Angular momentum

In brief

A figure skater spinning with arms outstretched pulls them tight against her body, and without any external push, her spin rate leaps upward, sometimes doubling or more. Nothing outside her has touched her to speed her up. Something about her rotation is being conserved, and pulling her mass inward trades one part of that conserved quantity for another: rotational inertia falls, so rotation rate must rise to compensate. That conserved quantity is angular momentum, the rotational counterpart of ordinary momentum, and its conservation, whenever no outside torque acts, governs everything from a spinning skater to a diver's twist to the changing speed of a planet along its orbit.

The full treatment

First look: the skater and the spinning stool

Sit on a freely spinning stool holding two weights at arm's length, spinning slowly. Pull the weights in to your chest and, with no other effort, you spin noticeably faster. Push the weights back out and you slow down again, all without touching the floor or anything else to push against. This cannot be explained by torque in the ordinary sense, since nothing external is twisting you: the torque from your own arm muscles pulling weights straight toward your body is close to zero, because a force directed straight toward the axis has no lever arm and produces no torque about it, exactly the zero-torque case from the torque entry. Yet your rotation rate changes substantially. Something is being held fixed while your rotational inertia visibly changes as the weights move inward, and rotation rate compensates to keep that something constant.

Building the idea: momentum, extended to rotation

Ordinary momentum, mass times velocity, measures quantity of straight-line motion and is conserved when no outside force acts. Rotational inertia is the rotational analog of mass, capturing how a body's mass is distributed relative to a spin axis. It is natural to ask whether a rotational analog of momentum exists, built from rotational inertia the way ordinary momentum is built from mass. Define angular momentum, written L, as rotational inertia times angular velocity: L = I times omega, where I is the body's rotational inertia about the chosen axis and omega is its angular velocity, how fast its angle changes, in radians per second. The units of L are kilogram meters squared per second.

To see why this combination is the right one to track, return to the rotational law: torque equals rotational inertia times angular acceleration, tau = I * alpha. Angular acceleration is the rate of change of angular velocity, so for a body whose rotational inertia stays fixed, tau = I times the rate of change of omega, the same as saying tau equals the rate of change of L. This holds even when I itself changes, provided the bookkeeping is done through L directly: net external torque equals the rate at which angular momentum changes, the direct rotational counterpart of net force equals the rate of change of ordinary momentum.

Conservation: what happens when torque is zero

The immediate and powerful consequence: if net external torque on a system, about a given axis, is zero, then angular momentum about that axis does not change over time. It is conserved, following directly from tau equals the rate of change of L: no torque means no change.

Return to the skater. Pulling weights straight toward the spin axis exerts, on the weights, a force directed along the radius, which by the torque relation, force times lever arm times sine of the angle between them, produces zero torque, since a force pointed straight at the axis makes an angle of zero with the line to the axis. With no external torque about that axis, angular momentum L = I * omega must stay the same before and after the weights move. Since I drops sharply as the weights come inward, and rotational inertia depends on mass times distance squared, omega must rise to keep the product I * omega fixed. This is the entire mechanism: not a mysterious boost of speed, but a fixed product redistributing itself between a shrinking I and a growing omega.

Formalizing the general case

For a single point mass moving with velocity, angular momentum about a chosen point can also be defined as mass times velocity times the perpendicular distance from that point to the line of the velocity, mirroring the force-times-perpendicular-distance structure of torque; for a rigid body rotating about a fixed axis this reduces exactly to I times omega. Conservation of angular momentum applies equally whether a body is rigid and simply changes shape internally, as with the skater, or is a collection of separate objects, such as two masses orbiting each other, exchanging angular momentum internally while the total is held fixed. This is why a planet moving along an elliptical orbit under the sun's gravity, a force always directed along the line to the sun and so producing zero torque about it, sweeps out its path faster near the sun and slower far away: its rotational inertia about the sun shrinks as it approaches, so its effective angular velocity must rise to keep angular momentum constant, precisely as with the skater's arms.

Lineage

The conservation of angular momentum emerged from Newtonian mechanics as a direct consequence of applying force and torque laws to systems of particles, though its most famous early observational trace predates the mechanics that explained it: Johannes Kepler's empirical law that a planet sweeps out equal areas in equal times is exactly a statement of angular momentum conservation, discovered from astronomical data decades before Newton supplied the underlying force law. The formal proof that zero net torque implies constant angular momentum, and the recognition of angular momentum as a conserved quantity on the same footing as ordinary momentum and energy, followed from the systematic rigid-body and orbital mechanics developed by Euler, Lagrange, and their successors.

The strongest case for it

Angular momentum conservation is trusted because it follows as a strict mathematical consequence of Newton's laws applied to any system where net external torque vanishes, with no separate assumption required. Its reach is extraordinary: it explains the skater's spin, the diver's twist, the stability of a spinning gyroscope and a thrown football, the equal-area law of planetary orbits, and a spinning top's fixed axis orientation absent disturbing torques. It survives, in modified form, even in the deepest revisions of physics; relativity and quantum mechanics both retain a conservation law for angular momentum, reinterpreted but never discarded, a strong sign that the underlying symmetry it reflects, the rotational symmetry of space itself, is genuinely fundamental. Every gyroscope, reaction wheel, and orbiting satellite engineered to date behaves exactly as this law predicts.

The strongest case against it

The conservation law applies strictly only when net external torque about the chosen axis is zero; real systems nearly always experience some external torque, from friction or air resistance, so angular momentum measured over a real skater or top slowly declines, and the claim of conservation is an idealization valid over the timescale where such torques are negligible. A second limit concerns the axis: conservation is a statement about a specific axis, and a system can conserve angular momentum about one axis while experiencing changing angular momentum about another if torques about that second axis do not vanish. A common misconception is believing the skater's spin speeds up because new energy has been added; her rotational kinetic energy does increase as she pulls her arms in, supplied by the work she does against the outward tendency of the spinning weights, but the rise in spin rate itself is explained by conservation of angular momentum, not an increase in torque. Another common error is applying L = I * omega to a body whose rotation axis is itself accelerating or changing direction in space, where the scalar treatment here is insufficient and the fuller vector treatment is required.

Where it stands now

Conservation of angular momentum in the absence of net external torque rests on broad consensus, provable directly from Newton's laws and confirmed continuously across mechanics, astronomy, and engineering, from Kepler's equal-area law to gyroscopic instruments and spacecraft reaction wheels. Its status was, if anything, strengthened by twentieth-century physics, which retained an angular momentum conservation law, tied to the rotational symmetry of space, in both relativistic and quantum settings, extending its reach well beyond the rigid bodies and orbits this entry builds it from.

Test yourself

A diver leaves the board with arms and legs extended, rotating slowly, then pulls into a tight tuck partway through the dive and straightens out again just before entering the water. Using angular momentum conservation, explain what happens to her rotation rate during the tuck and why, being explicit about what stays constant and what changes, and identify the moment when torque from the board's push can be treated as effectively zero so the argument applies cleanly. Then consider a second case: a horizontal turntable holding a small cart free to slide along a frictionless radial track from rim to center. Predict what happens to the turntable's spin rate as the cart slides inward, state which quantity you hold fixed to make that prediction, and identify what physical agent does work on the system as the rotation speeds up.