MEC-44

Rotational KE

Rigid-body rotational kinetic energy.

Reading speed

MachinesRotational KE

Governing equation

K=\tfrac12 I\omega^2

$\omega \in \left[1, 400\right]$

where

I: Inertia (kg·m²)
\omega: Rate (rad/s)
K: Energy (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.

00:0 / 00:08

①

Governing law — red

K=\tfrac12 I\omega^2

$\omega \in \left[1, 400\right]$

$\omega = 1$ $K = 0.2$

Along the curve

②

Domain of the variable — blue

\omega_{\min} = 1,\quad \omega_{\max} = 400

$\omega \in \left[1, 400\right]$

$\omega = 1$ $\omega = 1$

Along the curve

③

Response curve — green

K=\tfrac12 I\omega^2

$\omega \in \left[1, 400\right]$

$\omega = 1$ $K = 0.2$

Along the curve

④

Rate of change — amber

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

$\omega \in \left[1, 400\right]$

$\omega = 3.509$ $K^{\prime} = 1.404$

Along the curve

Lecture brief

Historical brief

Machine design grew from Coulomb torsion and Hertz contact (1881) through Soderberg fatigue and the heat-engine cycle. The lab writes shaft, bearing, contact and thermodynamic limits in SI. This sheet (MEC-44 — Rotational KE) is the form associated with Rotational KE. Working symbols: $I$ , $\omega$ $\rightarrow$ $K$ . Rigid-body rotational kinetic energy. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

K

from

I

\omega

in Mechanical via

K=\tfrac12 I\omega^2

Rigid-body rotational kinetic energy. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

I = 0.400\,\mathrm{kg·m^{2}}

\omega = 100.000\,\mathrm{rad/s}

, the governing relation

K=\tfrac12 I\omega^2

yields

K = 2000.00\,\mathrm{J}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Inertia I0.4 kg·m²Rate \omega100 rad/s

Outputs

Energy K2000.00 J

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Rigid-body rotational kinetic energy. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube