This angular momentum calculator computes L two ways — for a rigid body, L = Iω (moment of inertia × angular velocity), and for a point particle, L = mvr (perpendicular case) — and includes the conservation solver behind the figure-skater effect: when I changes and no external torque acts, I₁ω₁ = I₂ω₂ decides the new spin.

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The Quantity That Refuses to Change

Angular momentum is conserved whenever net external torque is zero — and that single sentence runs an astonishing range of phenomena: the skater’s spin-up (arms in, I falls, ω must rise so Iω holds), neutron stars collapsing into millisecond pulsars, gyrocompasses holding north, the Moon’s recession as Earth’s spin bleeds L into the lunar orbit, and Kepler’s equal-areas law (which is L conservation for a planet, stated geometrically 250 years before the algebra). The defaults put the rigid-body arithmetic on the table: our Batch-72 disk (I = 0.50) spinning at 10 rad/s carries L = 5.0 kg·m²/s, and squeezing its I to 0.25 (conservation mode) spins it to 20 rad/s — doubled, exactly as I halved.

The fine print belongs on the same page: L is a vector (direction along the axis, right-hand rule), τ_net = dL/dt is its Newton law, and “conserved” always means “in the absence of external torque about that axis” — internal rearrangements (the skater) never touch L, but a friction torque at the axle drains it steadily, which is why real flywheels spin down and why the conservation mode asks what CHANGED inside the system.

Worked Examples (Verified)

  • Rigid body (defaults): L = 0.5 × 10 = 5.0 kg·m²/s.
  • Point particle: 2 kg at 3 m/s on a 1.5 m radius → L = 2 × 3 × 1.5 = 9.0 kg·m²/s — a stone on a string carries more spin momentum than the spinning disk above, which surprises until the r²-vs-r weighting settles in.
  • Skater arithmetic: I₁ = 6 kg·m² at ω₁ = 2 rad/s (0.32 rev/s) → L = 12; tucked to I₂ = 2 → ω₂ = 6 rad/s (0.95 rev/s). A 3× speed-up for a 3× I-drop, exact.

Angular Momentum Discipline

  • ω in rad/s — convert rev/s × 2π and RPM × 2π/60 before multiplying; unit slips here are the genre’s most common error (the conservation output shows rev/s to make the magnitude legible).
  • Point-particle L depends on the reference point (origin); rigid-body L about the COM axis is the standard convention — mixing origins invalidates comparisons and conservation bookkeeping.
  • Perpendicular only for mvr: the general form carries sinθ between r and v; a particle moving radially toward the origin has L = 0 about it despite moving fast.
  • Conservation needs NO EXTERNAL torque about the axis — gravity through the COM is fine (no lever arm), axle friction isn’t; state the assumption when you invoke the law.
  • KE is not along for the ride: internal work changes rotational KE during I-changes (KE scales with I at fixed L: KE = L²/2I) — energy and momentum bookkeeping are separate ledgers.

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Frequently Asked Questions

How do you calculate angular momentum?

For a rigid body: L = Iω. For a point particle: L = mvr·sinθ (mvr when the motion is perpendicular to the radius). Defaults here: the rigid body 0.5 kg·m² × 10 rad/s = 5.0 kg·m²/s; the point particle 2 × 3 × 1.5 = 9.0 kg·m²/s. This calculator handles both forms plus the conservation solver I₁ω₁ = I₂ω₂.

Why does a figure skater spin faster with arms tucked?

Conservation of angular momentum. On (nearly) frictionless ice the external torque about the vertical axis is tiny, so L = Iω is fixed by the entry spin. Tucking the arms moves mass closer to the axis, cutting I — our example from 6 to 2 kg·m² — and ω must triple to keep the product constant. The extra rotational KE comes from the work the skater’s muscles do pulling the arms in against centrifugal effect.

What are the units of angular momentum?

kg·m²/s (equivalently J·s — the same unit as Planck’s constant, which is no accident: angular momentum is quantized in units of ħ in quantum mechanics). Check the two formulas agree: Iω gives (kg·m²)(1/s); mvr gives kg·(m/s)·m — same animal.

How is angular momentum different from linear momentum?

Linear p = mv is conserved absent net force; angular L = Iω (or r × p) is conserved absent net torque — about a chosen axis/point. A planet conserves both (in the two-body idealization), but systems routinely conserve one and not the other: a ball on a string shortened by pulling through a hole keeps L about the center while its linear momentum’s direction and magnitude both change. The torque/force pairing mirrors the L/p pairing exactly.

Does angular momentum conservation explain Kepler’s second law?

Yes — it’s the same theorem. Gravity is a central force (directed along r), so its torque about the Sun is zero; the planet’s L is constant. The area swept per unit time is dA/dt = L/(2m) — constant — which IS “equal areas in equal times.” The planet speeds up at perihelion because r shrinks and v must grow to hold mvr constant: orbital mechanics’ most elegant identity, and a fine check on any simulation.

Can angular momentum change without external torque?

The total, no — that’s the law. But it can move between parts of a system (Earth’s spin L transferring to the Moon’s orbital L via tides, slowing our day by ~2.3 ms/century while the Moon recedes 3.8 cm/yr) and between spin and orbital forms. Internal rearrangement is also how a falling cat reorients with L = 0 throughout — changing shape, not total. Conservation constrains the sum, not the distribution.

What is the torque–angular momentum relation?

τ_net = dL/dt — Newton’s second law in rotational form. Constant torque changes L linearly (a brake at fixed friction torque spins a wheel down on a predictable schedule); impulse form Δ L = τ·Δt mirrors linear impulse-momentum. It’s the equation that tells you how long a gyroscope battery of tricks can last against bearing friction.

Is this angular momentum calculator free?

Yes — free, instant, no sign-up. Rigid-body, point-particle and conservation modes in one tool, with the KE caveat printed where it belongs.