1. Copyright © 2007 Pearson Education, Inc. Slide 8-1 5.7appsSimple Harmonic Motion Other applications of this type of motion include sound, electric current, and electromagnetic waves. Part A shows a weight attached to a spring in its equilibrium (or rest) position. If the weight is pulled down and let go, its oscillatory motion, neglecting friction, is described by a sinusoid.

2. Copyright © 2007 Pearson Education, Inc. Slide 8-2 5.7 Simple Harmonic Motion As P moves around the circle from point (a,0), the point Q(0,y) oscillates between (0,a) and (0,–a). Similarly, the point R(x,0) oscillates between the points (a,0) and (–a,0). This oscillatory motion is called simple harmonic motion. Suppose point P(x,y) moves around the circle counterclockwise at a uniform angular speed . The coordinates of point P at time t are

3. Copyright © 2007 Pearson Education, Inc. Slide 8-3 5.7 Simple Harmonic Motion Amplitude of the motion is |a|. The period of the motion is One complete oscillation is a cycle per period. The number of cycles per unit of time is called the frequency and is equal to the reciprocal of the period. The position of a point oscillating about an equilibrium point at time t is modeled by either where a and  are constants, with  > 0, amplitude |a|, period and frequency

4. Copyright © 2007 Pearson Education, Inc. Slide 8-4 5.7Modeling the Motion of a Spring ExampleSuppose that an object is attached to a coiled spring. It is pulled down a distance of 5 inches from its equilibrium position, and then released. The time for one complete oscillation is 4 seconds. (a)Give an equation that models the position of the object at time t. (b)Determine the position at t = 1.5 seconds. (c)Find the frequency.

5. Copyright © 2007 Pearson Education, Inc. Slide 8-5 5.7 Modeling the Motion of a Spring Solution (a)At t = 0, the displacement is 5 inches below the equilibrium, so s(0) = –5. We use with a = –5. We choose the cosine function since cos  (0) = 1, and –5 · 1 = –5. The period is 4, so Thus, the motion is modeled by

6. Copyright © 2007 Pearson Education, Inc. Slide 8-6 5.7 Modeling the Motion of a Spring (b)After 1.5 seconds, the position is Since 3.54 > 0, the object is above the equilibrium position. (c)The frequency is the reciprocal of the period, or cycles per second.

7. Copyright © 2007 Pearson Education, Inc. Slide 8-7 5.7Analyzing Harmonic Motion ExampleSuppose that an object oscillates according to the model where t is in seconds and s(t) is in feet. Analyze the motion. SolutionThe motion is harmonic since the model is of the form Because a = 8, the object oscillates 8 feet in either direction of its starting point. The period is seconds it takes for one complete oscillation. The object completes oscillation per second.

8. Copyright © 2007 Pearson Education, Inc. Slide 8-8 5.7Damped Oscillatory Motion In the example of the stretched spring, friction was neglected, so oscillations go on and on. Friction causes the amplitude of the motion to diminish gradually until the weight comes to rest. We say that the motion has been damped. The figure shows how y = e -x sin x is bounded above by e -x and below by –e -x.

9. Copyright © 2007 Pearson Education, Inc. Slide 8-9 A typical example of damped oscillatory motion: Example Shock absorbers are put on automobiles to dampen oscillatory motion for a smoother ride. 5.7Damped Oscillatory Motion