Problem: An object is moving around a circle at `pi radians per second. How many seconds does it take the object to make a complete revolution? If the radius of the circle is 11 meters, then how fast is the object moving?
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Solution: Since there are 2 `pi radians in a revolution, it takes 2 seconds to complete at `pi radians per second.
If the radius is 11 meters, then the circumference of the circle is 2 `pi ( 11 meters) = 69.1 meters. If the object moves this far in 2 seconds, its speed must be 69.1/2 meters per second = 34.55 meters per second.
Alternatively, if the radius is 11 meters, then 1 radian corresponds to 11 meters on the circle. In that case, `pi radians corresponds to 11( `pi ) meters along the arc of the circle, and `pi radians per second corresponds to 11 `pi meters per second = 34.55 meters per second.
Generalized Response: In general if we are moving at `omega radians/second then since 2 `pi radians constitutes a revolution we require 2 `pi / `omega seconds to complete a revolution. This time is called the period of the motion.
If we are moving at `omega radians/second on a circle of radius r, then since each radian of angular motion corresponds to a distance r on the circle, we travel distance `omega * r every second. Our speed is therefore v = `omega * r.
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Figure Description: The figure below indicates motion through an angle of `pi radians, corresponding to 1 second's motion. Clearly two seconds will be required to complete the revolution.
