Mathematics
Elastic Strings and Springs
Elastic strings and springs are objects that can be stretched or compressed and then return to their original shape when the force is removed. The amount of stretch or compression is proportional to the force applied, and this relationship is described by Hooke's Law. This concept is used in various fields, including physics, engineering, and mathematics.
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6 Key excerpts on "Elastic Strings and Springs"
eBook - PDF- 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
eBook - PDFPhysics of Continuous Matter
Exotic and Everyday Phenomena in the Macroscopic World
- B. Lautrup(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
8 Hooke’s law When you bend a wooden stick, the reaction grows notably stronger the further you go—until it perhaps breaks with a snap. If you release the bending force before it breaks, the stick straightens out again and you can bend it again and again without it changing its reaction or its shape. That is what we call elasticity. Robert Hooke (1635–1703). En-glish biologist, physicist, and ar-chitect (no verified contemporary portrait exists). In physics he worked on gravitation, elasticity, built telescopes, and discovered diffraction of light. His famous law of elasticity goes back to 1660. First stated in 1676 as a Latin anagram ceiiinosssttuv , he revealed it in 1678 to stand for ut tensio sic vis , meaning “as is the extension, so is the force”. In elementary mechanics the elasticity of a spring is expressed by Hooke’s law , which says that the amount a spring is stretched or compressed beyond its relaxed length is proportional to the force acting on it. In continuous elastic materials, Hooke’s law implies that strain is a linear function of stress. Some materials that we usually think of as highly elastic, for example rubber, do not obey Hooke’s law except under very small deformations. When stresses grow large, most materials deform more than predicted by Hooke’s law and in the end reach the elasticity limit where they become plastic or break. The elastic properties of continuous materials are determined by the underlying molecular level but the relation is complicated, to say the least. Luckily, there are broad classes of mate-rials that may be described by a few material parameters that can be determined empirically. The number of such parameters depends on the how complex the internal structure of the ma-terial is. We shall almost exclusively concentrate on structureless, isotropic elastic materials, described by just two material parameters: Young’s modulus and Poisson’s ratio. In this chapter, the emphasis will be on matters of principle.
- Camille Duprat, Howard Stone(Authors)
- 2015(Publication Date)
- Royal Society of Chemistry(Publisher)
We can introduce yet another analogy: the energy just derived is that of a 2D elastica in the configuration shown in Figure 1.6c. The first term is the potential energy associated with a squeezing force r g applied to the ends, since Z cos q d S yields the projection of the vector joining the endpoints in the direction x of the applied force. The second term in eqn (1.47), propor-tional to the curvature squared, is the bending energy of the elastica when the surface tension g is identified with the bending modulus B Z g . To sum up, the nonlinear dynamics of the pendulum and the equilibria of a 2D hanging drop and of a 2D elastica are all governed by the same nonlinear differential equation (but different boundary conditions). 1.4 Solving the Linear 2D Elastica Linearizing the elastica model is relevant to small applied forces, when the deflection angle remains small everywhere. With the aim of illustrating this important approximation, we consider two specific geometries, as sketched in Figure 1.7, for which we derive explicit solutions. The undeformed configuration is straight, r 0 ( S ) Z S e x (the x axis is aligned with the rod in the undeformed configuration). We apply a small transverse load p Z p e y , and analyze the resulting deflection. The displacement from the undeformed configuration is assumed to be infinitesimal and purely transverse, r ( S ) Z r 0 ( S ) C y ( S ) e y , the deflection y ( S ) being a small quantity. 18 Chapter 1 Figure 1.7 Linear beam problems. (a) Linear cantilever beam near the natural con-figuration ( T 0 Z 0, B 6 Z 0) with clamped-free conditions, as studied in Section 1.4.1. (b) Stretched string ( T 0 6 Z 0, B Z 0) with pinned–pinned conditions, as studied in Section 1.4.2. The 2D elastica model of Section 1.2.3 and Section 1.3 is linearized as fol-lows.
eBook - PDF- 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
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
272 Chapter 10 | Simple Harmonic Motion and Elasticity Hooke’s Law for Stress and Strain Stress is directly proportional to strain. SI Unit of Stress: newton per square meter (N/m 2 ) 5 pascal (Pa) SI Unit of Strain: Strain is a unitless quantity. In reality, materials obey Hooke’s law only up to a certain limit, as Figure 10.32 shows. As long as stress remains proportional to strain, a plot of stress versus strain is a straight line. The point on the graph where the material begins to deviate from straight-line behavior is called the “proportionality limit.” Beyond the proportionality limit stress and strain are no longer directly proportional. However, if the stress does not exceed the “elastic limit” of the material, the object will return to its original size and shape once the stress is removed. The “elastic limit” is the point beyond which the object no longer returns to its original size and shape when the stress is removed; the object remains permanently deformed. Check Your Understanding (The answer is given at the end of the book.) 21. The block in the drawing rests on the ground. Which face—A, B, or C—experiences the largest stress and which face experi- ences the smallest stress when the block is resting on it? Stress Proportionality limit Elastic limit Strain Stress is directly proportional to strain Figure 10.32 Hooke’s law (stress is directly proportional to strain) is valid only up to the proportionality limit of a material. Beyond this limit, Hooke’s law no longer applies. Beyond the elastic limit, the material remains deformed even when the stress is removed. 20.0 cm 10.0 cm 30.0 cm B C A CONCEPT SUMMARY 10.1 The Ideal Spring and Simple Harmonic Motion The force that must be applied to stretch or com- press an ideal spring is given by Equation 10.1, where k is the spring constant and x is the displace- ment of the spring from its unstrained length. A spring exerts a restoring force on an object attached to the spring.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
Hooke’s Law for Stress and Strain Stress is directly proportional to strain. SI Unit of Stress: newton per square meter (N/m 2 ) 5 pascal (Pa) SI Unit of Strain: Strain is a unitless quantity. In reality, materials obey Hooke’s law only up to a certain limit, as Figure 10.32 shows. As long as stress remains proportional to strain, a plot of stress versus strain is a straight line. The point on the graph where the material begins to deviate from straight-line behavior is called the “proportionality limit.” Beyond the proportionality limit stress and strain are no longer directly proportional. However, if the stress does not exceed the “elastic limit” of the material, the object will return to its original size and shape once the stress is removed. The “elastic limit” is the point beyond which the object no longer returns to its original size and shape when the stress is removed; the object remains permanently deformed. Check Your Understanding (The answer is given at the end of the book.) 21. The block in the drawing rests on the ground. Which face—A, B, or C—experiences the largest stress, and which face experi- ences the smallest stress, when the block is resting on it? 20.0 cm 10.0 cm 30.0 cm B C A CONCEPT SUMMARY 10.1 The Ideal Spring and Simple Harmonic Motion The force that must be applied to stretch or com- press an ideal spring is given by Equation 10.1, where k is the spring constant and x is the displace- ment of the spring from its unstrained length. A spring exerts a restoring force on an object attached to the spring. The restoring force F x pro- duced by an ideal spring is given by Equation 10.2, where the minus sign indicates that the restoring force points opposite to the displacement of the spring. Simple harmonic motion is the oscillatory motion that occurs when a restoring force of the form F x 5 2kx acts on an object. A graphical record of position versus time for an object in simple har- monic motion is sinusoidal.
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