Mass for rotation
When you push a shopping cart, its mass resists the change in speed. When you spin a wheel, the resistance depends not only on the mass but on where it sits: mass far from the axis is much harder to set spinning, and much harder to stop. This rotational resistance is the moment of inertia, I, measured in kg·m² or lb·ft²:
I = Σ m r²
Each bit of mass counts with the square of its distance from the axis. Doubling the radius of a wheel of the same mass multiplies its moment of inertia by four.
Formulas for common shapes
| Shape | Moment of inertia |
|---|---|
| Solid cylinder or disk, about its axis | ½ m r² |
| Thin ring or thin-walled tube | m r² |
| Thick-walled tube | ½ m (r₁² + r₂²) |
| Solid sphere | ⅖ m r² |
| Thin spherical shell | ⅔ m r² |
| Rod about its center | m L² ÷ 12 |
| Rod about one end | m L² ÷ 3 |
| Rectangular plate about its center | m (a² + b²) ÷ 12 |
Worked example
A steel flywheel shaped like a solid disk weighs 20 lb and has a 6 inch radius:
- I = ½ × 20 lb × (0.5 ft)² = 2.50 lb·ft², or 0.1054 kg·m².
- Radius of gyration: 0.707 × 6 in = 4.24 in.
- Spinning at 3,000 rpm (314.2 rad/s), it stores ½ × 0.1054 × 314.2² ≈ 5,199 joules, enough to lift a 100 lb weight about 38 feet.
The same mass formed into a thin rim of the same radius would have twice the moment of inertia and store twice the energy.
Where it matters
- Engines and flywheels: a heavier flywheel smooths out the pulses of a piston engine; a lighter one lets the engine rev faster.
- Sports: a figure skater pulls in their arms to cut their moment of inertia and spin faster, because angular momentum I × ω is conserved.
- Bicycles and cars: rotating mass in wheels takes extra energy to accelerate, which is why light rims feel livelier.
- Engineering design: motor sizing for conveyors, fans and robots depends on the inertia of the load.
Notes
The formulas assume uniform density and an axis through the center of mass, except the rod about its end. For another parallel axis, add m d², where d is the distance between the axes (the parallel axis theorem). This calculator gives the mass moment of inertia, not the area moment of inertia used for beam bending.