Angular momentum and its conservation

Angular momentum for a particle

Angular momentum measures rotational motion. For a particle with position vector ec{r} relative to an origin and linear momentum ec{p}, angular momentum is

ec{L}=ec{r} imesec{p}

Its magnitude is

$L=rpsin heta$

where $heta$ is the angle between ec{r} and ec{p}. The direction follows the right-hand rule.

Angular momentum depends on the choice of origin, just as torque does.

Angular momentum for a rigid body

For a rigid body rotating about a fixed symmetry axis, angular momentum is often

ec{L}=Iec{omega}

This resembles linear momentum,

ec{p}=mec{v}

Here $I$ is moment of inertia and ec{omega} is angular velocity. In more advanced rigid body motion, ec{L} and ec{omega} do not always point in the same direction, but the simple form works for many introductory problems.

Torque and angular momentum

Net torque changes angular momentum:

ec{ au}_{net}=rac{dec{L}}{dt}

This is the rotational analog of

ec{F}_{net}=rac{dec{p}}{dt}

For fixed-axis rotation with constant $I$,

$au_{net}=Ialpha$

because ec{L}=Iec{omega}.

Conservation of angular momentum

If net external torque is zero, angular momentum is conserved:

ec{L}_i=ec{L}_f

This is one of the most important conservation laws in mechanics. Internal torques can redistribute angular momentum within a system, but they cannot change total angular momentum if external torque is zero.

Skater example

A spinning skater pulling in their arms decreases moment of inertia. If external torque is negligible,

$I_iomega_i=I_fomega_f$

When $I_f$ decreases, $omega_f$ increases. The skater spins faster.

Rotational kinetic energy increases in this process because the skater does work pulling in their arms. Angular momentum conservation does not require kinetic energy to stay constant.

Planetary motion

Angular momentum conservation explains why planets move faster when closer to the Sun in elliptical orbits. The gravitational force points nearly along the line from planet to Sun, so torque about the Sun is zero:

ec{ au}=ec{r} imesec{F}=0

Thus angular momentum is conserved. When $r$ is smaller, tangential speed must be larger.

This is related to Kepler's second law: equal areas are swept out in equal times.

Rolling and rotational collisions

Angular momentum conservation can be useful when external forces are present but exert no torque about a chosen point. For example, during a brief collision with a pivoted rod, angular momentum about the pivot may be conserved if the pivot force has zero lever arm.

This strategy is common in rotational collision problems.

Gyroscopic effects

Spinning objects resist changes in the direction of angular momentum. This leads to gyroscopic behavior in tops, bicycle wheels, spacecraft, and navigation devices. A torque perpendicular to angular momentum changes its direction, producing precession rather than simply tipping over.

The big idea

Angular momentum is rotational momentum. It is changed by net external torque and conserved when net external torque is zero. This principle explains spinning skaters, planetary motion, rotating machinery, gyroscopes, and many collision problems. Like linear momentum conservation, it is powerful because it can solve problems even when forces or torques are complicated.