1. Recall Radian measure:
One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle. Algebraically, this means that
where θ is measured in radians.
If s = r, then θ = 1 radian.

2. Linear & Angular Velocity
Definition:
Linear Velocity is distance/time:
Ex mph, 6 ft/sec, 27 cm/min, 4.5 m/sec
Definition:
Angular Velocity is central angle/time:
Ex. 360°/day, 2π rad/hour 30°/min

3. Linear & Angular Velocity
Deriving a formula which
relates the two velocities:
Definition of Linear Velocity:
Recall Arc Length Formula
Recognize definition of Angular Velocity to get:

4. Example 1
Find the radian measure of the central angle of a circle of radius 6 inches that intercepts an arc of length 12 inches.
Example 2
Find the radian measure of the central angle of a circle of radius 10 inches that intercepts an arc of length 6 inches.

5. Examples 3 & 4 Find the length of the arc on a circle of radius
9 feet intercepted by a central angle of 2.5 rad.
Find the radius of a circle if the intercepted arc is 18 cm and the central angle is 135°.

6. Example 5
Find the distance between the two cities. Assume that Earth is a sphere of radius 4000 miles and that the cities are on the same longitude (one city is due north of the other).
Dallas, TX 32°N
Omaha, NE 41°N

7. Example 6
Find the area of the sector of the circle with radius 4 inches and central angle 225°.

8. Example 7
Find the area of the sector of the circle with radius 12 m and central angle π/4.

9. Example 8
Consider the Earth which rotates on its axis once every 24 hours. Find the angular velocity of the Earth’s rotation in a) radians per hour and b) radians per minute.
Find the linear velocity of a point on the surface of the Earth. Assume that the radius of the Earth is 4000 miles.

10. Applications of Radian Measure
Example 9
Find the angular velocity of a ceiling fan if the fan rotates 40 times per minute.
Calculate the linear velocity of a point on the tip of the fan blade if the diameter of the fan blade is 4 feet.

11. Example 10
A carousel with a 50 ft diameter makes 4 revolutions per minute. a) Find the angular speed of the carousel in radians per minute b) Find the linear speed of the platform rim of the carousel.

12. Example 11
Patrick is riding a racing bike at a speed of 32 miles per hour. The wheels have a diameter of 17 inches.
a) Find the angular velocity of the wheels in radians per minute.
b) How many revolutions per minute is the wheel making at this speed?