Angular Momentum Calculator

Pull your arms in while spinning on a swivel chair and you instantly speed up — that's conservation of angular momentum in action. The calculator for angular momentum computes L from moment of inertia I and angular velocity ω, and also calculates the moment of inertia for common geometric shapes when given mass and dimensions. Angular momentum is one of the most conserved quantities in physics: in the absence of external torque, it never changes.

The Angular Momentum Formula

Angular momentum L (kg·m²/s) is defined as:

L = I × ω

where I is the moment of inertia (kg·m²) and ω is the angular velocity (rad/s). For a point mass m at distance r from the rotation axis: L = m × v × r = m × ω × r², which gives I = mr² for a point mass. The direction of L follows the right-hand rule — aligned with the rotation axis, pointing in the direction your thumb points when your fingers curl with the rotation. For composite objects, moments of inertia add: I_total = Σ(mᵢrᵢ²) for discrete masses, or ∫r²dm for continuous distributions. The angular velocity calculator converts between RPM, rad/s, and degrees/s when ω is given in non-SI units.

Conservation of Angular Momentum: Real-World Consequences

When net external torque is zero, L = constant. This single principle explains a surprising range of phenomena:

• Figure skater spin: pulling arms inward reduces I; since L = Iω = constant, ω must increase proportionally — a skater going from arms extended (I ≈ 4.5 kg·m²) to arms tucked (I ≈ 1.5 kg·m²) triples their spin rate
• Planetary orbits: Kepler's second law (equal areas in equal times) is a direct consequence of angular momentum conservation for a planet under a central gravitational force
• Neutron star pulsars: a stellar core collapsing from ~10⁸ m radius to ~10⁴ m (neutron star) reduces I by ~10⁸, increasing rotation rate from days to milliseconds
• Gyroscope stability: a spinning gyroscope resists changes to its rotation axis because a torque produces precession (rotation of the angular momentum vector) rather than changing its magnitude

Moment of Inertia for Common Shapes

I depends on both mass and how that mass is distributed relative to the rotation axis:

• Solid cylinder or disc (axis through center): I = ½MR²
• Hollow cylinder (thin-walled tube): I = MR²
• Solid sphere: I = ²⁄₅MR²
• Thin rod (axis through center): I = ¹⁄₁₂ML²
• Thin rod (axis through end): I = ¹⁄₃ML²

The parallel axis theorem extends these: I_off = I_cm + Md², where d is the distance from the center of mass to the new axis. The torque calculator and rotational motion calculators complete the rotational dynamics toolkit.

Angular Momentum in Quantum Mechanics

At the atomic scale, angular momentum is quantized — it can only take discrete values that are integer or half-integer multiples of ℏ (the reduced Planck constant, 1.055 × 10⁻³⁴ J·s). Electron orbital angular momentum has magnitude √(l(l+1))ℏ where l is the orbital quantum number. Electron spin is an intrinsic angular momentum with no classical analog — spin-½ particles have angular momentum ½ℏ. The addition of orbital and spin angular momenta (spin-orbit coupling) determines atomic energy levels and spectral line splitting, making angular momentum central to all of atomic and molecular spectroscopy.