# Angular momentum **Angular momentum** is the rotational analogue of [[Momentum|linear momentum]]: for a particle the vector product of position and momentum about a chosen origin, and for a rigid body spinning about a symmetry axis `L = I·w`, the [[Moment_of_inertia|moment of inertia]] times the angular velocity.[^idema-rot] One line governs it: `dL/dt = tau`, the rate of change of angular momentum equals the applied [[Torque|torque]].[^idema-prec] Because L is a vector, that line has a consequence with no linear counterpart — a torque at right angles to L turns it instead of lengthening it, which is why a spinning top does not fall over. In the microsim below the reader sets the spin of a top on a logarithmic slider from 20 to 2,000 rad/s and watches its angular-momentum arrow sweep a cone. Gravity applies a torque `m·g·r·sin(phi)` about the pivot, and in the fast-spin limit the axis precesses at `omega_p = m·g·r/(I·w)`: faster spin, *slower* [[Precession|precession]], and no fall.[^idema-prec] With the illustrative defaults — a 0.1 kg top, its centre of mass 0.04 m from the pivot, on a 0.03 m disk — a spin of 300 rad/s gives `omega_p` = 2.91 rad/s, a precession period of 2.16 s (derived). A second preset removes the torque and pulls a skater's arms in, halving I and doubling ω at fixed `L = I·w`; a third races a hoop, a disk and a sphere down a ramp under `a = g·sin(theta)/(1 + I/(m·R^2))`.[^idema-roll] The model is ILLUSTRATIVE: it is the fast-spin limit, omits nutation, and flags itself invalid when `omega_p/w` exceeds 0.1 — for these defaults, below about 93 rad/s (derived). On the [[Physics]] flagship this page serves Part II — Core theories, in the section *Rotation and angular momentum* (row P16), between [[Momentum|momentum]] and the quantum sections, where the same quantity reappears in multiples of ħ. ## Examples Three demonstrations cover the classical law, and they are the microsim's presets. A top leaning against gravity does not topple but walks its axis around a circle, because the torque is perpendicular to L. A skater in a torque-free spin pulls her arms in and speeds up, because I falls while L cannot change; the [[Rotational_energy|rotational energy]] `K = (1/2)·I·w^2` then *doubles* when I halves, the extra joule being the work of pulling inward (derived).[^idema-rot] A hoop, a disk and a sphere released together on a ramp arrive in the reverse order of their shape factor β = I/(mR²): at 30° the sphere accelerates at 3.50 m/s², the disk at 3.27 and the hoop at 2.45 (derived), mass and radius cancelling out.[^idema-roll] ## Definition in classical mechanics Angular momentum is always defined about a point, and changing the origin changes the value: a system can have none about its centre of mass and a great deal about a distant point. ### Orbital angular momentum in two dimensions In a plane, `L = m·v·r_perp`, where `r_perp` is the perpendicular distance from the origin to the line of motion — the lever arm. For a particle on a circle of radius r this is `L = m·r^2·w`, since `v = w·r`.[^idema-rot] The sign carries the sense of rotation, and that one number suffices for any planar problem: a [[Pendulum|pendulum]], an [[Orbit|orbit]] seen face-on. ### Scalar angular momentum from Lagrangian mechanics The Lagrangian of a particle in a central field, in polar coordinates, does not contain the angle θ, so its conjugate momentum `p_theta = m·r^2·thetadot` is a constant of the motion — with no mention of torque.[^cline-rolling] This is the cleanest definition available: angular momentum is whatever is conjugate to an angle, which is why the same words apply to a rigid body, a field and a [[Quantum_mechanics|quantum]] state. ### Orbital angular momentum in three dimensions In three dimensions the lever arm becomes a cross product, `L = r × p`, perpendicular to both position and momentum and of the planar magnitude; the torque is `tau = r × F`.[^idema-rot] The vector character is not decoration: the top precesses because L and τ point in different directions, and the content of `dL/dt = tau` here is that a perpendicular torque rotates L at constant length. ## Analogy to linear momentum Every translational statement has a rotational twin: mass becomes moment of inertia, velocity becomes `w`, force becomes torque, momentum becomes `L = I·w`, kinetic energy becomes `(1/2)·I·w^2`, work becomes `W = integral(tau·dtheta)`.[^idema-rot] The analogy is exact in form and treacherous in substance. Moment of inertia depends on the axis as well as the body, so a body has many values of I and no single one — there is no constant that converts I into m.[^os-inertia] And `L = I·w` holds only about a symmetry axis; otherwise L and ω point in different directions and the full inertia tensor is needed.[^idema-rot] ### Angular momentum and torque The law is `dL/dt = tau`, which about a fixed axis reduces to `tau = I·alpha`, the rotational form of Newton's second law.[^idema-rot] The top is instructive because that reduction fails. Gravity on the centre of mass, a distance r from the pivot along an axis tilted by φ, exerts a horizontal torque `m·g·r·sin(phi)` perpendicular to the horizontal projection of L. It cannot change |L|; it can only swing it, at the torque divided by that projection, `omega_p = m·g·r·sin(phi)/L_xy = m·g·r/(I·w)` — the sin φ cancels, so the precession rate does not depend on how far the top leans.[^idema-prec] Two cautions travel with it: it is a fast-spin limit that omits nutation, the periodic wobble in φ a real top shows, and it assumes a symmetric top.[^idema-prec] The microsim flags itself invalid above `omega_p/w` = 0.1, a design threshold rather than a book value. ## Conservation of angular momentum If the net external torque is zero, L is constant in magnitude *and direction*. The direction clause is the stronger half: a spinning wheel resists having its axis turned, the working principle of every [[Gyroscope|gyroscope]]. The law follows from [[Noether's_theorem|Noether's theorem]] as the quantity belonging to rotational symmetry, as [[Momentum|linear momentum]] belongs to translational symmetry.[^noether1918] Space having no preferred direction is why a skater can spin. ### General considerations Internal forces contribute no net torque: each action–reaction pair acts along the line joining the particles, so their moments cancel. Only external torques count, and only about the chosen origin. This is why the skater preset works — her muscles are internal, so whatever she does to her arms leaves L untouched and can only redistribute it between I and ω. ### Relation to Newton's second law of motion Crossing `r` into `F = dp/dt` gives `r × F = d(r × p)/dt` at once, since the extra term `v × p` vanishes for parallel vectors. So `dL/dt = tau` is not an independent postulate; it is [[Newton's_laws_of_motion|Newton's second law]] rewritten about a point.[^idema-rot] The rotational quantities make a class of problems easy rather than adding new physics, and the manipulation survives unchanged into [[Lagrangian_mechanics|Lagrangian]] and [[Hamiltonian_mechanics|Hamiltonian]] mechanics. ### Lagrangian formalism Here angular momentum is the momentum conjugate to an angular coordinate, and its conservation says the coordinate is cyclic — absent from the Lagrangian.[^cline-rolling] The advantage is that constraint forces never enter. Cline solves the rolling incline three ways and gets `xdotdot = (2/3)·g·sin(alpha)` for a disk each time, with a friction force of `(1/3)·M·g·sin(alpha)`; a yo-yo falls at `(2/3)·g` for the same reason.[^cline-rolling] ## Angular momentum in orbital mechanics Under a central force the torque about the centre vanishes, since `r` and `F` are parallel, so L is exactly conserved. The orbit therefore stays in a fixed plane, and the radius vector sweeps equal areas in equal times, the areal rate being `L/(2·m)`. That is [[Kepler's_laws_of_planetary_motion|Kepler's second law]], and it holds for *any* central force — an inverse square is needed for the ellipse, not for the area law.[^kepler1609] Conservation also keeps an [[Orbital_eccentricity|eccentric]] orbit from falling in: the centrifugal barrier `L^2/(2·m·r^2)` rises faster than the attraction as r shrinks. ## Solid bodies For a rigid body the sum over particles becomes `I = sum(m·r^2) = integral(rho·r^2·dV)`, with r measured from the axis.[^idema-rot] Standard shapes give I as a pure number times MR²: a hoop 1, a solid cylinder ½, a hollow sphere ⅔, a solid sphere ⅖, a cone 3/10; a thin rod takes 1/12 ML² about its centre.[^idema-rot] Two theorems move the axis without redoing the integral: the parallel-axis theorem adds `M·d^2`, and the perpendicular-axis theorem gives `I_z = I_x + I_y` for a flat body.[^idema-rot] Those numbers decide the microsim's rolling race. Rolling without slipping ties `v = w·R`, so β fixes both the acceleration `a = g·sin(theta)/(1 + beta)` and the energy split `K_rot/K = beta/(1 + beta)` — a third for a disk, a half for a hoop (derived).[^idema-roll] The arrival-time ratio is `sqrt(1 + beta)`, so a hoop takes 41 % longer than a sliding block and a solid sphere only 18 % longer (derived). ### Collection of particles Before the integral there is a sum, and it separates as linear momentum does: a system's total is that of its whole mass moving with the centre of mass, plus the angular momentum about that centre. A [[Helicopter|helicopter]] shows why the split matters. Four 4.00 m, 50.0 kg blades at 300 rpm on a 1,000 kg aircraft flying at 20.0 m/s hold 5.26×10⁵ J of rotational kinetic energy against 2.00×10⁵ J of translational — most of the machine's mechanical energy is in the rotor.[^os-heli] ## Angular momentum in general relativity In [[General_relativity|general relativity]] a rotating mass does not merely attract; it drags local inertial frames around with it, an effect Josef Lense and Hans Thirring derived in 1918.[^lense1918] A rotating [[Black_hole|black hole]], in the solution Roy Kerr found in 1963, is characterized by two numbers — mass and angular momentum — with a maximum spin above which no horizon exists.[^kerr1963] Conservation survives, but the definition becomes global and delicate: it is well defined for an isolated, asymptotically flat system and not in general. [[Tests_of_general_relativity|Tests]] of frame dragging use satellite gyroscopes and laser-ranged orbits. ## Angular momentum in quantum mechanics Quantum angular momentum keeps the algebra of the classical vector and discards its continuity: its components do not commute, so only the magnitude and one component can be known at once, and both come in steps set by ħ. ### Spin, orbital, and total angular momentum A quantum particle carries orbital angular momentum from its motion and an intrinsic [[Spin_(physics)|spin]] with no motional origin at all. Spin magnitudes run as `abs(S) = sqrt(J·(J + 1))·hbar` with 2J + 1 allowed orientations, J any multiple of ½; the two add vectorially, and it is the total that is conserved.[^schiller-spin] Massless particles have helicity only — a projection along the direction of motion.[^schiller-spin] ### Quantization Orbital angular momentum comes in integer units: for an [[Electron|electron]] in an [[Atom|atom]] the magnitude is `sqrt(l·(l + 1))·hbar` and the projection on a chosen axis is `m·hbar`, m running from −l to +l — the origin of the [[Quantum_number|quantum numbers]] labelling every [[Atomic_orbital|atomic orbital]].[^up3-atomic] Spin-½ particles take half-integer values instead, and the [[Stern–Gerlach_experiment|Stern–Gerlach experiment]] split silver atoms into exactly two beams. A corollary is that a 2π rotation is *not* the identity: a spin-J object returns to itself after 2π/J, so spin-½ needs 4π.[^schiller-spin] ### Uncertainty Because `L_x`, `L_y` and `L_z` fail to commute, a state with a definite projection on one axis has indefinite projections on the other two: the classical picture of an arrow pointing somewhere definite is wrong in principle, and what survives is a cone of orientations at fixed magnitude and fixed `L_z`.[^up3-atomic] The consequence is measurable in [[Nuclear_magnetic_resonance|nuclear magnetic resonance]] and in any spectrum split by a [[Magnetic_field|magnetic field]], where discrete projections appear as discrete [[Energy_level|energy levels]]. ## Angular momentum in electrodynamics An [[Electromagnetic_radiation|electromagnetic field]] carries angular momentum as well as [[Momentum|linear momentum]], as a density `r × (E × B)/(mu_0·c^2)`. The bookkeeping must include it: a charge spiralling in a magnetic field exchanges angular momentum with the field, and only the total is conserved. A current loop's magnetic moment is in turn proportional to its angular momentum through a gyromagnetic ratio — the bridge between the mechanical and magnetic quantities, and the basis of magnetic resonance. ## Angular momentum in optics Light carries both kinds. John Henry Poynting argued in 1909 that circularly polarized light carries ħ of angular momentum per [[Photon|photon]], and Richard Beth measured the torque it exerts on a suspended waveplate in 1936.[^poynting1909][^beth1936] A separate orbital contribution was identified in 1992: a [[Laser|laser]] beam with a helical phase front carries `l·hbar` per photon, independent of polarization and unbounded in l.[^allen1992] Both are now used to spin trapped particles and to multiply optical-fibre channels. ## Angular momentum in nature and the cosmos The [[Earth|Earth]] is itself the microsim's top. Its axis is tilted 23.4° and, under the torque the [[Moon|Moon]] and Sun exert on its equatorial bulge, precesses once in roughly 25,000 years, with a lunar nutation of period 18.6 years on top.[^idema-prec] Run in reverse, the same law explains why collapsing systems spin up: a shrinking cloud keeps L while I falls as r², so ω rises as r⁻² — the skater preset at astronomical scale (derived). It is also why matter falling toward a compact object forms a disk instead of dropping straight in. ## Angular momentum in engineering and technology Every device that resists being turned exploits the vector character of L. A [[Gyroscope|gyroscope]] holds its axis because changing the direction of L costs torque, and a rate gyro reads a vehicle's turn from the precession torque that turning produces — a second-order system with its own damping and natural frequency.[^hallauer-gyro] A [[Flywheel|flywheel]] stores energy in `(1/2)·I·w^2` while steadying a machine's speed, and [[Flywheel_energy_storage|flywheel energy storage]] scales that to grid service. Spacecraft reorient by spinning internal wheels at fixed total L, and an [[Inertial_navigation_system|inertial navigation system]] integrates rate-gyro outputs to hold a heading. Rotating machinery pays in reverse: a helicopter needs a tail rotor because the torque driving its main rotor must be resisted somewhere.[^os-heli] ## History The area law came first, the concept with rigid-body mechanics, the reason last. ### Law of Areas Johannes Kepler published in 1609, from Tycho Brahe's observations of Mars, that the line joining a planet to the Sun sweeps equal areas in equal times.[^kepler1609] He had no notion of angular momentum; the constancy of that areal rate is the constancy of `L/(2·m)`. [[Isaac_Newton|Newton]] then proved in the *Principia* that the area law follows from any centrally directed force, making it a statement about the geometry of forces rather than a fact about the Sun.[^newton1687] ### After Newton Leonhard Euler's treatise on the motion of rigid bodies, published in 1765, introduced the principal axes and the moments of inertia about them — the first general treatment of a spinning body.[^euler1765] Nineteenth-century mechanics turned that apparatus into the inertia tensor and the gyroscope. The modern reading arrived in 1918, when Emmy Noether showed that conservation of angular momentum follows from the isotropy of space; quantum theory soon found the same quantity again, in units of ħ.[^noether1918] ## See also - [[Precession]] - [[Gyroscope]] - [[Torque]] - [[Moment_of_inertia]] - [[Rolling]] - [[Momentum]] - [[Rotational_energy]] - [[Noether's_theorem]] ## References [^idema-rot]: Idema, Timon (2018). *Mechanics and Relativity*. Chapter 5, pp. 64–68 (`v = w·r`, `a_t = r·alpha`, `a_c = w^2·r`; `tau = r × F` and `tau = I·alpha`; `I = sum(m·r^2) = integral(rho·r^2·dV)`; Table 5.1 of shape factors — rod (1/12)ML² about its centre and (1/3)ML² about its end, hoop MR², solid cylinder (1/2)MR², hollow sphere (2/3)MR², solid sphere (2/5)MR², plate (1/12)M(a² + b²); the parallel- and perpendicular-axis theorems; `K_rot = (1/2)·I·w^2` and `W = integral(tau·dtheta)`; the warning that `L = I·w` holds only about a symmetry axis and that asymmetric bodies need the inertia tensor). The cone's 3/10 is at p. 77. The doubling of `K_rot` when I halves at fixed L is derived. Portal Book 080, https://open.umn.edu/opentextbooks/textbooks/mechanics-and-relativity [^idema-prec]: Idema (2018), *Mechanics and Relativity*, Chapter 5, pp. 68 and 71–72 (`dL/dt = tau`; the gravitational torque `abs(tau) = m·g·r·sin(phi)`; `w_p = m·g·r·sin(phi)/L_xy = m·g·r/(I·w)`; nutation as a periodic change in φ; the derivation assumes a symmetric top with its torque arm along L; the Earth's 23.4° tilt, ≈25,000-year precession and 18.6-year lunar nutation). The book gives no worked top example: the 0.1 kg, 0.04 m, 0.03 m defaults are the sub-manual's design choices, and `w_p` = 2.91 rad/s at a 300 rad/s spin, the 2.16 s period and the ≈93 rad/s validity threshold at `w_p/w` = 0.1 are derived from them. Portal Book 080. [^idema-roll]: Idema (2018), *Mechanics and Relativity*, Chapter 5, pp. 68–71 (the rolling condition `v = w·R`; the solid cylinder's `a = (2/3)·g·sin(theta)` derived three independent ways; the general `a = g·sin(theta)/(1 + I/(m·R^2))`). The 30° accelerations 3.50, 3.27 and 2.45 m/s² for sphere, disk and hoop, the energy split `beta/(1 + beta)` and the arrival-time ratio `sqrt(1 + beta)` are derived. Portal Book 080. [^cline-rolling]: Cline, Douglas (2018). *Variational Principles in Classical Mechanics*, revised second edition. Pp. 168–178 (the disk on an incline, `xdotdot = (2/3)·g·sin(alpha)` with a friction force of `(1/3)·M·g·sin(alpha)`; 7/5 in the denominator for a sphere at p. 174; the yo-yo falling at `(2/3)·g` at pp. 177–178; Atwood's machine at p. 171). The conjugate-momentum and cyclic-coordinate statements come from the same text's Lagrangian-dynamics chapters, pp. 155–198 (page to pin). Portal Book 073, https://open.umn.edu/opentextbooks/textbooks/variational-principles-in-classical-mechanics [^os-inertia]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 1*. OpenStax. Chapter 10, p. 484 (there is no constant that converts moment of inertia into mass) and p. 485 (the washer problem; the shape table of Fig. 10.20 was lost in the sub-manual's extraction). Chapters 10–11 span pp. 463–564 (pages to pin). Portal Book 077, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1 [^os-heli]: Sanny and Ling (2016), *University Physics Volume 1*, Chapter 10, Example 10.9, pp. 487–488 (four 4.00 m, 50.0 kg blades at 300 rpm on a 1,000 kg aircraft flying at 20.0 m/s; the book states the ratio K_trans/K_rot = 0.380, and K_rot = 5.26×10⁵ J with K_trans = 2.00×10⁵ J are derived from its inputs; rpm must be converted to rad/s first). Portal Book 077. [^schiller-spin]: Schiller, Christoph. *Motion Mountain, Volume IV: The Quantum of Change*. Pp. 125–142 (`abs(S) = sqrt(J·(J + 1))·hbar` with 2J + 1 orientations and J any multiple of ½, p. 126; massless particles have helicity only, p. 126; a spin-J object is unchanged after 2π/J, so spin-½ needs 4π, p. 127; the belt trick, pp. 131–132; composites of an odd number of fermions are fermions, p. 140). Portal Book 049. [^up3-atomic]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3*. OpenStax. Chapter 8 "Atomic Structure", pp. 347–392 (page to pin) — orbital angular momentum quantized as `sqrt(l·(l + 1))·hbar` with projections `m·hbar`, the resulting quantum numbers, and the non-commuting components that leave only the magnitude and one projection simultaneously definite. Portal Book 079, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^kepler1609]: Kepler, Johannes (1609). *Astronomia Nova*. Prague. (The equal-areas rule, derived from Tycho Brahe's observations of Mars.) [^newton1687]: Newton, Isaac (1687). *Philosophiæ Naturalis Principia Mathematica*. London: Royal Society. Book I, Proposition I, Theorem I (the area law follows from any centrally directed force). [^euler1765]: Euler, Leonhard (1765). *Theoria motus corporum solidorum seu rigidorum*. Rostock and Greifswald. [^noether1918]: Noether, Emmy (1918). "Invariante Variationsprobleme." *Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse*: 235–257. [^lense1918]: Lense, Josef; Thirring, Hans (1918). "Über den Einfluss der Eigenrotation der Zentralkörper auf die Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie." *Physikalische Zeitschrift* 19: 156–163. [^kerr1963]: Kerr, Roy P. (1963). "Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics." *Physical Review Letters* 11 (5): 237–238. [^poynting1909]: Poynting, John Henry (1909). "The Wave Motion of a Revolving Shaft, and a Suggestion as to the Angular Momentum in a Beam of Circularly Polarised Light." *Proceedings of the Royal Society A* 82 (557): 560–567. [^beth1936]: Beth, Richard A. (1936). "Mechanical Detection and Measurement of the Angular Momentum of Light." *Physical Review* 50 (2): 115–125. [^allen1992]: Allen, L.; Beijersbergen, M. W.; Spreeuw, R. J. C.; Woerdman, J. P. (1992). "Orbital Angular Momentum of Light and the Transformation of Laguerre-Gaussian Laser Modes." *Physical Review A* 45 (11): 8185–8189. [^hallauer-gyro]: Hallauer, William (2016). *Introduction to Linear, Time-Invariant, Dynamic Systems for Students of Engineering*. Pp. 185–217 (page to pin) — second-order damped response, within which the rate gyro is treated as a worked instrument. Portal Book 021, https://open.umn.edu/opentextbooks/textbooks/introduction-to-linear-time-invariant-dynamic-systems-for-students-of-engineering ## Further reading - Idema, Timon (2018). *Mechanics and Relativity*. Portal Book 080 — Chapter 5 is the cleanest short treatment of rotation, rolling and precession in the Portal Books, and supplies every classical equation on this page. - Cline, Douglas (2018). *Variational Principles in Classical Mechanics*, revised second edition. Portal Book 073 — the same rolling problems solved by forces, by energy and by variational methods. - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 1*. OpenStax. Portal Book 077 — Chapters 10–11 for moment of inertia, rotational dynamics and worked rotor problems. - Schiller, Christoph. *Motion Mountain, Volume IV: The Quantum of Change*. Portal Book 049 — spin, the 4π rotation and the spin–statistics connection, treated visually. ## External links - [Mechanics and Relativity](https://open.umn.edu/opentextbooks/textbooks/mechanics-and-relativity), Timon Idema — Portal Book 080 - [Variational Principles in Classical Mechanics](https://open.umn.edu/opentextbooks/textbooks/variational-principles-in-classical-mechanics), Douglas Cline — Portal Book 073 - [University Physics Volume 1](https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1), OpenStax — Portal Book 077 - The Wikipedia pair's external links list further resources, including demonstrations of the rotating-stool experiment <!-- MATTERSIM:BEGIN g24 — Matter & Energy Cluster microsim (framework build, specs/sims/Angular_momentum.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework), pending deploy:** *Angular momentum* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Angular_momentum.html` is live. <!-- pending: <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Angular_momentum.html" data-title="Angular momentum"></div> --> <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Angular_momentum) : [Wikitube](https://en.wikitube.io/wiki/Angular_momentum) · pinned revision [1363158474](https://en.wikipedia.org/w/index.php?oldid=1363158474) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P16 · sim pending (matter/Angular_momentum).*