CLS-10

Angular momentum of a rigid rotor

L = I ω, and τ = dL/dt.

Reading speed

RotationKepler / Newton

Governing equation

L=I\omega,\quad \tau=\dfrac{\mathrm{d}L}{\mathrm{d}t}

$\omega \in \left[-100, 100\right]$

where

I: Inertia (kg·m²)
\omega: Angular speed (rad/s)
L: Angular momentum (kg·m²/s)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

L=I\omega,\quad \tau=\dfrac{\mathrm{d}L}{\mathrm{d}t}

$\omega \in \left[-100, 100\right]$

$\omega = -100$ $L = -300$

Along the curve

②

Domain of the variable — blue

\omega_{\min} = -100,\quad \omega_{\max} = 100

$\omega \in \left[-100, 100\right]$

$\omega = -100$ $\omega = -100$

Along the curve

③

Response curve — green

L=I\omega,\quad \tau=\dfrac{\mathrm{d}L}{\mathrm{d}t}

$\omega \in \left[-100, 100\right]$

$\omega = -100$ $L = -300$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}L}{\mathrm{d}\omega}

$\omega \in \left[-100, 100\right]$

$\omega = -98.74$ $L^{\prime} = 3$

Along the curve

Lecture brief

Historical brief

Newtonian mechanics (1687) plus energy, angular momentum, Kepler and the ballistic parabola remain the first language of motion. Every sheet here is a closed-form orbit, throw, spin or oscillator. This sheet (CLS-10 — Angular momentum of a rigid rotor) is the form associated with Kepler / Newton. Working symbols: $I$ , $\omega$ $\rightarrow$ $L$ . For a rigid body about a principal axis, L = I ω. Vanishing torque conserves L, Kepler's areal law in disguise.

Purpose

Purpose: compute

L

from

I

\omega

in Classical mechanics via

L=I\omega,\quad \tau=\dfrac{\mathrm{d}L}{\mathrm{d}t}

L = I ω, and τ = dL/dt. Use it when a real classical mechanics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

I = 3.000\,\mathrm{kg·m^{2}}

\omega = 15.000\,\mathrm{rad/s}

, the governing relation

L=I\omega,\quad \tau=\dfrac{\mathrm{d}L}{\mathrm{d}t}

yields

L = 45.000\,\mathrm{kg·m^{2}/s}

. Fixed principal axis, rigid. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Inertia I3 kg·m²Angular speed \omega15 rad/s

Outputs

Angular momentum L45.000 kg·m²/s

Reading speed

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Narration of this film

Fixed principal axis, rigid.

For a rigid body about a principal axis, L = I ω. Vanishing torque conserves L, Kepler's areal law in disguise.

Reading speed

Watch on YouTube