667 6.7 Harmonic Motion (Modeling) Springs A weight on a spring has initial position s102 and period P. (a) To model displacement of the weight, find a function s given by s1t2 = a cos vt. (b) Evaluate s112. Is the weight moving upward, downward, or neither when t = 1? Support the results graphically or numerically. 27. s102 = 2 in.; P = 0.5 sec 28. s102 = 5 in.; P = 1.5 sec 29. s102 = -3 in.; P = 0.8 sec 30. s102 = -4 in.; P = 1.2 sec (Modeling) Music A note on a piano has given frequency F. Suppose the maximum displacement at the center of the piano wire is given by s102. Find constants a and v so that the function s1t2 = a cos vt models this displacement. Graph s in the viewing window 30, 0.054 by 3-0.3, 0.34. 31. F = 27.5; s102 = 0.21 32. F = 110; s102 = 0.11 33. F = 55; s102 = 0.14 34. F = 220; s102 = 0.06 (Modeling) Damped Oscillatory Motion Work each exercise. See Example 3. 39. Consider the damped oscillatory function s1x2 = 5e-0.3x cos px. (a) Graph the function y3 = 5e-0.3x cos px in the window 30, 34 by 3-5, 54. (b) The graph of which function is the upper envelope of the graph of y3? (c) For what values of x does the graph of y3 touch the graph of the function found in part (b)? 40. Consider the damped oscillatory function s1x2 = 10e-x sin 2px. (a) Graph the function y3 = 10e-x sin 2px in the window 30, 34 by 3-10, 104. (b) The graph of which function is the lower envelope of the graph of y3? (c) For what values of x does the graph of y3 touch the graph of the function found in part (b)? 35. How far was the weight pulled down from the equilibrium position before it was released? 36. How far, to the nearest hundredth of an inch, is the weight from the equilibrium position after 6 sec? 37. Graph the function on the interval 30, 124 by 3-12, 124, and determine the values for which the graph intersects the horizontal axis. 38. How many complete oscillations will the graph make during 12 sec? (Modeling) Damped Spring Motion A spring with a weight attached is pulled down and released. Because of friction and other resistive forces, the amplitude is decreasing over time, and t seconds after the spring is released, its position in inches is given by the function s1t2 = -11e-0.2t cos 0.5pt. t y
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