SECTION 3.3 EQ: How are linear and angular speeds calculated?

Warm-Up/Activator How does an odometer or speedometer on an automobile work? The transmission counts how many times the tires rotate (how many full revolutions take place) per second. A computer then calculates how far the car has traveled in that second by multiplying the number of revolutions by the tire circumference. Distance is given by the odometer, and the speedometer takes the distance per second and converts to miles per hour (or km/h). Realize that the computer chip is programmed to the tire designed for the vehicle. If a person were to change the tire size (smaller or larger than the original specifications), then the odometer and speedometer would need to be adjusted. Suppose you bought a Ford Expedition Eddie Bauer Edition, which comes standard with 17-inch rims (corresponding to a tire with inch diameter), and you decide to later upgrade these tires for 19- inch rims (corresponding to a tire with 28.2-inch diameter). If the onboard computer is not adjusted, is the actual speed faster or slower than the speedometer indicator? * *

Angular Speed & Linear Speed You will see in this section that the angular speed (rotations of tires per second), radius (of the tires), and linear speed(speed of the automobile) are related. In the context of a circle, we will first define linear speed, then angular speed, and then relate them using the radius.

Linear Speed It is important to note that although velocity and speed are often used as synonyms, speed is how fast you are traveling, whereas velocity is the speed in which you are traveling and the direction you are traveling. In this chapter, speed will be used but since the variable s is used for arc length, we will use the variable v for linear speed. Recall the general relationship between distance, rate, and time d = r*t Since our distance is an arc length, we’ll use s instead of d Since our rate is linear speed, we’ll use v instead of r So, s = v*t Dividing both sides of the equation by t leads to v = s/t

Example 1 A car travels at a constant speed around a circular track with circumference equal to 2 miles. If the car records a time of 15 minutes for 9 laps, what is the linear speed of the car in miles per hour?

Your Turn 1 A car travels at a constant speed around a circular track with circumference equal to 3 miles. If the car records a time of 12 minutes for 7 laps, what is the linear speed of the car in miles per hour?

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

Example 2 A lighthouse in the middle of a channel rotates its light in a circular motion with 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?

Your Turn 2 If the lighthouse in Example 2 is adjusted so that the beacon rotates one time every 40 seconds, what is the angular speed of the beacon in radians per minute?

Linear Speed and Angular Speed Relationship So… or

Example 3 A Ford F-150 comes standard with tires that have a diameter of 25.7 inches. If the owner decided to upgrade to tires with a diameter of 28.2 inches without having the onboard computer updated, how fast will the truck actually be traveling when the speedometer reads 75 miles per hour?

Your Turn 3 Suppose the owner of the F-150 in Example 3 decides to downsize the tires from their original 25.7-inch diameter to a 24.4-inch diameter. If the speedometer indicates a speed of 65 miles per hour, what is the actual speed of the truck?