Moment Of Inertia Disk⚠ unverified
Physics / Mechanics · Compute the moment of inertia of a solid disk or cylinder
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| mass | mass | kg | 1.0 | Mass of the disk or cylinder |
| radius | radius | m | 1.0 | Outer radius |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | I | kg*m^2 | Mass moment of inertia about the central axis, in kilogram-metres squared (kg*m^2) |
The science & history
Understanding the Parameters
- $m$ — scales $I$ linearly.
- $r$ — scales $I$ as $r^{2}$; mass far from the axis contributes more.
- $I$ — rotational analogue of mass in $\tau = I\alpha$ and $KE = \tfrac12 I\omega^{2}$.
Derivation (Approaching a Proof)
By definition $I = \int r_{\perp}^{2}\,dm$. For a uniform disk, integrate in polar rings: $dm = (m/(\pi R^{2}))\,2\pi r\,dr$, so
$$I = \int_0^{R} r^{2}\,\frac{2m}{R^{2}} r\,dr = \frac{2m}{R^{2}}\int_0^{R} r^{3}\,dr = \frac{1}{2} m R^{2}.$$
(Thin hoop about the same axis would be $m R^{2}$; solid disk is half that because mass is inward.)
History
Moments of inertia for standard shapes are classical rigid-body results (Euler, 18th century textbooks).
Related Concepts: Moment Of Inertia Point Mass, Parallel Axis Theorem, Rotational Kinetic Energy, Torque
Notes: Registry calculator moment-of-inertia-disk (unverified). Uniform solid disk/cylinder about
central axis only — not diameter axis ($I = \tfrac14 mr^{2} + \tfrac1{12}m h^{2}$ etc.).