Reference worksheet · fixed-axis rigid-body model
Uniform-Shape Axial Mass-Moment-of-Inertia Worksheet
Calculate mass moment of inertia about the stated centroidal spin axis for one uniform ideal solid cylinder/disc, annular cylinder, or solid sphere.
Controlled inputs
Use documented mass when available. The density helper is only for a uniform homogeneous shape. Changing a unit clears its paired numeric field and the previous result, so a number is never silently reinterpreted.
Reference result
Equations, units and model boundary
Mass moment of inertia about a specified fixed axis is J = ∫r⊥²dm, where r⊥ is perpendicular distance from each mass element to that axis. The same body generally has different inertia about a different axis.
Centroidal axial formulas used here
R = D/2 and Ri = d/2. The cylinder formulas are only for the central longitudinal symmetry axis. The sphere formula applies to a diameter through its center.
Optional uniform-density helper
The helper derives mass only for a homogeneous full solid/annular cylinder or solid sphere. Length does not appear in axial J once mass is known, but in density mode it changes mass and therefore J.
Exact unit normalization
The worksheet calculates in kg·m² and reports exact derived equivalents. In particular, 1 kg·m² = 10⁹ g·mm²; 1 in = 0.0254 m and 1 lbm = 0.45359237 kg exactly. “lbm·ft²” is used, not the ambiguous label “lb·ft²”.
Offset and composite rotors
The parallel-axis theorem is J = Jcm + m d⊥², where d⊥ is the perpendicular distance between parallel axes. It is not a substitute for defining the final common axis. Composite rotors may require signed material/void components, coordinate transformations, CAD mass properties or direct inertia measurement; this worksheet does not sum components.
Independent published check
University Physics (UCF/OpenStax) gives a coaxial system consisting of a 2.0 kg, 0.50 m-radius disk and a 1.0 kg annular cylinder with 0.20/0.30 m inner/outer radii; its published total is 0.315 kg·m². Two documented-mass worksheet runs give 0.250 kg·m² and 0.065 kg·m², whose controlled sum reproduces 0.315 kg·m². This checks the formulas; it does not authorize using this single-component worksheet as a general rotor assembly model.
Source classification
| Claim | Ταξινόμηση | Inspected source |
|---|---|---|
| J = ∫r⊥²dm; disk/cylinder ½mR²; sphere ⅖mR²; axis dependence | General classical mechanics; not an ISO formula | MIT OCW 8.01L lecture 28, rendered page 2 |
| Annular cylinder ½m(Ro²+Ri²) | Integrated uniform thick cylindrical shell | University of Illinois Engineering Dynamics Reference |
| Parallel-axis J = Jcm + md⊥² | General classical mechanics; perpendicular separation between parallel axes | MIT OCW 8.01L lecture 28, rendered page 2 |
| 0.315 kg·m² disk-plus-annulus check | Published university textbook example | UCF Pressbooks / OpenStax University Physics, section 10.4 |
| inch, pound-mass and lbm·ft² conversion basis | Metrology / unit conversion | NIST SP 811 Appendix B.8 |
- MIT OCW 8.01L Physics I, lecture 28 — definition, fixed-axis condition, common-shape formulas, axis dependence and parallel-axis theorem.
- University of Illinois Engineering Dynamics Reference, moments of inertia — thick cylindrical shell integration and axial annular-cylinder formula.
- UCF/OpenStax University Physics 10.4 — definition, axis dependence and published disk-plus-annular-cylinder example.
- NIST SP 811 Appendix B.8 — inch, pound-mass, pound-foot-squared and pound-inch-squared conversion basis.
Accessed: 14 July 2026. The cited MIT page was rendered and visually inspected. No ISO or API acceptance criterion is implemented.
Questions this worksheet does not answer
Is this the inertia of my complete rotor?
Only if its actual mass distribution about the stated axis is represented by this one ideal uniform shape. Real rotors commonly require CAD/FE mass properties, a controlled component sum or direct measurement.
Can I use the result for transverse bending or rocking?
No. The cylinder/disc result here is about the central longitudinal spin axis. Transverse moments contain different radius and length terms.
Why are material presets absent?
Density depends on the controlled alloy, porosity, processing state and temperature. Enter a value from the actual component specification or measurement record.
