590 CHAPTER 8 Applications of Trigonometric Functions Applications and Extensions In Problems 41–46, an object of mass m (in grams) attached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is T (in seconds) under simple harmonic motion. (a) Find a function that relates the displacement d of the object from its rest position after t seconds. (b) Graph the function found in part (a) for 5 oscillations using a graphing utility. 41. = = = = m a b T 25, 10, 0.7, 5 42. = = = = m a b T 20, 15, 0.75, 6 43. = = = = m a b T 30, 18, 0.6, 4 44. = = = = m a b T 15, 16, 0.65, 5 45. = = = = m a b T 10, 5, 0.8, 3 46. = = = = m a b T 10, 5, 0.7, 3 Concepts and Vocabulary 4. The motion of an object is given by ( ) ( ) = d t t 4cos 6 . Such motion is described as . The number 4 is called the . 5. When a mass hanging from a spring is pulled down and then released, the motion is called if there is no frictional force to retard the motion, and the motion is called if there is friction. 6. True or False If the distance d of an object from its rest position at time t is given by a sinusoidal graph, the motion of the object is simple harmonic motion. Skill Building In Problems 7–10, an object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period T, find a function that relates the displacement d of the object from its rest position after t seconds. Assume that the positive direction of the motion is up. 7. = = a T 5; 2 seconds 8. = = a T 10; 3 seconds 9. π = = a T 7; 5 seconds 10. π = = a T 4; 2 seconds 11. Rework Problem 7 under the same conditions, except that at time = t 0, the object is at its rest position and moving down. 12. Rework Problem 8 under the same conditions, except that at time = t 0, the object is at its rest position and moving down. 13. Rework Problem 9 under the same conditions, except that at time = t 0, the object is at its rest position and moving down. 14. Rework Problem 10 under the same conditions, except that at time = t 0, the object is at its rest position and moving down. In Problems 15–22, the displacement d (in meters) of an object at time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is the time required for one oscillation? (d) What is the frequency? 15. ( ) ( ) = d t t 5sin 3 16. ( ) ( ) = d t t 4 sin 2 17. π ( ) ( ) = d t t 8cos 2 18. π( ) ( ) = d t t 5cos 2 19. ( ) ( ) = − d t t 9 sin 1 4 20. ( ) ( ) = − d t t 2cos 2 21. π ( ) ( ) = + d t t 3 7cos 3 22. π ( ) ( ) = + d t t 4 3sin In Problems 23–26, graph each damped vibration curve for π ≤ ≤ t 0 2 . 23. ( ) ( ) = π − d t e t cos 2 t 24. ( ) ( ) = π − d t e t cos 2 t 2 25. ( ) = π − d t e t cos t 2 26. ( ) = π − d t e t cos t 4 In Problems 27–34, graph each function by adding y-coordinates. 27. ( ) = + f x x x cos 28. ( ) ( ) = + f x x x cos 2 29. ( ) = − f x x x sin 30. ( ) = − f x x x cos 31. ( ) = + f x x x sin cos 32. ( ) ( ) = + f x x x sin 2 cos 33. ( ) ( ) = + g x x x sin sin 2 34. ( ) ( ) = + g x x x cos 2 cos Mixed Practice In Problems 35–40, (a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding y-coordinates to graph each function on the interval π [ ] 0, 2 . 35. ( ) ( ) = f x x x sin 2 sin 36. ( ) ( ) = F x x x sin 3 sin 37. ( ) ( ) ( ) = G x x x cos 4 cos 2 38. ( ) ( ) = h x x x cos 2 cos 39. ( ) ( ) = H x x x 2 sin 3 cos 40. ( ) ( ) = g x x x 2 sin cos 3
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