The
rotational energy or
angular kinetic energy is the
kinetic energyThe kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately around an object's axis of rotation, one gets the following dependence on the object's
moment of inertiaIn classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
:
where
-
is the angular speed
-
is the moment of inertiaIn classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
around the axis of rotation.
-
is the kinetic energyThe kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
.
The
mechanical workIn physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...
required for / applied during rotation is the torque times the rotation angle. The instantaneous
powerIn physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...
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 massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
.
Note the close relationship between the results for linear (or translational) and rotational motion; the formula for the
In the rotating system, the
moment of inertiaIn classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
,
I, takes the role of the mass,
m, and the
angular velocityIn physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
,
, takes the role of the linear velocity,
v. The
rotational energy of a
rollingA wheel is a device that allows heavy objects to be moved easily through rotating on an axle through its center, facilitating movement or transportation while supporting a load, or performing labor in machines. Common examples found in transport applications. A wheel, together with an axle,...
cylinderA cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a 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. The Earth has a moment of inertia, I = 8.04×10
37 kg·m
2. Therefore, it has a rotational kinetic energy of 2.138×10
29 J.
Part of it can be tapped using
tidal powerTidal power, also called tidal energy, is a form of hydropower that converts the energy of tides into useful forms of power - mainly electricity....
. This creates additional friction of the two global tidal waves, infinitesimally slowing down Earth's angular velocity
ω. Due to conservation of
angular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
this process transfers angular momentum to the
MoonThe Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
's
orbitIn physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
al motion, increasing its distance from Earth and its orbital period (see
tidal lockingTidal locking occurs when the gravitational gradient makes one side of an astronomical body always face another; for example, the same side of the Earth's Moon always faces the Earth. A tidally locked body takes just as long to rotate around its own axis as it does to revolve around its partner...
for a more detailed explanation of this process).