Rotational energy
Rotation energy the kinetic energy of a rigid body (for example, flywheel ) which rotates about a fixed axis. This energy depends on the moment of inertia and the angular velocity of the body. The more weight is removed from the axis of rotation, the more energy is required to move a body in a particular rotational speed.
This can be illustrated by the following experiment: two identical heavy spheres with identical radii are placed on an inclined plane and roll down. A ball is made of a lightweight material such as plastic and is made solid. The other sphere but is hollow, but consists of a denser and therefore heavier material than plastic. The hollow ball is rolling slowly because with her the entire mass is distributed in a thin shell at a certain distance to the rotation axis. The massive sphere with the same mass rolls faster because a greater percentage of mass is close to the rotation axis and must therefore move more slowly on the circular path.
Rotational energy is at, among other things of importance: turbines, generators, wheels and tires, shafts, propellers.
Moment of inertia
A with the angular velocity around the x- axis rotary body having the rotational energy
With
- : Moment of inertia of the body about the x- axis
- : Angular velocity
In general, this can be expressed as
With
- : inertia tensor
- : Angular velocity
To indicate the energy of about any axis ( unit vector ), the rotating body, the angular speed will be expressed by its vector components:
Wherein the components are the components of n in the x-, y -and z- axis direction. Now applies to the rotational energy:
This is the moment of inertia with respect to an arbitrary axis
Examples
- A sphere with radius, the moment of inertia. If she rolls with the speed on the level, is its angular velocity, and consequently their total kinetic energy:
- A body that rotates around its diagonal xy surface, has the following angle speed:
Angular momentum
The rotational energy can be expressed by the angular momentum.
It should be noted that, in general, the angular momentum and the angular speed are not parallel to each other (except for rotation about a principal axis of inertia ); see also ellipsoid of inertia.