Angular velocity

The angular velocity is a vector quantity of physics, which indicates how quickly change with time, an angle about an axis. Your formula character ( little omega ). The SI unit of angular velocity is. It plays a role especially in rotations and is then referred to as rotation speed or rotational speed. In many cases, where the direction of the rotation axis in the reference system does not change, is sufficient to use as a scalar value of the vector.

Definition

The angular velocity is a pseudo vector indicating the rotation axis and speed of a rotational movement. The direction of the vector is perpendicular to the plane of rotation, and outputs the rotation direction ( similar to the corkscrew rule). Its amount is obtained by derivation of the rotation angle with respect to time:

The amount of angular velocity is used in processes in which the rotation axis does not change. The cause of non-constant angular velocity is an angular acceleration at a constant angular velocity, the expression simplifies to

Said angle being swept in time.

The angular velocity is independent of the radius, in contrast to the tangential velocity. The path velocity can then be written as a cross product of angular velocity and radius vector:

The cross product can be replaced by a matrix multiplication, as the angular velocity can be generalized to a second order tensor. This can be represented by a skew-symmetric matrix:

Standing angular velocity and radius vector perpendicular to each other, simplifying the cross product of the normal product of sums:

Forms the z -axis is the axis of rotation, so the angular velocity can be written as the following vector:

Demarcation to the angular frequency

Although the angular frequency and the angular velocity are denoted by the same symbols and although they are measured in the same unit, there are two different physical sizes.

The angular velocity is the rate of change of a geometric angle and is used in the context of rotational movements.

The angular frequency, however, is an abstract concept in the context of vibration. A vibration can be represented mathematically by a rotating link (see link model). The angle of the pointer will be referred to as the phase or phase angle. The rate of change of this phase angle is the angular frequency. Thus, it is - as well as the frequency - a measure of how fast a vibration and has - apart from the rotation of said imaginary pointer - nothing to do with a rotary motion.

Angular velocity of the line of sight

Motion in a plane

The velocity vector V of a particle P relative to an observer O can be decomposed in polar coordinates. The radial component of the velocity vector does not change the direction of the line of sight. The relationship between the tangential component and the angular velocity of the line of sight:

It should be noted that the angular velocity of the line of sight from the (arbitrarily ) chosen location of observer dependent.

Spatial movement

In three dimensions, the angular velocity is characterized by both its amount as well as by their direction.

As in the two -dimensional case, the particle has a component of its speed vector in the direction of the radius vector and another perpendicular to it. The plane which passes through the origin and lies in the vertical component of the velocity vector that defines a plane of rotation in which the behavior of the particle appears to a moment as in the two -dimensional case. The axis of rotation is then perpendicular to said plane and defines the direction of the vector of the angular velocity. Radius and velocity vector are assumed to be known. We then have:

Again, that the so calculated angular velocity depends on the (arbitrarily ) chosen location of the observer. One application is the relative movement of objects in astronomy (see self-motion (Astronomy) ).

Components of Euler angles

In the vehicle or aircraft, the orientation of the vehicle-mounted system is relative to the earth-fixed system in Euler angles. Standardized three successive rotations. First about the z- axis of the system g ( yaw angle ), then around the y- axis of the rotated system ( pitch angle ), and finally the x -axis of the body-fixed coordinate system ( roll / roll angle ).

The angular velocity of the body-fixed system results from the angular velocities about these axes.

This basis is not orthonormal. However, the unit vectors can be computed by means of the elementary rotations.

Angular velocity of the rigid body

The rigid body may rotate about any axis. It is shown that the angular velocity is independent of the choice of the reference point on that axis. This means that the angular velocity is an independent feature of the rotating rigid body.

The origin of the laboratory system is O, O1 and O2 during two points to the rigid body and with the speeds. Assume the angular velocity relative to O1 or O2 is or Because point P and O2 each have only one speed, the following applies:

The above two equations yield:

As the point P (and therefore ) is arbitrary, it follows that:

The angular velocity of the rigid body is independent of the choice of the reference point of the axis of rotation. In motor vehicles can thus be measured independently of the installation location of the yaw rate sensor, the yaw rate.

Applications and Examples

The angular velocity occurs in many applications of the equations of physics, astronomy or the art.

• A celestial body that moves at a distance R from the Earth with velocity perpendicular to the line of sight, the sky shows an apparent angular velocity. When meteors ( shooting stars ) it can account for up to 90 ° per second, very close to asteroids or comets can move a few degrees per hour in the sky. In stars, the angular velocity is given in seconds of arc per year and called proper motion.

• After the third Kepler's law, the squares of the periods T of the planet as the cubes of the semi-major axes a of their orbits behave. The angular velocities accordingly behave like ( " Keplerian rotation "). Kepler's law, according to the second, the angular velocity of the planet is dependent on an elliptical orbit with respect to the sun from the respective distance and thus varies along the path. It is greatest when the planet is at perihelion, and smallest when it is at aphelion.

• During the rotation of a rigid body, the angular velocity ω is in contrast to the velocity v of the radius -independent.

• The angular velocity of a rotor in an electric motor which rotates constantly at 3000 revolutions per minute,

• Is the angular frequency of the harmonic oscillation of a pendulum with the amplitude. Then the angular velocity of the pendulum is calculated as a function of time: