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424 13 CHAPTER Vibrations and Waves O U T L I N E 13.1 Hookes Law 13.2 Elastic Potential Energy 13.3 Comparing Simple Harmonic Motion with Uniform Circular Motion 13.4 Position, Velocity, and Acceleration as a Function of Time 13.5 Motion of a Pendulum 13.6 Damped Oscillations 13.7 Waves 13.8 Frequency, Amplitude, and Wavelength 13.9 The Speed of Waves on Strings 13.10 Interference of Waves 13.11 Reflection of Waves R i c k D o y l e / C o r b i s Periodic motion, from masses on springs to vibrations of atoms, is one of the most important kinds of physical behavior. In this chapter we take a more detailed look at Hookes law, where the force is proportional to the displacement, tending to restore objects to some equilibrium position. A large number of physical systems can be successfully modeled with this simple idea, including the vibrations of strings, the swinging of a pendulum, and the propagation of waves of all kinds. All these physical phenomena involve periodic motion. Periodic vibrations can cause disturbances that move through a medium in the form of waves. Many kinds of waves occur in nature, such as sound waves, water waves, seismic waves, and electromagnetic waves. These very different physical phenomena are described by common terms and concepts introduced here. 13.1 HOOKES LAW One of the simplest types of vibrational motion is that of an object attached to a spring, previously discussed in the context of energy in Chapter 5. We assume that the object moves on a frictionless horizontal surface. If the spring is stretched or compressed a small distance x from its unstretched or equilibrium position and then released, it exerts a force on the object as shown in Active Figure 13.1. From experiment this spring force is found to obey the equation [13.1] F s kx Ocean waves combine properties of both transverse and longitudinal waves. With proper balance and timing, a surfer can capture some of the waves energy and take it for a ride. Hookes law 13.1 Hookes Law 425 where x is the displacement of the object from its equilibrium position ( x 0) and k is a positive constant called the spring constant . This force law for springs was discovered by Robert Hooke in 1678 and is known as Hookes law . The value of k is a measure of the stiffness of the spring. Stiff springs have large k values, and soft springs have small k values. The negative sign in Equation 13.1 means that the force exerted by the spring is always directed opposite the displacement of the object. When the object is to the right of the equilibrium position, as in Active Figure 13.1a, x is positive and F s is negative. This means that the force is in the negative direction, to the left. When the object is to the left of the equilibrium position, as in Active Figure 13.1c, x is negative and F s is positive, indicating that the direction of the force is to the right.... View Full Document

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