Set 8 Problem number 3

• Important Note on Notation

• Problem

• Solution

• Generalized Solution

• Explanation in terms of Figure(s), Extension

• Figure(s)

Problem

An object is moving on a circle whose radius is 15.24 meters.

• At what rate, in radians/second, is the angle of the radial line from the center of the circle to the object changing if the object completes 73.99 revolutions per second?
• How fast is the object therefore moving?

Solution

Each revolution is 2 `pi radians, so

• 73.99 revolutions is 147.98 `pi radians, so
• 73.99 revolutions per second is 147.98 `pi radians per second, or
• 464.6572 radians per second.

Each revolution corresponds to moving 2 `pi ( 15.24 meters) of circumference.

• 73.99 revolutions would be 73.99 times this, or 7081 meters, so 73.99 revolutions per second would be 7081 meters per second.

Alternatively, since each radian on a 15.24 meter circle corresponds to 15.24 meters of arc distance, 464.6572 radians per second corresponds to ( 464.6572 )( 15.24 ) meters per second = 7081 meters per second. 

Generalized Solution

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.