Angular Momentum Calculator

Definition of angular momentum: from a point to a body

The angular momentum of a point mass about a chosen reference point is the cross product of its position vector and its momentum: L = r × p = r × m·v. Its magnitude is r·m·v·sin φ, where φ is the angle between r and v; for circular motion φ = 90°, so L = m·v·r = m·r²·ω. The vector L is perpendicular to the plane of motion and points along the axis of rotation according to the right-hand rule.

For a rigid body we add the angular momenta of all its points. When the body spins about a symmetry axis, every point contributes mᵢrᵢ²ω in the same direction, and the sum gives L = (Σmᵢrᵢ²)·ω = I·ω. The unit kg·m²/s is identical to J·s — the same unit in which the Planck constant h = 6.626·10⁻³⁴ J·s, the elementary quantum of angular momentum in atomic physics, is expressed.

Historical note: Kepler, Euler, Rankine, and Noether

The conservation of angular momentum is already hidden in Kepler’s second law (1609): the radius vector of a planet sweeps out equal areas in equal times, which is equivalent to the constancy of L = m·r²·ω along the orbit. Newton derived this law in the Principia (1687) for any central force. Leonhard Euler, between 1750 and 1765, formulated the general equations of rigid-body motion, in which the time derivative of angular momentum equals the torque, and introduced the moment of inertia.

The term moment of momentum was popularized by William Rankine in his Manual of Applied Mechanics (1858); the phrase angular momentum appeared somewhat later. A deeper justification of the conservation law was given by Emmy Noether in 1918: every continuous symmetry of the laws of physics corresponds to a conserved quantity, and the conservation of angular momentum follows from the invariance of the laws under rotations of space.

Deriving L = I·ω and the link to energy

Step 1. A point of mass mᵢ at distance rᵢ from the axis moves at speed vᵢ = ω·rᵢ, perpendicular to the radius. Its angular momentum about the axis is Lᵢ = rᵢ·mᵢ·vᵢ = mᵢ·rᵢ²·ω. Step 2. Sum over all points of the body: L = Σmᵢrᵢ²·ω = ω·Σmᵢrᵢ². The sum Σmᵢrᵢ² is the moment of inertia I, so L = I·ω.

Step 3. The rotational kinetic energy is E = ½Iω²; substituting ω = L/I gives E = L²/(2I) — at constant L the energy grows as I shrinks. Step 4. For I = 2 kg·m² and ω = 5 rad/s: L = 10 kg·m²/s, E = ½·2·25 = 25 J. Step 5. Newton’s second law for rotation, M = dL/dt, says that stopping this body in 1 s needs an average torque of 10 N·m, and in 0.1 s, 100 N·m.

Conservation of angular momentum

If the net external torque on a system is zero, its total angular momentum stays constant: I₁ω₁ = I₂ω₂. Hence the skater’s spin: pulling in the arms reduces I from about 3 to 1 kg·m², so ω triples, from 12.6 to 37.7 rad/s. The same mechanism speeds up a diver in the tuck position, a neutron star after the collapse of a stellar core (rotation periods of milliseconds), and a falling cat, which reorients itself without any external torque by counter-rotating the front and rear of its body.

Conservation of angular momentum also explains the stability of a gyroscope and of a bicycle wheel: the vector L tends to keep its direction in space, and an external torque — from the weight, for example — does not topple the axis but makes it precess at angular speed Ω = M/L. In the Earth–Moon system the tides transfer angular momentum from the Earth’s spin to the Moon’s orbit: the day lengthens by about 2 ms per century and the Moon recedes by 3.8 cm per year, while the sum of the two angular momenta stays constant.

Assumptions: principal axis, rigid body, L versus E

The scalar formula L = I·ω is exact when the body spins about one of its principal axes — for example the symmetry axis of a wheel, a cylinder, or a sphere. For rotation about an arbitrary axis the angular momentum is the vector L = ηω, where Î is the inertia tensor, and in general it is not parallel to ω; this shows up as the vibration of an unbalanced wheel and the need to balance rotors. The calculator assumes a principal axis and a rigid body with constant I.

It pays to distinguish angular momentum from rotational energy. Both depend on I and ω, but L = I·ω is linear in ω, has the unit kg·m²/s, and is a vector, while E = ½Iω² is quadratic, has the unit joule, and is a scalar. Two wheels with the same L can have different energies: at fixed L the energy is L²/(2I), so the wheel with the smaller I has more energy. The result panel shows both quantities side by side so that the difference is visible.

Units, orders of magnitude, and applications

The SI unit of angular momentum is kg·m²/s; J·s and N·m·s are dimensionally identical, and US units use lb·ft²/s (1 kg·m²/s = 23.73 lb·ft²/s). The calculator shows kg·m²/s and J·s, and after the unit switch lb·ft²/s. Orders of magnitude: a hard-drive platter about 0.02 kg·m²/s, a bicycle wheel 2–3 kg·m²/s, a skater in a spin 30–40 kg·m²/s, a helicopter rotor of the order of 10⁵ kg·m²/s, the Earth about its own axis 5.9·10³³ kg·m²/s, the Moon in its orbit 2.9·10³⁴ kg·m²/s — five times more than the Earth’s spin.

In engineering the angular momentum of a rotor governs navigation gyroscopes, reaction wheels in satellites (changing their ω turns the satellite without propellant), and the reaction torque in helicopters, which the tail rotor counteracts. In atomic physics angular momentum is quantized in units of ħ = h/2π = 1.055·10⁻³⁴ J·s — the same kg·m²/s this calculator displays, only thirty-four orders of magnitude smaller than in the bicycle-wheel example.

Sources

The definition, derivation, and applications come from mechanics textbooks and historical sources: D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, chapter on rolling, torque, and angular momentum; H. D. Young, R. A. Freedman, University Physics, chapter “Dynamics of Rotational Motion” (angular momentum, conservation, gyroscopes); R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. I, chapters 18–20 (rotation, angular momentum, the gyroscope); L. Euler, Theoria motus corporum solidorum seu rigidorum, 1765; W. J. M. Rankine, A Manual of Applied Mechanics, 1858; E. Noether, “Invariante Variationsprobleme,” Nachrichten der Gesellschaft der Wissenschaften zu Göttingen, 1918; J. Kepler, Astronomia nova, 1609; BIPM, SI Brochure, 9th edition. This page is for educational purposes.