614 CHAPTER 6 The Circular Functions and Their Graphs EXAMPLE 6 Finding Angular Speed of a Pulley and Linear Speed of a Belt A belt runs a pulley of radius 6 cm at 80 revolutions per min. See Figure 19. (a) Find the angular speed of the pulley in radians per second. (b) Find the linear speed of the belt in centimeters per second. 6 cm Figure 19 SOLUTION (a) The angular speed 80 revolutions per min can be converted to radians per second using the following facts. 1 revolution = 2p radians and 1 min = 60 sec We multiply by the corresponding unit fractions. Here, just as with the unit circle, the word unit means 1, so multiplying by a unit fraction is equivalent to multiplying by 1. We divide out common units in the same way that we divide out common factors. v = 80 revolutions 1 min # 2p radians 1 revolution # 1 min 60 sec v = 160p radians 60 sec Multiply. Divide out common units. v = 8p 3 radians per sec Angular speed (b) The linear speed v of the belt will be the same as that of a point on the circumference of the pulley. v = rv = 6 a 8p 3 b = 16p≈50 cm per sec Linear speed S Now Try Exercise 121. 6.2 Exercises CONCEPT PREVIEW Fill in the blanks to complete the coordinates for each point indicated in the first quadrant of the unit circle in Exercise 1. Then use it to find each exact circular function value in Exercises 2 – 5, and work Exercise 6. 1. (__, __) (__, __) (__, __) (__, __) (__, __) 0 0 8 608 908 458 308 P 3 P 2 P 4 P 6 x y 0 2. cos 0 3. sin p 4 4. sin p 3 5. tan p 4 6. Find s in the interval C 0, p 2 D if cos s = 1 2 .
RkJQdWJsaXNoZXIy NjM5ODQ=