--- 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: 2026-09-12 --- > **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 on a fixed axle it is simpler: L equals I times omega. Direction matters, so L points along the axle by the right-hand rule. A net torque changes L the way a net force changes momentum. Push off centre and the spin changes; push straight through the axle and nothing happens. **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 almost no outside torque, 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 top refuses to fall over, 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. But the torque nudges the direction of the L vector sideways instead of shrinking it, and those nudges add up to a steady sweep. Gyroscopes in ships, planes, and phones use exactly this response to sense turns. **Tip.** Run every spin problem through one checklist. Name the point you measure L about. Add up outside torques about that point. If none act, set L before equal to L after and solve. Use it for skaters pulling in, divers tucking, and stars collapsing. Never assume L is conserved when an outside twist is present. **Recap.** Angular momentum is I times omega for a spinning body, it stays fixed when no outside torque acts, and a torque on a 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)