CLS-04

Rotational kinetic energy

K = ½ I ω² for a rigid body about a fixed axis.

Reading speed

RotationEuler

Governing equation

K=\tfrac12 I\omega^2

$\omega \in \left[0, 200\right]$

where

I: Moment of inertia (kg·m²)
\omega: Angular speed (rad/s)
K: Rotational KE (J)

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

K=\tfrac12 I\omega^2

$\omega \in \left[0, 200\right]$

$\omega = 0$ $K = 0$

Along the curve

②

Domain of the variable — blue

\omega_{\min} = 0,\quad \omega_{\max} = 200

$\omega \in \left[0, 200\right]$

$\omega = 0$ $\omega = 0$

Along the curve

③

Response curve — green

K=\tfrac12 I\omega^2

$\omega \in \left[0, 200\right]$

$\omega = 0$ $K = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}K}{\mathrm{d}\omega}

$\omega \in \left[0, 200\right]$

$\omega = 1.258$ $K^{\prime} = 3.145$

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-04 — Rotational kinetic energy) is the form associated with Euler. Working symbols: $I$ , $\omega$ $\rightarrow$ $K$ . Summing ½ m v² over a rigid body with v = ω r produces ½ I ω², I being the moment of inertia about the axis.

Purpose

Purpose: compute

K

from

I

\omega

in Classical mechanics via

K=\tfrac12 I\omega^2

K = ½ I ω² for a rigid body about a fixed axis. 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 = 2.500\,\mathrm{kg·m^{2}}

\omega = 12.000\,\mathrm{rad/s}

, the governing relation

K=\tfrac12 I\omega^2

yields

K = 180.000\,\mathrm{J}

. Fixed axis, rigid, I is an input. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Moment of inertia I2.5 kg·m²Angular speed \omega12 rad/s

Outputs

Rotational KE K180.000 J

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Narration of this film

Fixed axis, rigid, I is an input.

Summing ½ m v² over a rigid body with v = ω r produces ½ I ω², I being the moment of inertia about the axis.

Reading speed

Watch on YouTube