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 makes 7 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 19.49 meters?

Solution

Each revolution is 2 `pi radians, so

• 7 revolutions is 14 `pi radians, so
• 7 revolutions per second is 14 `pi radians per second, or
• 43.96 radians per second.

Each revolution corresponds to moving 2 `pi ( 19.49 meters) of circumference.

• 7 revolutions would be 7 times this, or 856.7 meters, so 7 revolutions per second would be 856.7 meters per second.

Alternatively, since each radian on a 19.49 meter circle corresponds to 19.49 meters of arc distance, 43.96 radians per second corresponds to ( 43.96 )( 19.49 ) meters per second = 856.7 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.