What this calculator does
Moment of inertia of standard shapes works out rotational inertia of common rigid shapes, and the resulting rotational kinetic energy. Enter your own figures above and the answer updates as you type: nothing is fixed in the code, so the result reflects exactly the numbers you supply.
The formula this calculator evaluates is printed under the tool and explained below, so you can check the working by hand or reuse it in a spreadsheet.
The formula
FormulaI = c·m·r² with shape coefficient c: solid sphere 2/5, thin spherical shell 2/3, solid cylinder/disk 1/2, thin rod about centre 1/12, thin rod about end 1/3, thin hoop 1
The inputs explained
| Field | What to enter |
|---|---|
| Shape | Choose from Solid sphere (about a diameter), Thin spherical shell (about a diameter), Solid cylinder or disk (about central axis), Thin rod (about its centre), Thin rod (about one end), Thin hoop or ring (about its centre). |
| Mass (kg) | A number, measured in kg. Starts at 2. |
| Radius (or length, for a rod) (m) | A number, measured in m. Starts at 0.3. |
| Angular velocity (0 to skip) (rad/s) | A number, measured in rad/s. Starts at 0. |
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 the answer changes with mass
Every other input is held at the calculator’s starting values while mass varies. Select any row to load that scenario into the calculator.