661 6.7 Harmonic Motion Given y1 and y2, define their sum to be y3 = y1 + y2. Evaluate y1 and y2 at the given value of x and show that their sum is equal to y3 evaluated at x. Use the method of addition of ordinates. See Example 4. 45. y1 = sin x, y2 = sin 2x; x = p 6 46. y1 = cos x, y2 = cos 2x; x = 2p 3 47. y1 = tan x, y2 = sec x; x = p 4 48. y1 = cot x, y2 = csc x; x = p 3 These summary exercises provide practice with the various graphing techniques presented in this chapter. Graph each function over a one-period interval. 1. y = 2 sin px 2. y = 4 cos 3 2 x 3. y = -2 + 1 2 cos p 4 x 4. y = 3 sec p 2 x 5. y = -4 csc 1 2 x 6. y = 3 tan a p 2 x + pb Graph each function over a two-period interval. 7. y = -5 sin x 3 8. y = 10 cos a x 4 + p 2b 9. y = 3 - 4 sin a 5 2 x + pb 10. y = 2 - sec3p1x - 324 Summary Exercises on Graphing Circular Functions 6.7 Harmonic Motion ■ Simple Harmonic Motion ■ Damped Oscillatory Motion Simple Harmonic Motion In part A of Figure 65, a spring with a weight attached to its free end is in equilibrium (or rest) position. If the weight is pulled down a units and released (part B of the figure), the spring’s elasticity causes the weight to rise a units 1a 702 above the equilibrium position, as seen in part C, and then to oscillate about the equilibrium position. a y 0 –a A. B. C. Figure 65 If friction is neglected, this oscillatory motion is described mathematically by a sinusoid. Other applications of this type of motion include sound, electric current, and electromagnetic waves.
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