What Is Angular Velocity?
The angular velocity of an object rotating about an axis is a quantity that describes the angular speed of that object. Angular velocity is considered to be a vector quantity, though it is often referred to as a pseudovector. The SI unit for angular velocity is the radian per second, and the symbol for angular velocity is typically ω (lower case Greek letter omega). However, in many cases, angular velocity is put into terms that are more relevant to the system in question. These units typically include degrees per hour, revolutions per second, and rotations per minute.
The angular velocity vector always runs perpendicular to the plane in which the object is rotating. One can determine the direction of a single angular velocity vector by using the right hand rule convention. That is to say, if we know the direction of the angular velocity vector, ω, we can then find out whether the object is rotating clockwise or counter-clockwise through the application of this rule. Curl your fingers (almost like a fist) and imagine that the vector ω points upward toward the ceiling and runs through your right hand with your thumb also pointed toward the ceiling. According to the right hand rule, the object will be rotating counter-clockwise about the origin going in the direction of your curled fingers. Similarly, if you were to do the same and point your right thumb downwards, the vector ω will point downward and the object will be rotating clockwise. You can also use this rule to determine the direction of the vector ω so long as you assume the direction of rotation begins from the center of the hand and moves outward to the tips of the fingers.
In two dimensions, for a given particle, p, that is rotating about some kind of origin with an angular position ϕ, that particle has two different components of its velocity V. There is a V_perpendicular that is always perpendicular to the radius between p and the origin, and there is also a V_parallel that runs in the same direction as the radius. These two velocity vectors are commonly referred to as the cross-radial and radial components respectively. The rate of change of the angular position of the particle p, (dϕ/dt) is functionally related to vperpendicular by the following rule:
vperpendicular = r*(dϕ/dt)
In our system, there is also an angle θ such that θ is the angle between the vectors V and v_parallel (there is also another angle between vperpendicular and V such that this angle and θ adds to 90 degrees). According to this definition, vperpendicular is related to θ by:
vperpendicular = |V|*sin(θ)
where |V| is the magnitude of the velocity. We also assume that the vector quantity ù representing the angular velocity of p is also the same as the change in angular position with respect to time (a.k.a dϕ/dt). In other words:
ù = dϕ/dt
Now, knowing this relationship, we can define ω in terms of the magnitude of V by substitution, yielding the following:
ω = (|V|*sin(θ))/|r|
In two dimensions, ω has no direction. That said, if the x and y axes are interchanged, ω changes its sign as well, meaning it is a pseudoscalar. In three dimensions, however, ω is a pseudovector and the right hand rule applies.