Rotational Kinematics and Angular Momentum
Rotational kinematics describes orientation and angular velocity, while angular momentum and torque provide the rotational analogues of momentum and force that govern how bodies spin.
Definition
Angular momentum is the rotational counterpart of linear momentum, defined as the moment of momentum about a reference point; its time rate of change equals the net torque, and it is conserved when no external torque acts.
Scope
This topic covers the description of rotational motion through angular displacement, angular velocity, and angular acceleration, the definition of torque and angular momentum about a point or axis, and the rotational form of Newton's second law stating that torque equals the rate of change of angular momentum. It includes conservation of angular momentum for systems free of external torque.
Core questions
- How are angular velocity and angular momentum defined and related?
- What is the rotational analogue of Newton's second law?
- Under what conditions is angular momentum conserved?
Key concepts
- Angular displacement, velocity, and acceleration
- Torque
- Angular momentum
- Moment of inertia about an axis
- Conservation of angular momentum
Key theories
- Torque-angular-momentum relation
- The net torque on a body about a point equals the time rate of change of its angular momentum about that point, the rotational analogue of force equals rate of change of momentum.
- Conservation of angular momentum
- When no net external torque acts, the total angular momentum of a system is constant, explaining why spinning bodies speed up as they contract and the planarity of orbits.
Clinical relevance
Conservation of angular momentum explains the spin-up of figure skaters and divers, the stability of rotating wheels and tops, the behavior of reaction wheels for spacecraft attitude control, and the fixed orientation of gyroscopes used in navigation.
History
The concept of angular momentum and its conservation emerged from Kepler's law of equal areas, recognized by Newton as a consequence of central forces, and was generalized by Euler into the rotational laws of motion for extended bodies. Its full role as an independently conserved vector quantity was clarified through eighteenth- and nineteenth-century mechanics.
Key figures
- Leonhard Euler
- Isaac Newton
- Jean-Baptiste Biot
Related topics
Seminal works
- taylor2005
- kleppner2014
Frequently asked questions
- Why does a skater spin faster when pulling in their arms?
- With no external torque, angular momentum is conserved; pulling the arms in lowers the moment of inertia, so the angular velocity must rise to keep the product constant, and the skater spins faster.
- Is angular momentum always parallel to angular velocity?
- Only for rotation about a principal axis or for a symmetric body. In general the inertia tensor makes angular momentum point in a different direction from angular velocity, which is the source of wobble and precession.