--- title: "Angular Momentum and Its Conservation" description: "Angular momentum is defined for a particle by its position and momentum and for a rigid body by its moment of inertia and angular velocity, and it changes only when an external torque acts. A skater p" canonical: https://lightmysky.com/learn/science/angular-momentum-and-its-conservation-mt_9v3S9-rhwc source: https://lightmysky.com/learn/science/angular-momentum-and-its-conservation-mt_9v3S9-rhwc.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Angular Momentum and Its Conservation Angular momentum is defined for a particle by its position and momentum and for a rigid body by its moment of inertia and angular velocity, and it changes only when an external torque acts. A skater pulling in her arms and a gyroscope precessing are the same law seen twice. Subject: Science · Area: Forces & Motion · Ages 20 to 22 Page: https://lightmysky.com/learn/science/angular-momentum-and-its-conservation-mt_9v3S9-rhwc ## Ready when they can - Computes the angular momentum of a particle about a stated point and of a rigid body about its axis - Uses conservation of angular momentum where the net external torque is zero - Explains gyroscopic precession as the response of an angular momentum vector to a torque ## Lesson: Spin is conserved: skaters, tops, and gyroscopes Angular momentum measures how much rotation a thing carries. For a single particle it is position cross momentum, and only the part of the motion that circles your chosen point counts. For a rigid body spinning about a principal axis, L equals I omega and points along the axle. For an arbitrary fixed axis, L equals I omega gives the axial component, not the full vector. Net external torque about a fixed inertial origin changes L. Torque is r cross F, so a push changes spin only when the line of action of the push misses the axis; a push whose line of action passes through the axle gives zero torque. Angular velocity ω = dθ/dt and angular acceleration α = dω/dt are signed about the fixed axis with units radians per second and radians per second squared. **Example.** Worked example: a 5 kg particle moves at 2 metres per second along a straight line whose closest approach to the chosen origin is 3 m. At closest approach, r is the 3 m vector from the origin to the particle and p equals m v equals 10 kilogram metres per second, perpendicular to r. Then L equals r cross p has size r p sine 90 degrees equals 3 times 10 equals 30 kilogram metres squared per second, directed perpendicular to the plane of the motion by the right-hand rule. *(drawing: Angular momentum is position cross momentum.)* **Example.** Watch a skater spin with arms wide, then pull them in. Her spin rate jumps up, though nothing twisted her around. Pulling in shrinks her moment of inertia, so omega must grow to keep the product I times omega fixed. With negligible external torque about the spin axis, total L stays constant before and after. The same trade explains divers tucking to spin faster. Note that her kinetic energy still rises, since her muscles do work pulling in. A fast symmetric top, tilted and pivoted at its tip, can hold its tilt, and instead its axis slowly circles the vertical. This circling is called precession. Gravity still pulls on the centre of mass and still applies a torque about the tip. For a fast spinner this torque is perpendicular to the L vector, so it turns the direction of L instead of changing its size, and those turns add up to a steady sweep. This steady precession is an approximation for a fast spinning symmetric top, tilted and pivoted at its tip; a slow or unbalanced top can still fall. Name the point you measure L about, a fixed inertial origin. Add up the net external torque about that point. If it is zero, set L before equal to L after and solve. Never assume L is conserved when a net external torque acts. **Recap.** Angular momentum is I times omega for a spinning body about a principal axis, it stays fixed when the net external torque about the chosen origin is zero, and a torque on a fast gyroscope turns the spin axis sideways. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Rotational Dynamics and Rolling Without Slipping](https://lightmysky.com/learn/science/rotational-dynamics-and-rolling-without-slipping-mt_1mV6ukwDxK) - [The Cross Product and Oriented Area](https://lightmysky.com/learn/mathematics/the-cross-product-and-oriented-area-mt_QEBM_Ms-SK) ## Opens up - [Generalised Coordinates and the Lagrangian](https://lightmysky.com/learn/science/generalised-coordinates-and-the-lagrangian-mt_9QPk1zI9jE) - [The Two-Body Problem, Orbital Energy and Kepler's First Two Laws](https://lightmysky.com/learn/science/the-two-body-problem-orbital-energy-and-keplers-first-two-laws-mt_CjHFftkB6M) - [Orbital Angular Momentum and the Quantum Numbers](https://lightmysky.com/learn/science/orbital-angular-momentum-and-the-quantum-numbers-mt_OXlbat0vat)