What is conservation of angular momentum?
The conservation of angular momentum states that when the net external torque on a system is zero, the system's total angular momentum stays constant. It follows from extending Newton's laws of motion to rotation, and along with conservation of energy and linear momentum, it's one of the three big conservation laws in mechanics.
How do you calculate angular momentum?
- Point particle: L = r × p (the cross product of position from the pivot and momentum)
- Rigid body rotating about a fixed axis: L = Iω (moment of inertia × angular velocity)
- Link to torque: τ = dL/dt. Zero torque means L doesn't change.
The moment of inertia I gets bigger the farther mass is spread from the axis. Since L = Iω stays fixed, shrinking I makes ω go up, and increasing I makes it go down.
Examples on Earth and in space
| Example | What's happening |
|---|---|
| Figure skating spin | Pulling the arms in reduces I, so the skater spins faster |
| Kepler's second law | Planets move faster near the Sun and slower far away, sweeping equal areas in equal times |
| Neutron stars (pulsars) | A collapsing star shrinks dramatically and ends up spinning extremely fast |
| Spinning chair demo | Pulling in your arms while holding dumbbells makes the chair spin faster |
| Gyroscopes and bicycles | Spinning wheels resist changes to their axis, which adds stability |
How is it different from linear momentum conservation?
Linear momentum is conserved when net external force is zero; angular momentum is conserved when net external torque is zero. A planet feels only a central force pointing toward the Sun, so the torque is zero and angular momentum is conserved. That's the physics behind Kepler's second law. At a deeper level, Noether's theorem ties angular momentum conservation to the rotational symmetry of space. For energy-based conservation laws, see the laws of thermodynamics and entropy.
FAQ
- When a skater pulls in her arms, is kinetic energy conserved too?
- No. Angular momentum is conserved, but rotational kinetic energy (½Iω²) actually increases. The extra energy comes from the work her muscles do pulling her arms inward.
- Does friction break conservation of angular momentum?
- If friction acts as an external torque, angular momentum decreases. The law holds exactly only when net external torque is zero.
Sources reviewed: 2026-10-25
References
- Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol. I. Reading, MA: Addison-Wesley.
- Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics (10th ed.). Hoboken, NJ: Wiley.
- Kepler, J. (1609). Astronomia Nova. Heidelberg: Voegelin.