663 6.7 Harmonic Motion (b) Substitute the given value of t in the equation found in part (a). s1t2 = -5 cos p 2 t Equation from part (a) s11.52 = -5 cos c p 2 11.52d Let t = 1.5. s11.52 ≈3.54 in. Use a calculator. Because 3.54 70, the object is above the equilibrium position. (c) The frequency is the reciprocal of the period, or 1 4 oscillation per sec. S Now Try Exercise 7. EXAMPLE 2 Analyzing Harmonic Motion Suppose that an object oscillates according to the model s1t2 = 8 sin 3t, where t is in seconds and s1t2 is in feet. Analyze the motion. SOLUTION The motion is harmonic because the model is s1t2 = a sin vt. Because a = 8, the object oscillates 8 ft in either direction from its starting point. The period 2p 3 ≈2.1 is the time, in seconds, it takes for one complete oscillation. The frequency is the reciprocal of the period, so the object completes 3 2p ≈0.48 oscillation per sec. S Now Try Exercise 17. Damped Oscillatory Motion In the example of the stretched spring, we disregard the effect of friction. Friction causes the amplitude of the motion to diminish gradually until the weight comes to rest. In this situation, we say that the motion has been damped by the force of friction. Most oscillatory motions are damped. For instance, shock absorbers are put on an automobile in order to damp oscillatory motion. Instead of oscillating up and down for a long while after hitting a bump or pothole, the oscillations of the car are quickly damped out for a smoother ride. The decrease in amplitude of a damped oscillatory motion usually follows the pattern of exponential decay. EXAMPLE 3 Analyzing Damped Oscillatory Motion A typical example of damped oscillatory motion is provided by the function s1x2 = e-x cos 2px. (The number e ≈2.718 is the base of the natural logarithm function.) We use x rather than t to match the variable for graphing calculators. (a) Provide a calculator graph of y3 = e-x cos 2px, along with the graphs of y1 = e-x and y 2 = -e-x for 0 … x … 3. (b) Describe the relationships among the three graphs drawn in part (a). (c) For what values of x does the graph of y3 touch the graph of y1? (d) For what values of x does the graph of y3 intersect the x-axis?
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