1. Linear and Angular Speeds (3.3)
JMerrill, 2009

2. Linear Speed Linear speed is the speed of a car
Recall the relationship between distance, rate and time: d = rt
Rate is speed so
Think of Nascar: the distance is the arc length, s. If v = speed and t = time, we have

3. Definition of Linear Speed
If a point, P, moves along the circumference of a circle at a constant speed, then the linear speed is denoted by

4. Example – Linear Speed
A car travels at a constant speed around a circular track with circumference of 2 miles. If the car records a time of 15 minutes for 9 laps, what is the linear speed of the car?

5. Your Turn: Linear Speed
A car travels at a constant speed around a circular track with circumference of 3 miles. If the car records a time of 12 minutes for 7 laps, what is the linear speed in miles per hour?
Solution: t = 12 minutes, or hour and s = 7 X 3.
V = = = 105 mph.
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6. Angular Speed
In linear speed, we find how fast a position along the circumference is changing. In angular speed, we find how fast the central is changing.
Angular speed is the rotation of the tires per second.

7. Definition of Angular Speed
If a point, P, moves along the circumference of a circle at a constant speed, then the central angle, θ, that is formed with the terminal side passing through the point P also changes over time, t, at a constant speed. The angular speed, ω (omega), is given by
θ is in radians

8. Example – Angular Speed
A lighthouse in the middle of a channel rotates its light in a circular motion with a constant speed. If the beacon of light completes one rotation every 10 seconds, what is the angular speed of the beacon in radians per minute?

9. Example
One rotation, in radians, is 2π
Using we have

10. You Do – Angular Speed
If the lighthouse is adjusted so that the beacon rotates 1 time every 40 seconds, what is the angular speed of the beacon in radians per minute?

11. Relating Linear and Angular Speed
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An important application of this is if you decide to increase the size of your tires. If you increase the size and do not upgrade the computer, your speedometer will be off maybe 5-10mph! Read example 3 p149.