Chapter 1: Oscillatory Motion PDF

Chapter 1 Oscillatory Motion Periodic motion is motion of an object that regularly repeats—the object returns to a given position after a fixed time interval. With a little thought, we can identify several types of periodic motion in everyday life. Your car ret...

Chapter 1 Oscillatory Motion Periodic motion is motion of an object that regularly repeats—the object returns to a given position after a fixed time interval. With a little thought, we can identify several types of periodic motion in everyday life. Your car returns to the driveway each after-noon. You return to the dinner table each night to eat. A bumped chandelier swings back and forth, returning to the same position at a regular rate. The Earth returns to the same position in its orbit around the Sun each year, resulting in the variation among the four seasons. The Moon returns to the same relationship with the Earth and the Sun, resulting in a full Moon approximately once a month. In addition to these everyday examples, numerous other systems exhibit periodic motion. For example, the molecules in a solid oscillate about their equilibrium positions; electromagnetic waves, such as light waves, radar, and radio waves, are characterized by oscillating electric and magnetic field vectors; and in alternating-current electrical circuits, voltage, current, and electric charge vary periodically with time. A special kind of periodic motion occurs in mechanical systems when the force acting on an object is proportional to the position of the object relative to some equilibrium position. If this force is always directed toward the equilibrium position, the motion is called simple harmonic motion, which is the primary focus of this chapter. 1.1 Motion of an Object Attached to a Spring As a model for simple harmonic motion, consider a block of mass mattached to the end of a spring, with the block free to move on a horizontal, frictionless surface (Fig.1.1). When the spring is neither stretched nor compressed, the block is at the position called the equilibrium positionof the system, which we identify as x = 0. We know from experience that such a system oscillates back and forth if disturbed from its equilibrium position. We can understand the motion in Figure 1.1 qualitatively by first recalling that when the block is displaced to a position x, the spring exerts on the block a force that is proportional to the position and given by Hooke’s law Fs = −kx (1.1) We call this a restoring force because it is always directed toward the equilibrium position and therefore opposite the displacement from equilibrium. That is, when the block is displaced to the right of x= 0 in Figure 1.1, then the position is positive and 1 the restoring force is directed to the left. When the block is displaced to the left of x= 0, then the position is negative and the restoring force is directed to the right. Applying Newton’s second law Fx = max to the motion of the block, with Equation 1.1 providing the net force in the x direction, we obtain −kx = max k ax = − x (1.2) m Figure 1.1 A block attached to a spring moving on a frictionless surface. (a) When the block is displaced to the right of equilibrium (x>0), the force exerted by the spring acts to the left. (b) When the block is at its equilibrium position (x=0), the force exerted by the spring is zero. (c) When the block is displaced to the left of equilibrium (x