613 6.2 The Unit Circle and Circular Functions As an example of linear and angular speeds, consider the following. The human joint that can be flexed the fastest is the wrist, which can rotate through 90°, or p 2 radians, in 0.045 sec while holding a tennis racket. The angular speed of a human wrist swinging a tennis racket is v = u t Formula for angular speed v = p 2 0.045 Let u = p 2 and t = 0.045. v≈35 radians per sec. Use a calculator. If the radius (distance) from the tip of the racket to the wrist joint is 2 ft, then the speed at the tip of the racket is v = rv Formula for linear speed v ≈21352 Let r = 2 and v = 35. v = 70 ft per sec, or about 48 mph. Use a calculator. In a tennis serve the arm rotates at the shoulder, so the final speed of the racket is considerably greater. (Data from Cooper, J., and R. Glassow, Kinesiology, Second Edition, C.V. Mosby.) EXAMPLE 5 Using Linear and Angular Speed Formulas Suppose that point P is on a circle with radius 10 cm, and ray OP is rotating with angular speed p 18 radian per sec. (a) Find the angle generated by P in 6 sec. (b) Find the distance traveled by P along the circle in 6 sec. (c) Find the linear speed of P in centimeters per second. SOLUTION (a) To find the angle generated by P, solve for u in the angular speed formula v = u t . Substitute the known quantities v = p 18 radian per sec and t = 6 sec. u = vt Angular speed formula solved for u u = p 18 162 Let v = p 18 and t = 6. u = p 3 radians Multiply. (b) To find the distance traveled by P, use the arc length formula s = r u with r = 10 cm and, from part (a), u = p 3 radians. s = r u = 10 a p 3b = 10p 3 cm Let r = 10 and u = p 3 . (c) Use the formula for linear speed with r = 10 cm and v = p 18 radians per sec. v = rv = 10 a p 18b = 5p 9 cm per sec Linear speed formula S Now Try Exercise 81.
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