Rotational kinetic energy

Rotational energy or angular kinetic energy is kinetic energy 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, the following dependence on the object's moment of inertia is observed:





E


r
o
t
a
t
i
o
n
a
l



=


1
2


I

ω

2




{\displaystyle E_{\mathrm {rotational} }={\frac {1}{2}}I\omega ^{2}}
where




ω



{\displaystyle \omega \ }
is the angular velocity




I



{\displaystyle I\ }
is the moment of inertia around the axis of rotation




E



{\displaystyle E\ }
is the kinetic energyThe mechanical work required for or 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 relationship between the result for rotational energy and the energy held by linear (or translational) motion:





E


t
r
a
n
s
l
a
t
i
o
n
a
l



=


1
2


m

v

2




{\displaystyle E_{\mathrm {translational} }={\frac {1}{2}}mv^{2}}
In the rotating system, the moment of inertia, I, takes the role of the mass, m, and the angular velocity,



ω


{\displaystyle \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).
An example is the calculation of 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 = 8.04×1037 kg·m2. Therefore, it has a rotational kinetic energy of 2.138×1029 J.
A good example of actually using earth's rotational energy is the location of the European spaceport in French Guiana. This is within about 5 degrees of the equator, so space rocket launches (for primarily geo-stationary satellites) from here to the east obtain nearly all of the full rotational speed of the earth at the equator (about 1,000 mph, sort of a "sling-shot" benefit). This saves significant rocket fuel per launch compared with rocket launches easterly from Kennedy Space Center (USA), which obtain only about 900 mph added benefit due to the lower relative rotational speed of the earth at that northerly latitude of 28 degrees.
Part of the earth's rotational energy can also be tapped using tidal power. Additional friction of the two global tidal waves creates energy in a physical manner, infinitesimally slowing down Earth's angular velocity ω. Due to the 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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