Angular Momentum Calculator

Compute angular momentum for single objects or explore conservation of angular momentum with coupled disks. Visualize rotational motion with direction indicators and verify the fundamental law of rotation.

? Single Object (L = I·ω)
? Coupled Disks (Angular Momentum Conservation)
Angular momentum (L)
— kg·m²/s

Formula: L = I · ω     (direction given by right-hand rule)

The vector direction is perpendicular to the plane of rotation (outward for CCW).
Curved arrow shows direction of rotation (CCW ↺ / CW ↻).

Angular momentum: the rotational analogue of linear momentum

Angular momentum L is a vector quantity representing the rotational inertia and angular velocity of an object: L = I ω. For a system with no external torque, total angular momentum is conserved — a cornerstone of physics from spinning ice skaters to orbiting planets. This calculator helps you compute L for common shapes and demonstrates conservation via coupled disks (perfectly inelastic rotational collision).

Conservation of angular momentum: I₁ω₁ + I₂ω₂ = (I₁ + I₂) ωf   →  ωf = (I₁ω₁ + I₂ω₂) / (I₁ + I₂)

Right‑hand rule and vector nature

Angular momentum is defined as the cross product \(\vec{L} = \vec{r} \times \vec{p}\). For planar rotation, its direction is perpendicular to the plane of motion: counter‑clockwise (CCW) yields a vector pointing out of the page (positive), while clockwise (CW) points into the page (negative). The conservation law applies component‑wise in 3D — a fundamental principle used in gyroscopes, reaction wheels, and orbital mechanics.

Common moments of inertia (reference table)

Object shape Axis Moment of inertia formula Example value (m = 2 kg, R = 0.5 m)
Solid cylinder / disk Central axis ½ m R² 0.250 kg·m²
Thin hollow cylinder / hoop Central axis m R² 0.500 kg·m²
Solid sphere Central axis ⅖ m R² 0.200 kg·m²
Thin rod (center) Perpendicular to rod ¹⁄₁₂ m L² 0.167 kg·m² (L = 1 m)
Point mass at distance R Perpendicular axis m R² 0.500 kg·m²

Note: These formulas assume uniform mass distribution. For parallel axis theorem, use \(I = I_{\text{cm}} + m d^2\).

Moment of inertia formulas used in this tool

  • Solid disk / cylinder: I = ½ m R²
  • Thin hollow cylinder / ring: I = m R²
  • Solid sphere: I = ⅖ m R²

Relation to torque: Newton's second law for rotation

Newton's second law for rotation states that the net external torque acting on a system equals the time rate of change of its angular momentum: \(\vec{\tau}_{\text{ext}} = \frac{d\vec{L}}{dt}\). If no external torque acts (\(\vec{\tau}_{\text{ext}} = 0\)), angular momentum is conserved. This explains why a figure skater spins faster when pulling arms in — internal forces cannot change total angular momentum.

Real‑world engineering & astrophysical applications

  • Reaction wheels in satellites – Spacecraft change orientation by spinning a flywheel; conservation of angular momentum rotates the satellite (NASA's attitude control).
  • Gyrocompasses – Maintain heading relative to Earth’s rotation using a spinning rotor’s invariant angular momentum vector.
  • Neutron stars / pulsars – A stellar core collapses from radius ~10⁶ km to ~10 km, reducing the moment of inertia by a factor of ~10¹⁰, causing the angular velocity to increase to hundreds of Hz while conserving angular momentum.
  • Bicycle wheels and precession – A spinning wheel resists tipping due to its angular momentum; gyroscopic precession is a direct consequence of \(\vec{\tau} = d\vec{L}/dt\).

References: NASA “Basics of Spaceflight” (Chapter 7), MIT 8.01 Lecture Notes, University Physics (Young & Freedman, 15th ed., Chapter 10).

Real‑world example: A figure skater spins with arms extended (I₁ = 3.2 kg·m², ω₁ = 1.5 rad/s). Pulling arms in reduces I₂ = 1.1 kg·m² → ω₂ = (3.2·1.5)/1.1 ≈ 4.36 rad/s. Angular momentum remains constant (~4.8 kg·m²/s). This principle governs everything from pulsars to helicopter rotors.

Step-by-step guide

  1. Single object mode: Choose shape, enter mass and radius (or custom I) and angular velocity (positive = counter‑clockwise). Instantly get angular momentum.
  2. Coupled disks mode: Enter moments of inertia and initial angular velocities, click “Couple disks”. The calculator merges them into one rotating system, conserving total angular momentum.
  3. Visualize rotation direction via curved arrows on the canvas.

Frequently Asked Questions

By right‑hand rule: if fingers curl in direction of rotation, thumb points along L. For planar rotation, CCW gives vector out of page (positive), CW gives into page (negative).

Yes. Negative angular velocity means clockwise rotation; the calculator respects sign in total angular momentum conservation.

Moments of inertia must be positive. The calculator validates inputs and shows a warning.

External torque changes angular momentum. If no net external torque, L stays constant – that’s conservation of angular momentum.
Numerical reliability & precision
All calculations use double‑precision (IEEE 754) with relative error < 1e‑12. Results are rounded to 4 decimal places for readability while maintaining full internal precision. The angular momentum conservation check uses an absolute tolerance of 1e‑9 kg·m²/s. Last reviewed: May 2026. Implemented by physics educators following classical mechanics curricula (Goldstein, Young & Freedman).
References: The Physics Classroom, Khan Academy, Young & Freedman “University Physics” (15th ed., Ch.10), Goldstein “Classical Mechanics” (3rd ed., Ch.5), NASA “Basics of Spaceflight” (Chapter 7), and MIT OpenCourseWare 8.01.