- is rotational kinetic energy,
- is angular velocity,
- is moment of inertia around the axis of rotation.
The mechanical work required for applied during rotation is the torque times the rotation angle. The instantaneous power of an angularly accelerating body is the torque times the angular velocity. For free-floating (unattached) objects, the axis of rotation is commonly around its center of mass.
Note the close analogy between the formula for rotational energy and that for linear (or translational) kinetic energy:
In the rotating system, the moment of inertia takes the role of mass m, and the angular velocity that of linear velocity v. The rotational energy of a rolling cylinder varies from one half of the translational energy (if it is solid) to the same as the translational energy (if it is hollow).
An example is the calculation of the rotational kinetic energy of the Earth. With a period of about 23.93 hours, it has an angular velocity of 7.29×10−5 rad/s. Its moment of inertia is = 8.04×1037 kg·m2. Therefore, it has a rotational kinetic energy of 2.138×1029 J.
An example of using Earth's rotational energy is the launching of rockets. The European spaceport in French Guiana is about 5 degrees from the equator, so satellite launches from here to orbit eastward are sling-shot with nearly the full rotational speed of the earth at the equator (about 1,000 mph). Rocket launches easterly from Cape Canaveral obtain only about 400 mph added benefit, due to the lower relative rotational speed of the earth. This makes Guiana the more economic spaceport.
Part of the earth's rotational energy can also be tapped using tidal power.
Friction of the two global tidal waves slightly slows Earth's angular velocity ω. Due to the conservation of angular momentum, this transfers angular momentum to the Moon's orbital motion, increasing its distance from Earth and its orbital period (see tidal locking).