What this calculator does
Moment of inertia is the rotational equivalent of mass: it measures how hard something is to spin up or slow down. Unlike mass it depends not just on how much material there is but on how far that material sits from the axis.
That distance matters enormously, because it enters squared. A hoop with all its mass at the rim has more than twice the moment of inertia of a solid disk of the same mass and radius, which is why a hoop and a disk released together down a slope do not arrive at the same time.
The formula
Each shape has a coefficient that multiplies mass times radius squared: two fifths for a solid sphere, two thirds for a thin shell, a half for a solid cylinder, one twelfth for a rod about its centre, a third for a rod about one end, and one for a hoop.
| Term | Meaning |
|---|---|
| Moment of inertia (I) | Resistance to angular acceleration, in kilogram metres squared. |
| Shape coefficient | The fraction in front of mr², determined by how the mass is distributed relative to the axis. |
| Rotational kinetic energy | Half the moment of inertia times angular velocity squared, the rotational counterpart of ½mv². |
The inputs explained
| Field | What to enter |
|---|---|
| Shape | The shape and the axis it rotates about. The same object has different moments of inertia about different axes. |
| Mass (kg) | The mass in kilograms. |
| Radius (or length, for a rod) (m) | The radius in metres, or the length for a rod. |
| Angular velocity (0 to skip) (rad/s) | Angular velocity in radians per second, used only for the kinetic energy figure. Leave at zero to skip it. |
When to use it
Comparing shapes that roll
Objects with more mass near the rim roll more slowly down a slope, because more of the available energy goes into spinning rather than moving.
Sizing a flywheel
A flywheel stores energy in rotation, and a large moment of inertia is exactly what makes it effective.
Working out the torque needed
Angular acceleration is torque divided by moment of inertia, so the inertia has to be known before a motor can be sized.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does shape change the moment of inertia?
The same mass and radius distributed in different shapes.
| Shape | Moment of inertia | Rotational kinetic energy |
|---|---|---|
| Rod about centre | 0.015000 kg·m² | 0.7500 J |
| Solid sphere | 0.072000 kg·m² | 3.600 J |
| Solid cylinder | 0.090000 kg·m² | 4.500 J |
| Thin shell | 0.120000 kg·m² | 6.000 J |
| Thin hoop | 0.180000 kg·m² | 9.000 J |
Questions
Why does a hoop roll down a slope more slowly than a disk?
Because more of its mass sits far from the axis, so a larger share of the gravitational energy goes into spinning rather than into forward motion. The disk converts more of it into speed down the slope.
Why does the axis matter so much?
Because the distances are measured from that axis. A rod about its centre has a moment of inertia four times smaller than the same rod about one end, since the mass is on average much closer to the pivot.
Does it depend on how fast it is spinning?
No. Moment of inertia is a property of the object and the chosen axis alone. Speed only enters when calculating angular momentum or rotational kinetic energy.
What is the parallel axis theorem?
A rule for shifting the axis: the moment of inertia about any parallel axis equals the value about the centre of mass plus mass times the square of the offset. It is how the rod-about-the-end figure follows from the rod-about-the-centre one.
For the angular momentum that inertia produces, see the angular momentum calculator. For the torque needed to spin it up, see the torque calculator.