SECTION 8.5 Simple Harmonic Motion; Damped Motion; Combining Waves 585 NOTE In the solution to Example 1, =− a 5 because the object is initially pulled down. (If the initial direction is up, then use =a 5. ) j THEOREM Simple Harmonic Motion An object that moves on a coordinate axis so that the displacement d from its rest position at time t is given by either ω ω ( ) ( ) ( ) ( ) = = d t a t d t a t cos or sin where a and ω > 0 are constants, moves with simple harmonic motion. The motion has amplitude a and period π ω = T 2 . The frequency f of an object in simple harmonic motion is the number of oscillations per unit time. Since the period is the time required for one oscillation, it follows that the frequency is the reciprocal of the period; that is, ω π ω = = > f T 1 2 0 Figure 46 Rest position t 5 0 d 5 0 25 Building a Model for an Object in Simple Harmonic Motion Suppose that an object attached to a coiled spring is pulled down a distance of 5 inches from its rest position and then released. If the time for one oscillation is 3 seconds, develop a model that relates the displacement d of the object from its rest position after time t (in seconds). Assume no friction. Solution EXAMPLE 1 The motion of the object is simple harmonic. See Figure 46. When the object is released ( ) = t 0 , the displacement of the object from the rest position is −5 units (since the object was pulled down). Because = − d 5 when = t 0, it is easier to use the cosine function ω ( ) ( ) = d t a t cos to describe the motion.* The amplitude is − = 5 5 and the period is 3, so π ω ω π = − = = = a 5 and 2 period 3, so 2 3 A function that models the motion of the object is π ( ) ( ) = − d t t 5cos 2 3 Now Work PROBLEM 7 2 Analyze Simple Harmonic Motion Analyzing the Motion of an Object Suppose that the displacement d (in meters) of an object at time t (in seconds) is given by the function ( ) ( ) = d t t 10 sin 5 (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? EXAMPLE 2 (continued) *No phase shift is required if a cosine function is used, since a maximum displacement occurs at = t 0.
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