Problem: An object makes a complete revolution around a circle in 6 seconds. Through how many radians per second is the object moving? How fast is the object traveling if it move on a circle of radius 2 meters?
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Solution: A revolution is 2 `pi radians. A revolution in 6 seconds is 2 `pi / 6 radians per second = 1.047 radians per second.
A 2 meter radius implies circumference 2 `pi ( 2 meters); traveling this distance in 6 seconds implies a speed of 2.094 meters per second.
Alternatively, since each radian on a 2 meter circle corresponds to 2 meters of arc distance, 1.047 radians per second corresponds to ( 1.047)( 2) meters per second = 2.094 meters per second.
Generalized Response: If a complete revolution requires time T, then 2 `pi radians of angular motion are completed in time T. The rate at which angular motion proceeds is therefore angular velocity = `omega = 2 `pi rad / T. On a circle of radius r, the 2 `pi rad corresponds to distance 2 `pi r, and the speed of the object is speed = 2 `pi r / T.
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