What this calculator does
Rotational kinetic energy is the energy stored in a spinning object because of its rotation, separate from any kinetic energy it has from moving in a straight line. A flywheel sitting still on a fixed axle can still hold a large amount of energy purely from spinning, and this calculator works out how much from its moment of inertia and angular velocity.
The formula, KE = ½Iω², mirrors the familiar linear kinetic energy formula ½mv², but swaps mass for moment of inertia (how the object's mass is distributed around the axis of rotation) and velocity for angular velocity (how fast it is spinning, in radians per second). Because ω is squared, doubling the spin rate quadruples the stored energy, which is why flywheels and turbines are so sensitive to running speed.
The formula
Multiply the moment of inertia by the square of the angular velocity, then take half of that product. Moment of inertia depends on both the mass of the object and how far that mass sits from the axis of rotation, so the same mass spinning further from the axis stores more energy at the same angular velocity.
| Term | Meaning |
|---|---|
| I | Moment of inertia, in kg·m², a measure of how hard the object is to spin, given its mass distribution around the axis. |
| ω (omega) | Angular velocity, in radians per second, how fast the object is rotating. |
| KE | Rotational kinetic energy, in joules: KE = ½Iω². |
The inputs explained
| Field | What to enter |
|---|---|
| Moment of inertia (I) (kg·m²) | The moment of inertia of the rotating object about its axis, in kilogram metres squared. This depends on shape as well as mass, for example a solid disc and a thin ring of the same mass have different values. |
| Angular velocity (ω) (rad/s) | The angular velocity, in radians per second. To convert from rpm, multiply by 2π and divide by 60. |
When to use it
Sizing a flywheel for energy storage
Flywheel energy-storage systems are designed around exactly this formula: for a given moment of inertia, spinning faster stores far more energy, because the energy grows with the square of angular velocity rather than in direct proportion to it.
Checking a physics problem
Rotational kinetic energy questions commonly give a moment of inertia (sometimes needing to be derived from a standard shape formula first) and an angular velocity, then ask for the stored energy or how it changes if the spin rate changes.
Comparing spin-up energy needs
Working out the rotational kinetic energy at a target angular velocity shows how much energy a motor needs to supply to bring a rotating part up to that speed from rest.
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 rotational kinetic energy changes with angular velocity
A fixed moment of inertia of 2 kg·m², across a range of angular velocities.
| Angular velocity | Rotational kinetic energy |
|---|---|
| 5 rad/s | 25.000 J |
| 10 rad/s | 100.000 J |
| 15 rad/s | 225.000 J |
| 20 rad/s | 400.000 J |
| 30 rad/s | 900.000 J |
| 50 rad/s | 2,500.000 J |
How rotational kinetic energy changes with moment of inertia
A fixed angular velocity of 10 rad/s, across a range of moment of inertia values.
| Moment of inertia | Rotational kinetic energy |
|---|---|
| 1 kg·m² | 50.000 J |
| 2 kg·m² | 100.000 J |
| 4 kg·m² | 200.000 J |
| 6 kg·m² | 300.000 J |
| 10 kg·m² | 500.000 J |
| 20 kg·m² | 1,000.000 J |
Questions
How is rotational kinetic energy different from linear kinetic energy?
Linear kinetic energy (½mv²) is the energy of an object moving from one place to another. Rotational kinetic energy (½Iω²) is the energy of an object spinning in place. A rolling object, such as a wheel, has both at once, and its total kinetic energy is the sum of the two.
Why does moment of inertia depend on shape, not just mass?
Moment of inertia depends on how far each bit of mass sits from the axis of rotation, and mass further out contributes more. A ring and a solid disc of identical total mass and radius have different moments of inertia because the ring has all its mass at the rim.
How do I find the rotational kinetic energy formula for a specific shape?
Standard reference tables give the moment of inertia formula for common shapes about a given axis, such as ½MR² for a solid disc or MR² for a thin ring. Once you have that value of I, this calculator applies KE = ½Iω² directly.
What happens to rotational kinetic energy if angular velocity doubles?
It quadruples, since the formula depends on ω squared. Doubling the spin rate of a flywheel or rotor takes roughly four times the energy input to reach, all else being equal.
For the related quantity that depends on ω rather than ω², see the angular momentum calculator. To convert between rpm, frequency and angular velocity in rad/s, use the angular frequency calculator.