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AG24_Oscillations

AG24_Oscillations - CHAPTER 24 OSCILLATIONS Near the top of...

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1 CHAPTER 24: OSCILLATIONS * Near the top of the 241-m tall Hancock Tower in Boston, two steel boxes filled with lead are part of a system designed to reduce the swaying and twisting of the building caused by the wind. The mass of each box is nearly 300,000 kg (weight 300 tons). It might seem that adding a large mass to the top of the building would make it more ―top heavy‖ and might increase the amount of swaying. Why is such a large mass used and how does it reduce the swaying of the building? 24.1 VIBRATIONS ARE CAUSED BY RESTORING FORCES Vibration, one of the most common kinds of motion, is repeated motion back and forth along the same path. Vibratory motion is periodic if the same motion repeats over and over an object goes back and forth over the same path in exactly the same way. Each time the object repeats its original motion, we say that it has completed another cycle . To complete one cycle of motion, the object must be at the same point and heading in the same direction as it was at the start of the cycle. The period and frequency of vibration are defined exactly as for uniform circular motion , which is another kind of periodic motion. The period T is the time to complete one cycle. The frequency f = 1/ T is the number of cycles per unit time. The SI unit of frequency is the hertz (1 Hz = 1 s 1 = 1 cycle per second). Checkpoint The pendulum in a grandfather clock swings from its extreme leftmost position to its extreme rightmost position in 1.0 s. What is the frequency of its periodic motion? Vibrations (also called mechanical oscillations) can occur in the vicinity of a point of stable equilibrium . An equilibrium point is stable if the net force on the object is directed toward the equilibrium point when the object is near (but not at) the equilibrium point (Fig. 24.1). Such a force is called a restoring force since it tends to restore equilibrium. The amplitude of the motion is the maximum displacement from the equilibrium point. -------------- * © 2007, Alan Giambattista. Answer 0.50 Hz Figure 24.1 (a) A point of stable equilibrium for a roller-coaster car. If the car is displaced slightly from its position at the bottom of the track, the net force pulls the car back toward the equilibrium point. (b) A point of unstable equilibrium for a roller-coaster car. If the car is displaced slightly from the very top of the track, the net force pushes the car away from the equilibrium point.

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2 A special kind of oscillatory motion called simple harmonic motion (or SHM ) occurs whenever the restoring force is proportional to the displacement from equilibrium . The ideal spring is a favorite model used by physicists because the restoring force it provides is proportional to the displacement from equilibrium. Hooke’s law , eq ( ) x F k x x applies to small deformations of many kinds of objects, not just springs. Thus, simple harmonic motion occurs in many situations as long as the vibrations are not too large.
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