Angular Momentum — Core Principles

Core Principles

Angular momentum is the rotational equivalent of linear momentum, quantifying an object's 'spinning tendency'. For a point particle, it's defined as $\vec{L} = \vec{r} \times \vec{p}$, where $\vec{r}$ is the position vector from a chosen origin and $\vec{p}$ is the linear momentum.

Its direction is given by the right-hand rule. For a rigid body rotating about a fixed axis, angular momentum simplifies to $L = I\omega$, where $I$ is the moment of inertia and $\omega$ is the angular velocity.

The SI unit is J\cdot s or kg\cdot m^2/s. A crucial principle is the conservation of angular momentum: if the net external torque ($\vec{\tau}_{ext}$) acting on a system is zero, its total angular momentum ($\vec{L}_{total}$) remains constant.

This means $I\omega = \text{constant}$ for a rigid body. This principle explains phenomena like a figure skater speeding up when pulling in her arms or planetary motion. The rate of change of angular momentum is equal to the net external torque: $\vec{\tau}_{ext} = d\vec{L}/dt$.

Important Differences

vs Linear Momentum

Aspect	This Topic	Linear Momentum
Definition	Angular Momentum ($\vec{L}$)	Linear Momentum ($\vec{p}$)
Formula (Point Particle)	$\vec{L} = \vec{r} \times \vec{p}$	$\vec{p} = m\vec{v}$
Formula (System/Body)	$L = I\omega$ (for rigid body, fixed axis)	$\vec{P}_{total} = M\vec{V}_{CM}$ (for system of particles)
Type of Motion	Rotational motion	Translational motion
Reference Point	Always defined with respect to an origin/axis	Independent of reference point (for a given inertial frame)
Rate of Change	$\vec{\tau}_{ext} = d\vec{L}/dt$	$\vec{F}_{ext} = d\vec{p}/dt$
Conservation Condition	Net external torque is zero ($\vec{\tau}_{ext} = 0$)	Net external force is zero ($\vec{F}_{ext} = 0$)
Unit	kg\cdot m^2/s or J\cdot s	kg\cdot m/s or N\cdot s
Analogue of Mass	Moment of Inertia ($I$)	Mass ($m$)

Angular momentum describes rotational inertia in motion, while linear momentum describes translational inertia. Angular momentum is inherently dependent on a chosen origin and involves the cross product of position and linear momentum, or the product of moment of inertia and angular velocity for rigid bodies. Its change is governed by external torque. Linear momentum, on the other hand, is independent of the origin and is simply the product of mass and linear velocity. Its change is governed by external force. Both are vector quantities and are conserved under specific conditions (zero net external torque for angular, zero net external force for linear).