Anything that spins — a flywheel, a wheel rolling down a hill, a figure skater pulling in their arms — stores energy in its rotation. This calculator computes that rotational kinetic energy using KE_rot = ½Iω², lets you solve the formula in reverse for moment of inertia or angular velocity, and adds the translational term for objects that both spin and move forward.
How the Rotational Kinetic Energy Calculator works
The calculator applies KE_rot = ½ × I × ω² directly: it multiplies the moment of inertia (I, in kg·m²) by the square of the angular velocity (ω, in rad/s) and halves the result. The formula is the direct rotational analog of the familiar linear kinetic energy formula KE = ½mv² — moment of inertia plays the role of mass, and angular velocity plays the role of linear velocity.
The squared term in ω means rotational kinetic energy grows much faster than angular velocity itself. Doubling the spin rate of a flywheel quadruples the energy it stores, which is why energy-storage flywheels are engineered to spin as fast as their materials safely allow rather than simply being made heavier.
Inputs and what they mean
Moment of inertia (kg·m²): how an object's mass is distributed relative to its axis of rotation. A solid disk, a hoop, and a sphere of the same mass and radius all have different moments of inertia because their mass sits at different average distances from the axis — use the Moment of Inertia calculator to compute I for common shapes before entering it here.
Angular velocity (rad/s): how fast the object is spinning, in radians per second. To convert from RPM, multiply by 2π/60 ≈ 0.1047.
For the Rolling Total tab — mass (kg) and linear velocity (m/s): the object's total mass and the speed of its center of mass moving forward, used to compute the translational term ½mv² that adds to the rotational term.
Limits and edge cases
This calculator assumes a rigid body rotating about a fixed axis (or rolling without slipping, for the Rolling Total tab). It does not account for energy lost to friction, air resistance, or slippage between a rolling object and its surface — real systems dissipate some of this energy as heat, so KE_rot = ½Iω² represents the maximum stored energy, not necessarily the energy recoverable in practice.
On the Solve tab, solving for angular velocity requires a non-negative energy value and a positive moment of inertia; solving for moment of inertia requires a nonzero angular velocity, since a stationary object (ω = 0) stores zero rotational kinetic energy no matter its mass distribution.