Rotational Kinetic Energy: Work, Energy, and Power

The rotational kinetic energy is the kinetic energy due to the rotation of an object and is part of its total kinetic energy.

Learning Objective

Express the rotational kinetic energy as a function of the angular velocity and the moment of inertia, and relate it to the total kinetic energy

Key Points

Rotational kinetic energy can be expressed as: $E_{rotational} = \frac{1}{2}I\omega ^{2}$where $\omega$ is the angular velocity and $I$is the moment of inertia around the axis of rotation.
The mechanical work applied during rotation is the torque times the rotation angle: $W = \tau \theta$.
The instantaneous power of an angularly accelerating body is the torque times the angular velocity: $P = \tau \omega$.
There is a close relationship between the result for rotational energy and the energy held by linear (or translational) motion.

Terms

inertia
The property of a body that resists any change to its uniform motion; equivalent to its mass.
torque
A rotational or twisting effect of a force; (SI unit newton-meter or Nm; imperial unit foot-pound or ft-lb)
angular velocity
A vector quantity describing an object in circular motion; its magnitude is equal to the speed of the particle and the direction is perpendicular to the plane of its circular motion.

Full Text

Rotational 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 around an object's axis of rotation yields the following dependence on the object's moment of inertia:

Kinetic Energy of Rotation

Things that roll without slipping have some fraction of their energy as translational kinetic and the remainder as rotational kinetic. The ratio depends on the moment of inertia of the object that's rolling.

$E_{rotational} = \frac{1}{2}I\omega ^{2}$,

where $\omega$ is the angular velocity and $I$ is the moment of inertia around the axis of rotation.

The mechanical work applied during rotation is the torque ($\tau$) times the rotation angle ($\theta$): $W = \tau \theta$.

The instantaneous power of an angularly accelerating body is the torque times the angular velocity: $P = \tau \omega$.

Note the close relationship between the result for rotational energy and the energy held by linear (or translational) motion:

$E_{translational} = \frac{1}{2}mv ^{2}$.

In the rotating system, the moment of inertia takes the role of the mass and the angular velocity takes the role of the linear velocity.

As an example, let us calculate the rotational kinetic energy of the Earth (animated in Figure 1 ). 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.

The Rotating Earth

The earth's rotation is a prominent example of rotational kinetic energy.

This can be partially 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. 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.

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