Angular Velocity

For rotational motion about a point or axis, angular velocity is the rate of change of the angular position with time, or in other words the derivative of the angular position with time:

$ \omega = \frac{d\theta}{dt}  $ = angular velocity, where θ = the angular position
t = time

The direction of the angular velocity vector is perpendicular to the plane of rotation as given by the right-hand rule. The angular velocity is expressed in units of [angular distance/time], often radians per second.

For an object moving in a curved path it can be useful to describe the motion using both angular and linear velocities. Using a fundamental relation for circular geometry:

$ \theta = \frac{s}{r} $
where θ = angle
s = arc subtended by θ
r = radius of the circle

The magnitudes of the linear and angular velocities are related by:

$ v = \frac{ds}{dt} , \omega = \frac{d\theta}{dt} \rightarrow v = r\omega  $ where v = linear velocity
r = distance from the axis of rotation
ω = angular velocity

Note that we are not using vector notation in this expression, rather it is the magnitude of the velocities that follow the relation.

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