Rotation & Angular Dynamics · Physics

For rigid bodies, L is not always parallel to ω.

In the simplest cases, angular momentum looks like L = Iω. But in more general rigid-body motion, the direction of L can differ from the spin axis—an idea that leads into gyroscopic behavior.

This topic

Angular Momentum of a Rigid Body

Learn when L = Iω is valid, how axis choice matters, and why asymmetric spinning can behave unexpectedly.

Fixed axis

When L = Iω works

For rotation about a fixed symmetry axis (or a principal axis), the angular momentum aligns with the angular velocity and has magnitude Iω.

Applies for fixed-axis rotation with a defined I
Symmetry/principal-axis intuition
Direction set by right-hand rule
Why “about this axis” must be stated

General

Axis dependence and distributions

A rigid body’s angular momentum depends on how mass is distributed relative to the axis. Different axes can produce different L even for the same ω.

Same ω, different L about different axes
Using correct moment of inertia for the axis
Interpreting I as a geometric property
Why shifting the axis changes L

Key idea

When L is not parallel to ω (intro)

For rotations not aligned with symmetry axes, angular momentum can point in a different direction than the angular velocity. This is the conceptual doorway to precession.

Asymmetric mass distribution intuition
Why L can “tilt” relative to ω
Consequences: torque causes direction changes
How this shows up in gyroscopes (preview)

Tool

Using Στ = dL/dt with rigid bodies

The most general statement is that external torque changes angular momentum. Even when the spin rate stays nearly constant, torque can change the direction of L.

Torque can change direction more than magnitude
Visualizing dL as a vector change
Small torques causing gradual reorientation
Common scenario: gravity torque on a spinning top

Practice

Practice & Exercises

Practice recognizing when L = Iω is valid, choosing axes, and reasoning about direction changes under torque.

Identify principal/symmetry axis situations
Compute L using correct I about an axis
Conceptual: magnitude vs direction changes of L
Vector reasoning prompts for dL/dt
Exam-style rigid-body angular momentum sets