mistakes
- Moment of inertia I: Use the correct moment of inertia about the axis of rotation.
- Angular velocity ω: In rad/s – not rpm.
- Units: I in kg·m², ω in rad/s → L in kg·m²/s.
- Vector nature: Angular momentum is a vector (axial) – direction matters.
- Conservation: Angular momentum is conserved only if no external torque acts.
Applications
Angular momentum is the rotational analogue of linear momentum, given by the product of moment of inertia and angular velocity. It is a conserved quantity in isolated systems, making it crucial in many engineering disciplines. In aerospace, angular momentum is used to control satellite attitude through reaction wheels. In mechanical systems, it governs the behaviour of gyroscopes, which are used in navigation and stabilisation. The concept also applies to the design of spinning objects like tops, wheels, and turbine rotors, where stability and control depend on angular momentum. In astrophysics, it explains the rotation of celestial bodies. Engineers use angular momentum to design systems that require precise rotational control or energy storage.
- Satellite attitude control systems
- Gyroscope design for navigation and stabilisation
- Flywheel energy storage systems
- Design of spinning machinery (turbines, compressors)
- Astronomical and planetary motion analysis
Related calculators · Dynamics
Recently updated formulas
Suggest an improvement
Found a mistake? Have a better example? Let us know.
You may send up to 3 suggestions per 24 hours.