Set 8 Problem number 4

• Important Note on Notation

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

An object travels around a circle of radius 18.99 meters, completing one revolution in 3 seconds.

• How many radians per second does the line from the center to the object sweep through?
• How fast is the object moving?

Solution

A revolution is 2 `pi radians.

• A revolution in 3 seconds is 2 `pi / 3 radians per second = 2.093 radians per second.
• This quantity is called the angular velocity of the rotational motion.

A 18.99 meter radius implies circumference 2 `pi ( 18.99 meters).

• Traveling this distance in 3 seconds implies a speed of 39.74 meters per second.

Alternative reasoning:

• Since each radian on a 18.99 meter circle corresponds to 18.99 meters of arc distance, 2.093 radians per second corresponds to

• ( 2.093 ) ( 18.99 ) meters per second = 39.74 meters per second.

Generalized Solution

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.