Problem: An object makes 97 complete revolutions around a circle in a second. How many radians per second is this? How fast is the object moving if it moves on a circle of radius 14.5 meters?
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Solution: Each revolution is 2 `pi radians, so 97 revolutions is 194 `pi radians, so 97 revolutions per second is 194 `pi radians per second, or 609.4 radians per second.
Each revolution corresponds to moving 2 `pi ( 14.5 meters) of circumference; 97 revolutions would be 97 times this, or 8836 meters, so 97 revolutions per second would be 8836 meters per second.
Alternatively, since each radian on a 14.5 meter circle corresponds to 14.5 meters of arc distance, 609.4 radians per second corresponds to ( 609.4)( 14.5) meters per second = 8836 meters per second.
Generalized Response: Since a revolution is 2 `pi radians, n revolutions per second corresponds to n * 2 `pi = 2 `pi * n radians per second. On a circle of radius r, since each radian corresponds to distance r on the circle, the velocity of a point on the circle will be 2 `pi * n * r.
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