Angular speed and linear speed

The angle swept over time

Rather than tracking the distance travelled, we can track the angle theta (in radians) swept by the radius connecting the center of the circle to the moving point. In UCM, this angle increases steadily: we define the angular speed omega as the change in angle per unit of time.

omega = theta / t (in rad/s), for a motion at constant angular speed.

Link with the period

In one complete revolution, the angle swept is 2*pi radians, over a duration T. So:

omega = 2pi / T = 2pi*f

Relation between v and omega

The linear speed and the angular speed are linked by the radius of the circle:

v = omega * R

Example: a wheel of radius R = 0.5 m turns at omega = 4 rad/s. Its linear speed at a point on the rim is v = 4 * 0.5 = 2 m/s. Its period is T = 2pi/omega = 2pi/4 ~ 1.57 s.

Summary table

Relation	Formula
Angular speed - period	omega = 2*pi/T
Angular speed - frequency	omega = 2pif
Linear speed - angular	v = omega * R

Classic pitfall: omega is measured in radians per second (rad/s), not in revolutions per second or degrees per second. If a problem gives a rotation speed in revolutions/minute, you must first convert it to rad/s before using the formulas above.