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Set 8 Problem number 4
An object makes a complete revolution around a
circle in 4 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 5 meters?
A revolution is 2 `pi radians.
- A revolution in 4 seconds is 2 `pi / 4 radians per
second = 1.57 radians per second.
- This quantity is called the angular velocity
of the rotational motion.
A 5 meter radius implies circumference 2 `pi ( 5
meters).
- Traveling this distance in 4 seconds implies a
speed of 7.85 meters per second.
Alternative reasoning:
- Since each radian on a 5 meter circle corresponds
to 5 meters of arc distance, 1.57 radians per second corresponds to
- ( 1.57 ) ( 5 ) meters per second = 7.85 meters
per second.
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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