Moment of Inertia Calculator
Moment of inertia (I) measures how much an object resists rotational acceleration, just like mass resists linear acceleration. A 5 kg solid disk with a 0.5 m radius has I = 0.625 kg*m^2. If you rearrange that same mass into a hollow ring, I jumps to over 1.0 kg*m^2 because more mass is far from the axis. This is why figure skaters spin faster when they pull their arms in.
The moment of inertia formulas for common shapes about their standard axes:
| Shape | Formula | Notes |
|---|---|---|
| Solid Cylinder/Disk | I = (1/2)MR2 | About the central axis |
| Hollow Cylinder | I = (1/2)M(R12 + R22) | R1 = inner, R2 = outer |
| Solid Sphere | I = (2/5)MR2 | About any diameter |
| Hollow Sphere | I = (2/3)MR2 | Thin spherical shell |
| Thin Rod (center) | I = (1/12)ML2 | About the midpoint |
| Thin Rod (end) | I = (1/3)ML2 | About one end |
The radius of gyration (k = sqrt(I/M)) tells you the effective distance at which you could concentrate all the mass and get the same moment of inertia. It is useful for comparing objects of different shapes and sizes.
For compound objects, the parallel axis theorem lets you calculate I about any axis: I = Icm + Md2, where d is the distance from the center of mass to the new axis. This is how engineers calculate the moment of inertia for complex machine parts.