How to use it
- Choose the shape and the axis it turns about.
- Enter the mass and the dimensions the shape needs: radius, inner radius, length or width.
- To use an axis parallel to the usual one, enter its distance d from the centre of mass (0 keeps the usual axis).
- Read the moment of inertia in kg·m², g·cm² and lb·ft², and the radius of gyration.
Formulas
| Shape | Moment of inertia |
|---|---|
| Solid disk or cylinder, central axis | ½ m r² |
| Thin ring or thin-walled tube | m r² |
| Thick-walled tube, radii r₁ and r₂ | ½ m (r₁² + r₂²) |
| Solid sphere | ⅖ m r² |
| Thin spherical shell | ⅔ m r² |
| Rod about its centre | m L² ÷ 12 |
| Rod about one end | m L² ÷ 3 |
| Plate a × b, axis perpendicular through the centre | m (a² + b²) ÷ 12 |
| Point mass at distance r | m r² |
Parallel axis (Steiner) theorem: I = I_cm + m d². Radius of gyration: k = √(I ÷ m).
Worked example
A 2 kg solid disk, radius 50 cm: I = ½ × 2 × 0.5² = 0.25 kg·m² (2,500,000 g·cm², 5.93259 lb·ft²); k = √(0.25 ÷ 2) = 0.3536 m.
A 3 kg rod 2 m long about its centre has I = 3 × 2² ÷ 12 = 1 kg·m². Move the axis 1 m to the end: I = 1 + 3 × 1² = 4 kg·m², the same as mL²/3.
Tips
- Use the radius, not the diameter. Entering the diameter in the radius box makes the result four times too large.
- The offset applies only to axes through the centre of mass. For a rod about its end, pick that shape directly.
- Mass far from the axis dominates. A flywheel stores more energy with a heavy rim than with the same mass spread over a disk.