Definitions

Translational energy

Rotational energy

The rotational energy or angular kinetic energy is the kinetic energy due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately in an object's centre of mass frame, one gets the following dependence on the object's moment of inertia:

E_{rotation} = frac{1}{2} I omega^2
where
omega is the angular velocity
I is the moment of inertia.

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 frequency.

Note the close relationship between the results for linear (or translational) and rotational motion; the formula for the

E_{translational} = frac{1}{2} m v^2

In the rotating system, the moment of inertia, I, takes the role of the mass, m, and the angular velocity, omega , takes the role of the linear velocity, v. The rotational energy of a rolling cylinder varies from one half of the translational energy (if it is massive) to the same as the translational energy (if it is hollow).

As an example, let us calculate the rotational kinetic energy of the Earth. As the Earth has a period of about 23.93 hours, it has an angular velocity of 7.29×10-5 rad·s-1. Assuming that the Earth is perfectly spherical and uniform in mass density, it has a moment of inertia, I = 9.72×1037 kg·m2. Therefore, it has a rotational kinetic energy of 2.58×1029 J.

Part of it can be tapped using tidal power. This creates additional friction of the two global tidal waves, infinitesimally slowing down Earth's angular velocity ω. Due to conservation of angular momentum this process transfers angular momentum to the Moon's orbital motion, increasing its distance from Earth and its orbital period (see tidal locking for a more detailed explanation of this process).

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