What Is Angular Velocity?

A circle in radians (a circle is 2, that is: 360 degrees = 2), the arc radian traveled in unit time is the angular velocity. The formula is: = / t ( is the radian traveled, t is time) The unit of is: radians per second.

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The Greek name (uppercase) or (lowercase) is usually used as the omega international accent / o'miga /.

Object motion

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Generally speaking, the angular velocity in high-dimensional space is a second-order obliquely symmetrical angular displacement.

Main article: Rigid body dynamics

In order to deal with the problem of rigid body motion, it is best to use a coordinate system fixed on the rigid body, and then learn the coordinate conversion between this coordinate system and the laboratory coordinate system. As shown on the right, O is the origin of the laboratory coordinate system, and O 'is the origin of the rigid body coordinate system, and the vector R between O and O'. The mass point (') is at the position of point P on the rigid body. The vector position of this mass point in the laboratory coordinates is Ri, and the vector position in the rigid body coordinates is ri. We can see that the position of this particle can be written as:

The most important feature of a rigid body is that the distance between any two points does not change with time. This means that the length of the vector is constant. According to the Euler rigid body's finite rotation theorem, we can use it instead, which represents the rotation matrix, but the position of the particle at the initial moment. This substitution seems very meaningful, and only changes over time, not relative vectors. For a rigid body that rotates O ', the position of the mass point can be written as:

Differentiating the speed of the particle with time, you can get the speed of the particle:

Where Vi is the velocity of the particle in laboratory coordinates, and V is the velocity of the O 'point (origin of rigid body coordinates) in laboratory coordinates, so the velocity of the particle can be written as:

is the angular velocity tensor. If we take the dual of the angular velocity tensor, we can get the pseudo vector of angular velocity.

The multiplication of a matrix can be replaced by an outer product, which leads to:

It can be seen that the velocity of a particle in a rigid body can be decomposed into two terms-the velocity of a fixed reference point in the rigid body plus an outer product containing the angular velocity of the particle relative to this reference point. This angular velocity is the "spin" angular velocity compared to the angular velocity of the O 'point with respect to the O point.

It is important that each particle in a rigid body has the same spin angular velocity, which is independent of the choice of origin on the rigid body or in the laboratory coordinate system. In other words, this is the real physical quantity of a rigid body trait, regardless of the choice of the coordinate system. However, the angular velocity of the reference point on the rigid body relative to the origin of the laboratory coordinates is related to the choice of the coordinate system. For convenience, the centroid of the rigid body is usually selected as the origin of the rigid body coordinate system. This will greatly simplify the mathematical form Expression of angular momentum.