# Angular and Linear Speed Notes

```Trigonometry
Name _________________________________________
Section 2.1 – Linear and Angular Speed
Date ________________________ Pd _____________
1.
We will be describing speed of an object traveling on a circular path in two different ways. What
are some units of measure for speed?
2.
The Angular Speed (or Angular Velocity) of an object traveling on a circular path is the ratio of
the angle through which the object travels to the time the object is traveling.
In short:
angular speed 
time
In symbols:


t
3.
The Linear Speed (or Linear Velocity) of an object traveling on a circular path is the ratio of the

arc length through
which the object travels to the time the object is traveling.

In short:
arc length
linear speed 
time
In symbols:
v
s r

 r
t
t
4.
Angular Speed is measured in (radians/minute) or (revolutions/minute)

5.
Linear Speed is measured in (centimeters/second) or (meters/minute) or (feet/sec)
6.
Useful abbreviations:
7.
Useful equivalences:
8.
Hints for solving:





s = arc length
v = linear speed
t = time
seconds
1 minute = 60

1 hour = 60 minutes
1 day = 24 hours
A = Area of a Sector
 = angular speed
 = angle (in radians)
1 mile = 5280 feet
1 revolution = 2 radians
Read the question carefully to help determine which formula to use 
Use angular speed to calculate linear speed
Pay attention to the required units of measure for the answer
Use the equivalences (above, and others as needed) to help with conversions
Examples:
A.
Consider the Earth, with a radius of approximately 4000 miles, which rotates on its axis once
every 24 hours. What is the angular velocity of the Earth’s rotation in radians per hour? In degrees per
hour?
B.
Consider a ceiling fan with radius of 2 feet that rotates 30 times per minutes. What is the angular
velocity of the fan’s rotation in radians per hour? In degrees per hour?
C.
Back to the Earth … what is the linear velocity (in miles per hour) of the Earth’s rotation?
D.
What is the linear velocity (in miles per hour) of the ceiling fan?
Note: Despite a much larger angular velocity, the tip of the blade of a ceiling fan has a much smaller
linear velocity than the Earth because of the vast difference in the lengths of their respective radii.
Homework: Page 453, #’s 95, 97. Use a calculator, and show all work!
```