Calculator
Example data table
| Case | Body | Inputs | Formula | I (kg·m²) |
|---|---|---|---|---|
| 1 | Solid disk | m = 10 kg, r = 0.20 m | I = 1/2 m r² | 0.20000000 |
| 2 | Slender rod (center) | m = 4 kg, L = 1.50 m | I = 1/12 m L² | 0.75000000 |
| 3 | Solid sphere | m = 6 kg, r = 0.12 m | I = 2/5 m r² | 0.03456000 |
| 4 | Rectangular plate (perpendicular) | m = 8 kg, a = 0.40 m, b = 0.30 m | I = 1/12 m (a² + b²) | 0.16666667 |
| 5 | Two point masses | m₁=2 kg at 0.50 m, m₂=3 kg at 0.20 m | I = Σ mᵢ rᵢ² | 0.59000000 |
Formulas used
Mass moment of inertia measures how strongly a body resists angular acceleration about an axis.
- Point masses: I = Σ(mᵢ rᵢ²)
- Parallel-axis theorem: Iref = Icm + m d²
- Solid cylinder (axis): I = 1/2 m r²
- Solid cylinder (diameter): I = (1/12) m (3r² + L²)
- Thin-walled cylinder (axis): I = m r²
- Thin-walled cylinder (diameter): I = (1/12) m (6r² + L²)
- Solid sphere: I = 2/5 m r²
- Thin spherical shell: I = 2/3 m r²
- Slender rod: Icenter = 1/12 m L², Iend = 1/3 m L²
- Rectangular plate: Iz = 1/12 m (a² + b²), Ix = 1/12 m b², Iy = 1/12 m a²
- Rectangular solid: I = 1/12 m (sum of squares of the other two sides)
Tip: If your axis is not through the center, enter an offset distance d.
How to use this calculator
- Select Calculation mode: shape, point masses, or custom inertia.
- Pick output units to match your workflow.
- For shapes, choose a shape and axis, then enter dimensions.
- Enter mass directly, or switch to density mode to compute mass.
- If your axis is shifted, enter offset distance d to apply the parallel-axis theorem.
- Press Calculate. The result appears above the form.
- Use Download CSV or Download PDF to save outputs.
What moment of inertia represents
Moment of inertia (I) is the rotational “mass distribution” about a chosen axis. It links applied torque to angular acceleration through τ = Iα. For the same mass, placing material farther from the axis increases I and makes the object harder to spin up or slow down. A higher I also means more rotational kinetic energy at the same speed. Overall.
Units and scaling rules
SI inertia uses kg·m². If you double every length dimension, inertia increases by the square of that scale, even if mass stays constant. If mass doubles with the same geometry, I also doubles. These scaling rules help you sanity‑check results before exporting a report.
Common shapes and quick formulas
Many rotating parts match standard bodies: solid cylinders and disks, thin rings, spheres, rods, plates, and rectangular solids. The calculator selects the correct closed‑form formula for the chosen body and axis, then applies unit conversion so your output is consistent.
Choosing the rotation axis
The axis matters as much as the shape. A cylinder about its symmetry axis has a much smaller inertia than the same cylinder about a central diameter. If your axis is shifted away from the centroid, use the parallel‑axis theorem with offset distance d to account for the extra m d² term.
Using density to estimate mass
When mass is unknown, density mode estimates mass from material density and geometric volume. This is useful for early design, quick comparisons, or when you only have dimensions and a material spec. It also helps compare candidate materials without rebuilding the model each time.
Engineering uses and checks
Use inertia in motor sizing, flywheel energy storage, robotics joints, drivetrain dynamics, vibration studies, and braking calculations. Combine I with ω and α to estimate torque demand and transient response. For gears and pulleys, reflect inertia through ratios to see what the motor “feels” at its shaft.
Common mistakes to avoid
Common errors include mixing radius and diameter, using centimeters while expecting meters, selecting the wrong axis, or forgetting an offset for a non‑center axis. For point masses, make sure r is the perpendicular distance to the axis. Recalculate after any unit change, and keep significant figures realistic.
FAQs
Is this the same as area moment of inertia?
No. Mass moment of inertia uses mass distribution for rotational dynamics. Area moment of inertia uses cross‑section geometry for bending and deflection in beam theory.
When do I use the parallel-axis theorem option?
Use it when the rotation axis does not pass through the object’s centroid. Enter the offset distance d to add the m d² term.
Can it handle multiple parts or assemblies?
Yes, approximate an assembly by summing inertias of parts about the same axis. Compute each part separately, then add results. Use offsets to shift each part’s inertia to the common axis.
Why does the numeric value change when I switch units?
The physical inertia is the same, but the numeric value depends on units. For example, lbm·ft² uses larger length units than kg·m², so the number differs after conversion.
How accurate is density mode?
It is only as accurate as your dimensions and material density. It ignores holes, fillets, and non‑uniform density unless you model them as separate parts and sum their inertias.
What radius should I enter for point masses?
Enter the perpendicular distance from the axis of rotation to each mass. If the mass moves on a circle around the axis, r is that circle’s radius.