584 CHAPTER 8 Applications of Trigonometric Functions The swinging of a pendulum, the vibrations of a tuning fork, and the bobbing of a weight attached to a coiled spring are examples of vibrational motion. In this type of motion, an object swings back and forth over the same path. In Figure 43, the point B is the equilibrium (rest) position of the vibrating object.The amplitude is the distance from the object’s rest position to its point of greatest displacement (either point A or point C in Figure 43).The period is the time required to complete one vibration—that is, the time it takes to go from, say, point A through B to C and back to A . Figure 43 Coiled spring Compressed A B C Rest Stretched Amplitude Amplitude Simple harmonic motion is a special kind of vibrational motion in which the acceleration a of the object is directly proportional to the negative of its displacement d from its rest position. That is, = − > a kd k , 0. Figure 44 x y u (2a, 0) (a, 0) (0, a) (0, 2a) Q95 (0, y) Q 5 (x, 0) P 5 (x, y) O Tuning fork Credit: dionisvero/iStock/Getty Images For example, when the mass hanging from the spring in Figure 43 is pulled down from its rest position B to the point C , the force of the spring tries to restore the mass to its rest position. Assuming that there is no frictional force to retard the motion, the amplitude will remain constant. The force increases in direct proportion to the distance that the mass is pulled from its rest position. Since the force increases directly, the acceleration of the mass of the object must do likewise, because (by Newton’s Second Law of Motion) force is directly proportional to acceleration. As a result, the acceleration of the object varies directly with its displacement, and the motion is an example of simple harmonic motion. Simple harmonic motion is related to circular motion. To see this relationship, consider a circle of radius a , with center at ( ) 0, 0 . See Figure 44. Suppose that an object initially placed at ( ) a, 0 moves counterclockwise around the circle at a constant angular speed ω. Suppose further that after time t has elapsed the object is at the point ( ) = P x y , on the circle.The angle θ, in radians, swept out by the ray OP in this time t is θ ω = t t ω θ = The coordinates of the point P at time t are θ ω θ ω ( ) ( ) = = = = x a a t y a a t cos cos sin sin Corresponding to each position ( ) = P x y , of the object moving about the circle, there is the point ( ) = Q x, 0 , called the projection of P on the x -axis . As P moves around the circle at a constant rate, the point Q moves back and forth between the points ( ) a, 0 and ( ) −a, 0 along the x -axis with a motion that is simple harmonic. Similarly, for each point P there is a point ( ) ′ = Q y 0, , called the projection of P on the y -axis . As P moves around the circle, the point ′Q moves back and forth between the points ( )a 0, and ( ) −a 0, on the y -axis with a motion that is simple harmonic. Therefore, simple harmonic motion can be described as the projection of constant circular motion on a coordinate axis. To illustrate, again consider a mass hanging from a spring where the mass is pulled down from its rest position to the point C and then released. See Figure 45(a). The graph shown in Figure 45(b) describes the displacement d of the object from its rest position as a function of time t , assuming that no frictional force is present. Figure 45 (a) (b) A d t B C
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