Let a particle move in a circle around the particle O in the plane Oxy. If at time t, the particle is at point A, and the radius OA forms an angle θ with the Ox axis, which is called the angular position. At time t+Δ t, the particle reaches point B, and the radius OB forms an angle θ+Δ θ with the Ox axis. That is to say, in Δ t time, the particle turns over the angle Δθ, which is called the angle Δθ of the particle to point O angular displacement 。 Angular displacement has not only magnitude but also steering. Generally, it is specified that the angular displacement in counterclockwise direction shall be taken as positive value, and the angular displacement in clockwise direction shall be taken as negative value. The limit value of the ratio of angular displacement Δθ to time Δ t when Δ t approaches zero is
ω It is called the instantaneous angular velocity of t particle to O point at a certain time (referred to as angular velocity for short). [3]
When circular radius When the same, Central angle The larger θ is, the longer the arc of the circle it corresponds to is proportional to the two. Therefore, the ratio of the arc length to the radius can be used to represent the size of the center angle of the circle. For example, if the arc length is 0.12m and the radius is 0.1m, then θ=0.12m ÷ 0.1m=1.2
The unit of arc length and radius is meters, which should be eliminated when calculating the ratio of the two. For the convenience of expression, θ has a unit: radian , represented by the symbol rad. Thus, the angle θ calculated above is 1.2 radians, recorded as θ=1.2 rad [1] For a circle, θ=2 π rad=360 °, then
The unit of angular displacement is rad, and the unit of angular velocity is s -1 Or rad/s [3]
Angular velocity is a vector. Press Right hand spiral rule , thumb direction is ω Direction. When the particle makes Rotate counterclockwise When, ω Upward; do Clockwise rotation When, ω Down. Set the linear speed as v , take the center of the circle as the origin, and set the position vector（ Position vector ）For r , then v = ω × r
This formula can be used as a general definition of angular velocity. [2]
The angular coordinate φ and the angular displacement Δφ are not vectors. Let Δ t → 0, then the angular displacement Δφ takes zero as the limit, which is called infinitesimal angular displacement. The infinitesimal angular displacement is called differential angular displacement after ignoring the high-order infinitesimal quantity, which is recorded as dφ . It can be proved that, dφ Is a vector. Further, angular velocity ω = dφ /Dt is also a vector. [4]
angular velocity ω yes Pseudo vector 。 [5] When the right-handed system is changed to the left-handed system, the angular velocity is reversed. Its essence is second-order tensor （ Ω ）The essence of general vector is a first-order tensor, so vector is a simple expression of angular velocity, and tensor is an accurate expression of angular velocity. [6] 
