The moment of inertia is the rotational equivalent of mass: it tells you how hard it is to change an object’s rotation. It depends not only on mass but on how far that mass is from the axis – a hoop is twice as hard to spin up as a disk of the same mass and radius.
How to use the Mass Moment of Inertia Calculator
- Shape
- Mass m
- Second mass m₂
- Separation d
- Radius r (outer / major radius R for the torus)
- Inner radius (tube radius r for the torus)
- Length L
- Side or semi-axis a
- Side or semi-axis b
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose the shape; only its dimensions appear. Axes pass through the centre of mass unless stated (the rod can also be turned about its end).
Inputs explained
| Field | What to enter |
|---|---|
| Shape | Choose one: Point mass at distance r, Two point masses (separated by d, about the centre of mass), Thin rod about its centre, Thin rod about one end … |
| Mass m | Units: kg, g, mg, t, lb, oz |
| Second mass m₂ | Units: kg, g, mg, t, lb, oz |
| Separation d | Units: m, cm, mm, km, in, ft … |
| Radius r (outer / major radius R for the torus) | Units: m, cm, mm, km, in, ft … |
| Inner radius (tube radius r for the torus) | Units: m, cm, mm, km, in, ft … |
| Length L | Units: m, cm, mm, km, in, ft … |
| Side or semi-axis a | Units: m, cm, mm, km, in, ft … |
| Side or semi-axis b | Units: m, cm, mm, km, in, ft … |
Example values
How the main result changes when Mass m changes and everything else stays at the example values above:
| Mass m (kg) | Moment of inertia I (kg·m²) |
|---|---|
| 1 | 0.045 |
| 1.5 | 0.0675 |
| 2 | 0.09 |
| 2.5 | 0.1125 |
| 3 | 0.135 |
| 4 | 0.18 |
Formula
Point mass m·r² · Rod (centre) mL²/12 · Disk ½mr² · Hoop mr² · Solid sphere ⅖mr² · Thin shell ⅔mr²
Cuboid m(a² + b²)/12 · Solid cone 3/10·mr² · Torus m(R² + ¾r²) · Ellipsoid m(a² + b²)/5
Worked example
Inputs — Shape: Solid cylinder or disk about its axis; Mass m: 2 kg; Radius r (outer / major radius R for the torus): 0.3 m.
Results — Moment of inertia I = 0.09 kg·m²; Radius of gyration k = √(I/m) = 0.21213 m.
Key terms
- Moment of inertia
- Resistance to angular acceleration, ∫r² dm.
Frequently asked questions
Why is a hoop harder to spin than a disk?
All its mass is at the full radius.
What is the radius of gyration?
The distance at which all the mass could be concentrated to give the same I.
How do I find I about another axis?
Use the parallel axis theorem: I = I_cm + m·d².
What are the units?
kg·m².