✓ Sources checked · 10 September 2026 · View verification
Moment of inertia I = ∫r² dm measures mass distribution relative to a specified rotation axis. It controls angular acceleration for a given net torque in fixed-axis rigid-body rotation.
By Matan · Updated
State the axis with each formula
- Uniform slender rod, axis through its centre perpendicular to its length: I = ML²/12.
- Uniform disk, central axis perpendicular to its face: I = MR²/2.
- Uniform thick cylindrical tube, symmetry axis: I = M(Router² + Rinner²)/2.
- Thin hoop, symmetry axis: I = MR².
The units are kg·m². “Around the centre” is insufficient without the axis direction. The tube expression approaches the hoop result for a thin wall. [1]
Worked disk-versus-hoop comparison
Take M = 2 kg and R = 0.30 m. The disk has I = 0.09 kg·m² and the hoop I = 0.18 kg·m². Under the same 0.36 N·m net torque, their angular accelerations are 4 rad/s² and 2 rad/s² respectively. These calculations assume rigid bodies about fixed symmetry axes.
What inertia does not establish
It resists changes of angular motion; no continuing torque is needed solely to sustain constant angular velocity without losses. Spacecraft orbit stability or bicycle handling cannot be inferred from one inertia value alone. For a compound body, add each part’s inertia about the same axis, shifting axes with I = Icm + Md² when appropriate. [1]
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How this article was checked
Sources checked; calculations checked where shown. The definitions, model assumptions and main explanatory claims were cross-checked against the sources below. Worked-example arithmetic and units were checked separately. These examples are illustrative calculations, not experimental measurements.
Source cross-check and editing assisted by AI. No independent human scientific review is recorded for this revision. The check applies to the version dated 10 September 2026; it is not a guarantee that every possible interpretation is correct.
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Sources and further reading
- OpenStax: Calculating moments of inertia — Calculating moments of inertia.
Cite this article
Matan. “Moment of Inertia Examples.” Your Physicist. Originally published 2023-03-17; updated 2026-09-10. https://your-physicist.com/moment-of-inertia-examples/
Revision history
: Replaced the incorrect hollow-cylinder formula and added explicit axes and a checked comparison. Added linked sources and a transparent verification record.
