Springs Physics

6 Key excerpts on "Springs Physics"

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  280 CHAPTER 10 Sarah Reinertsen is a professional athlete who holds numerous world records in her disability division. Her athletic performance is made possible by a high-tech prosthetic leg made of carbon fiber, which flexes and stores elastic potential energy, like a spring does. The elastic potential energy stored by a spring is one of the topics in this chapter. Simple Harmonic Motion and Elasticity LEARNING OBJECTIVES After reading this module, you should be able to... 10.1 Apply Hooke’s law to simple harmonic motion. 10.2 Apply simple harmonic motion relations to the reference circle. 10.3 Apply conservation-of-energy principles to solve simple harmonic motion problems involving springs. 10.4 Analyze pendulum motion. 10.5 Define damped harmonic motion. 10.6 Define driven harmonic motion. 10.7 Apply elastic deformations to define stress and strain. 10.8 Relate Hooke’s law to stress and strain. Don Bartletti/Los Angeles Times/Getty Images 10.1 The Ideal Spring and Simple Harmonic Motion Springs are familiar objects that have many applications, ranging from push-button switches on electronic components, to automobile suspension systems, to mattresses. In use, they can be stretched or compressed. For example, the top drawing in Figure 10.1 shows a spring being stretched. Here a hand applies a pulling force F x  Applied to the spring. The subscript x reminds us that F x  Applied lies along the x axis (not shown in the drawing), which is parallel to the length of the spring. In response, the spring stretches and undergoes a displacement of x from its original, or “unstrained,” length. The bottom drawing in Figure 10.1 illustrates the spring being compressed. Now the hand applies a pushing force to the spring, and it again undergoes a displacement from its unstrained length. Experiment reveals that for relatively small displacements, the force F x  Applied required to stretch or compress a spring is directly proportional

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  LEARNING OBJECTIVES After reading this module, you should be able to... 10.1 Apply Hooke’s law to simple harmonic motion. 10.2 Apply simple harmonic motion relations to the reference circle. 10.3 Apply conservation-of-energy principles to solve simple harmonic motion problems involving springs. 10.4 Analyze pendulum motion. 10.5 Define damped harmonic motion. 10.6 Define driven harmonic motion. 10.7 Apply elastic deformations to define stress and strain. 10.8 Relate Hooke’s law to stress and strain. Don Bartletti/Los Angeles Times/Getty Images CHAPTER 10 Simple Harmonic Motion and Elasticity Sarah Reinertsen is a professional athlete who holds numerous world records in her disability division. Her athletic performance is made possible by a high-tech prosthetic leg made of carbon fiber, which flexes and stores elastic potential energy, like a spring does. The elastic potential energy stored by a spring is one of the topics in this chapter. 10.1 The Ideal Spring and Simple Harmonic Motion Springs are familiar objects that have many applications, ranging from push-button switches on electronic components, to automobile suspension systems, to mattresses. In use, they can be stretched or compressed. For example, the top drawing in Figure 10.1 shows a spring being stretched. Here a hand applies a pulling force F x Applied to the spring. The subscript x reminds us that F x Applied lies along the x axis (not shown in the drawing), which is parallel to the length of the spring. In response, the spring stretches and undergoes a displacement of x from its original, or “unstrained,” length. The bot- tom drawing in Figure 10.1 illustrates the spring being compressed. Now the hand applies a pushing force to the spring, and it again undergoes a displacement from its unstrained length. Experiment reveals that for relatively small displacements, the force F x Applied required to stretch or compress a spring is directly proportional to the displacement x, 257

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  10.1 | The Ideal Spring and Simple Harmonic Motion Springs are familiar objects that have many applications, ranging from push-button switches on electronic components, to automobile suspension systems, to mattresses. In use, they can be stretched or compressed. For example, the top drawing in Figure 10.1 shows a spring being stretched. Here a hand applies a pulling force F x Applied to the spring. The subscript x reminds us that F x Applied lies along the x axis (not shown in the drawing), which is parallel to the length of the spring. In response, the spring stretches and undergoes a displacement of x from its original, or “unstrained,” length. The bottom drawing in Figure 10.1 illustrates the spring being compressed. Now the hand applies a pushing force to the spring, and it again undergoes a displacement from its unstrained length. Experiment reveals that for relatively small displacements, the force F x Applied required to stretch or compress a spring is directly proportional to the displacement x, or F x Applied ~ x. As is customary, this proportionality may be converted into an equation by introducing a proportionality constant k: F Applied x 5 kx (10.1) The constant k is called the spring constant, and Equation 10.1 shows that it has the dimen- sions of force per unit length (N/m). A spring that behaves according to F x Applied 5 kx is said to be an ideal spring. Example 1 illustrates one application of such a spring. Sarah Reinertsen is a professional athlete who holds numerous world records in her disability division. Her athletic performance is made possible by a high-tech prosthetic leg made of carbon fiber, which flexes and stores elastic potential energy, like a spring does. The elastic potential energy stored by a spring is one of the topics in this chapter. 10 | Simple Harmonic Motion and Elasticity 251 Chapter | 10 LEARNING OBJECTIVES After reading this module, you should be able to...

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  College Physics, Global Edition

  • Raymond Serway, Chris Vuille(Authors)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  WCN 02-300 5.5 | Spring Potential Energy 137 Unless otherwise noted, all content on this page is © Cengage Learning. 5.5 Spring Potential Energy Springs are important elements in modern technology. They are found in machines of all kinds, in watches, toys, cars, and trains. Springs will be introduced here, then studied in more detail in Topic 13. Work done by an applied force in stretching or compressing a spring can be recov- ered by removing the applied force, so like gravity, the spring force is conservative, as long as losses through internal friction of the spring can be neglected. That means a potential energy function can be found and used in the work–energy theorem. Figure 5.21a shows a spring in its equilibrium position, where the spring is nei- ther compressed nor stretched. Pushing a block against the spring as in Figure 5.21b compresses it a distance x. Although x appears to be merely a coordinate, for springs it also represents a displacement from the equilibrium position, which for our pur- poses will always be taken to be at x 5 0. Experimentally, it turns out that doubling a given displacement requires twice the force, and tripling it takes three times the force. This means the force exerted by the spring, F s , must be proportional to the displacement x, or F s 5 2kx [5.16] where k is a constant of proportionality, the spring constant, carrying units of new- tons per meter. Equation 5.16 is called Hooke’s law, after Sir Robert Hooke, who discovered the relationship. The force F s is often called a restoring force because the spring always exerts a force in a direction opposite the displacement of its end, tending to restore whatever is attached to the spring to its original position. For positive values of x, the force is negative, pointing back towards equilibrium at x 5 0, and for negative x, the force is positive, again pointing towards x 5 0.

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  Vibrations and Waves

  • A.P. French(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

    3The free vibrations of physical systems

  IN MAKING THE STATEMENT quoted opposite about the elastic properties of objects, Robert Hooke rather overstated the case. The restoring forces in any actual physical system are only approximately linear functions of displacement, as we noted near the beginning of Chapter 1 . Nevertheless, it is remarkable that a vast variety of deformations of physical systems, involving stretching, compressing, bending, or twisting (or combinations of all of these) result in restoring forces proportional to displacement and hence lead to simple harmonic vibration (or a superposition of harmonic vibrations). In this chapter we shall consider a number of examples of such motions, with particular emphasis on the way in which we can relate the kinematic features of the motion to properties that can often be found by purely static measurement. We shall begin with a closer look at the system that forms a prototype for so many oscillatory problems—a mass undergoing one-dimensional oscillations under the type of restoring force postulated by Hooke. Much of the discussion in the next section will probably be familiar ground, but it is important to be quite certain of it before proceeding further.

  The Basic Mass-Spring Problem

  In our first reference to this type of system in Chapter 1 , we characterized it as consisting of a single object of mass m acted on by a spring [Fig. 3-1 (a) ] or some equivalent device, e.g., a thin wire [Fig. 3-1(b) ], that supplies a restoring force equal to some constant k times the displacement from equilibrium. This identifies, in terms of a system of a particularly simple kind, the two features that are essential to the establishment of oscillatory motions:

  Fig. 3-1 (a ) Mass-spring system . (b ) Mass-wire system

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  Applied Structural and Mechanical Vibrations

  Theory and Methods, Second Edition

  • Paolo L. Gatti(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  1 Chapter 1 Review of some fundamentals 1.1 INTRODUCTION Basic physics teaches us that force and motion are strictly related and are inseparable by nature. This fact – which is by no means obvious – is at present a key result with deep ramifications in almost every branch of physics and engineering. One of these branches is dynamics : the study that relates the motion of physical bodies to the forces acting on them. Within certain limitations, this is the realm of Newton’s laws (see Chapter 3, Section 3.1), in the framework of the discipline generally referred to as classical physics . In mathematical form, the fact that force causes a change in the state of motion of a body is expressed by Newton’s second law F t m = ( 29 d d ν (1.1) which defines the unit of force once the units of mass and distance are given. An important part of dynamics, in turn, considers the analysis and pre-diction of vibratory motion , in which a physical system oscillates about a stable equilibrium position as a consequence of a disturbance that sets it in motion. This type of behaviour and many of its aspects and consequences – wanted or unwanted, expected or unexpected – are common in everyday experience for all of us and are the subject of this book. However, it must be clear from the outset that we shall restrict our attention to linear vibra-tions or, more precisely, to situations in which vibrating systems can be modelled as linear, so that the principle of superposition applies. Future sections of this chapter and future chapters will clarify this point in stricter detail.

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