MEC-46

Parallel-axis theorem

Second-moment shift to a parallel axis.

Reading speed

MachinesParallel-axis theorem

Governing equation

I=I_{cm}+md^2

$d \in \left[0, 2\right]$

where

I_{cm}: Icm (kg·m²)
m: Mass (kg)
d: Offset (m)
I: Inertia (kg·m²)

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

I=I_{cm}+md^2

$d \in \left[0, 2\right]$

$d = 0$ $I = 0.12$

Along the curve

②

Domain of the variable — blue

d_{\min} = 0,\quad d_{\max} = 2

$d \in \left[0, 2\right]$

$d = 0$ $d = 0$

Along the curve

③

Response curve — green

I=I_{cm}+md^2

$d \in \left[0, 2\right]$

$d = 0$ $I = 0.12$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}I}{\mathrm{d}d}

$d \in \left[0, 2\right]$

$d = 0.013$ $I^{\prime} = 0.201$

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-46 — Parallel-axis theorem) is the form associated with Parallel-axis theorem. Working symbols: $I_{cm}$ , $m$ , $d$ $\rightarrow$ $I$ . Second-moment shift to a parallel axis. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

I

from

I_{cm}

m

d

in Mechanical via

I=I_{cm}+md^2

Second-moment shift to a parallel axis. 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_{cm} = 0.120\,\mathrm{kg·m^{2}}

m = 8.000\,\mathrm{kg}

d = 0.250\,\mathrm{m}

, the governing relation

I=I_{cm}+md^2

yields

I = 0.6200\,\mathrm{kg·m^{2}}

. 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

Icm I_{cm}0.12 kg·m²Mass m8 kgOffset d0.25 m

Outputs

Inertia I0.6200 kg·m²

Reading speed

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Narration of this film

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

Second-moment shift to a parallel axis. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube