Conservation of Energy

12 Key excerpts on "Conservation of Energy"

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  eBook - ePub

  Newtonian Dynamics

  An Introduction

  • Richard Fitzpatrick(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  CHAPTER 5 Conservation of Energy DOI: 10.1201/9781003198642-5 5.1 Introduction Nowadays, the Conservation of Energy is undoubtedly the single most important idea in physics. Strangely enough, although the basic idea of energy conservation was familiar to scientists from the time of Newton onward, this crucial concept only moved to center stage in physics in about 1850 (i.e., when scientists first realized that heat was a form of energy). According to the ideas of modern physics, energy is the fundamental substance that makes up all things in the universe. Energy can take many different forms; for instance, potential energy, kinetic energy, electrical energy, thermal energy, chemical energy, nuclear energy, etcetera. In fact, everything that we observe in the world around us represents one of the multitudinous manifestations of energy. There exist processes in the universe that transform energy from one form into another; for instance, mechanical processes (which are the focus of this book), thermal processes, electrical processes, nuclear processes, etcetera. However, all of these processes leave the total amount of energy in the universe invariant. In other words, whenever, and however, energy is transformed from one form into another, it is always conserved. For a closed system (i.e., a system that does not exchange energy with the rest of the universe), the previous law of universal energy conservation implies that the total energy of the system in question must remain constant in time. 5.2 Energy Conservation During Free-Fall Consider a mass, m, that is falling vertically under the influence of gravity. We already know how to analyze the motion of such a mass. Let us employ this knowledge to search for an expression for the conserved energy during this process. (Note that this is clearly an example of a closed system, involving only the mass and the gravitational field.) The physics of free-fall under gravity is summarized by the three equations (2.16)–(2.18)

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  Exploring Integrated Science

  • Belal E. Baaquie, Frederick H. Willeboordse(Authors)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  64 Exploring Integrated Science A powerful idea in physics is that of invariants , that is, things that do not change as a system evolves in time (see also Chapter 2). The existence of invariants and their use allows us to place constraints on the possible dynamics of a system even though we might be ignorant of the details. Denoting total energy by E we have the fundamental (classical) relation Total energy of system D Kinetic Energy C Potential Energy ) E D T C U: (3.40) Suppose the system has energy E 1 at time t 1 and energy E 2 at a later time t 2 , then, the change in energy is E D E 2 NUL E 1 . Conservation of Energy implies that E D 0 (3.41) ) T C U D 0: (3.42) Note an important fact that since all we know is that E D 0 , the absolute value of E has not been fixed. Hence, energy is only defined up to a constant, since E and E C constant would both be equally conserved. 3.8 Free energy Energy is everywhere! Einstein’s famous formula E D mc 2 tells us that even the mass of a body is a form of congealed energy (see Chapter 22). All material things are different forms of energy. If energy is indeed everywhere, why is there always a fear that our society is “running out energy”, or that there is a shortage of fuel? Why are we asked to reduce, reuse and recycle? We all intuitively know that energy is precious and that possessing energy is of high value. So we need to wonder: Is all energy equal? Or is there a certain energy that is more desirable than another? Fig. 3.20: Coal provides us with “useful” energy. One of the main developments of science in the nineteenth century was the realization by Sadi Carnot, Rudolf Clausius and others that useful energy — energy that can do mechanical work, energy that can be “controlled” and directed — is a very special kind of energy. This special form of energy is called “free energy” to differentiate it from energy in general. Let us consider the forms of energy that we find useful.

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  Physics for Students of Science and Engineering

  • A. L. Stanford, J. M. Tanner(Authors)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  This chapter introduces one of the most important and perhaps the most far-reaching of all principles in the sciences—the Conservation of Energy. A conservation principle is a rule or a natural law that specifies that the value of a physical quantity does not change during the course of a physical process but remains constant. The quantity that does not change is said to be conserved. The simplicity of conservation principles makes them concise expressions of natural law and powerful tools of scientific analysis. Conservation of Energy is only one of a number of conservation principles that students of science or engineering will encounter. The significance and the usefulness of these principles should become apparent to the student as they are used to analyze and interpret physical phenomena.

  Of course, we have not yet defined energy. This chapter will introduce several physical quantities and concepts that are necessary for the understanding of energy and its associated conservation principle. First we will define work and describe how it may be accomplished and calculated. Then we will briefly consider power, a quantity that is useful in many practical applications. Next we will encounter energy in its mechanical forms, namely, kinetic and potential energies, and see how these quantities are related to work. Finally, we will see how kinetic and potential energies are used to express the Conservation of Energy principle, the essential physical relationship of this chapter.

  5.1 Work

  In the nonscientific world, work is often thought of in terms of some physical or mental effort. In physics, however, the term work is defined precisely. Doing work requires the use of force, and work on a body does not take place without displacement of that body. We will begin with a simple situation in which work occurs: work done by a constant force.

  Work by a Constant Force

  Suppose a constant force F is applied to a particle that moves in a straight line, say along the x axis from x 1 to x 2 through a displacement of Δr = (x 2 – x 1 )î Δx î. The work done by the force F on the particle is defined to be the product of the magnitude of the displacement and the magnitude of that component of the force that is in the direction of the displacement. In Figure 5.1 the force F is applied to a particle at an angle θ measured from the x axis. As the particle is displaced by Δr = Δx î, the vector component of force in the direction of the displacement is

  Fx

  î. Then the work W done by the force F on the particle in displacing it by Δr = Δx

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  The Mechanical Universe

  Mechanics and Heat, Advanced Edition

  • Steven C. Frautschi, Richard P. Olenick, Tom M. Apostol, David L. Goodstein(Authors)
  • 2008(Publication Date)
  • Cambridge University Press

    (Publisher)

  CHAPTER ENERGY: CONSERVATION AND CONVERSION . .. You see, therefore, that living force [energy] may be converted into heat, and that heat may be converted into living force, or its equivalent attraction through space. All three, therefore -namely, heat, living force, and attraction through space (to which I might also add light, were it consistent with the scope of the present lecture) - are mutually convertible into one another. In these conversions nothing is ever lost. The same quantity of heat will always be converted into the same quantity of living force. We can therefore express the equivalency in definite language applicable at all times and under all circumstances. James Prescott Joule, On Matter, Living Force, and Heat (1847) 10.1 TOWARD AN IDEA OF ENERGY The law of Conservation of Energy is one of the most fundamental laws of physics. No matter what you do, energy is always conserved. So why do people tell us to conserve energy? Evidentally the phrase conserve energy has one meaning to a scientist and quite a different meaning to other people, for example, to the president of a utility company or to a politician. What then, exactly, is energy? 219 220 ENERGY: CONSERVATION AND CONVERSION The notion of energy is one of the few elements of mechanics not handed down to us from Isaac Newton. The idea was not dearly grasped until the middle of the nineteenth century. Nevertheless, we can find its germ even earlier than Newton. The essence of the idea of the Conservation of Energy can be seen in the incredibly fertile experiments that Galileo performed with balls rolling down inclined planes. It is astonishing how many results Gaiileo squeezed out of his simple experiments. Bodies fall much too fast to be timed by the crude water clocks of the seventeenth century, but by slowing down the failing motion with his inclined planes, Galileo showed that uniformly accelerated motion was a part of nature. That alone was an achievement to crown him a genius.

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  Thermodynamics

  Concepts and Applications

  • Stephen R. Turns(Author)
  • 2006(Publication Date)
  • Cambridge University Press

    (Publisher)

  (See Ref. [4].) Rudolf Clausius (1822–1888) wrote in 1850 one of the most succinct and modern-sounding statement of the energy conservation principle [6]: “The energy of the universe is constant.” (Clausius also named the property entropy and presented clear statements of the second law of thermodynamics. We will consider these concepts in Chapter 6.) 5.2 ENERGY CONSERVATION FOR A SYSTEM We begin our study of the Conservation of Energy principle by considering a system of fixed mass. We start by explicitly transforming the generic conservation principles from Chapter 1 (Eqs. 1.1 and 1.2) to statements of energy conservation. By defining X to be energy E, Eq. 1.1, which applies to the time interval t 2  t 1 , becomes E in  E out  E generated  E stored  E sys (t 2 )  E sys (t 1 ). (5.1) Similarly, by defining to be the time rate of energy Eq. 1.2, which applies to an instant, becomes    (5.2) Figure 5.1 shows a system with superimposed arrows representing the various terms in these equations. Our task now is to associate each term in these Conservation of Energy equations with particular forms of energy for various physical situations. We consider rather general statements of energy conservation in which all forms of energy and their interconversions are allowed. Since we include all forms of energy in our energy accounting, no generation term appears. Our only restriction is the exclusion of nuclear transformations; the proper treatment of nuclear transformations and the relationship between mass and energy are beyond the scope of this book. E # stored . E # generated E # out E # in E # , X # At this point, you may find it useful to review the detailed discussion of systems and control volumes in Chapter 1.

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  Heat Transfer

  Thermal Management of Electronics

  • Younes Shabany(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  29 3 Principle of Conservation of Energy 3.1 FIRST LAW OF THERMODYNAMICS In Chapter 2 it was shown that energy transfers from one system to another as either work or heat. Energy transfer from one system to another is the only way that a system gains or loses energy. The first law of thermodynamics states that energy is not generated or destroyed; it only changes from one form to another or transfers from one system to another. This important physical principle can be expressed by a simple mathematical equation. Consider a system with total initial energy E i as shown in Figure 3.1. The total energy of this system after some time, during which total energy E in enters the system and total energy E out leaves the system, will be E f . The first law of thermodynamics requires that the difference between the total energy entering this system and the total energy leaving it be equal to the difference between its final and initial energies [1–3]: E E E E f i in out − = − . (3.1) The difference between final and initial energies of a system is called change of energy of that system or energy accumulation in that system; ∆ E E E f i system = − . Therefore, the first law of thermodynamics can be expressed by the following equation: E E E in out system − = ∆ . (3.2) This is a simple yet powerful equation that is the basis of all the energy and heat transfer analyses. It is also known as energy balance equation . The energy balance equation, Equation 3.2, deals with initial and final states of a system without being concerned with what happens between these two states or how long it takes to reach from an initial state to a final state. However, these are impor-tant design parameters in most engineering applications. Let’s assume that the state of a system changes during a time interval ∆ t as shown in Figure 3.2. The energy of the system at times t and t + ∆ t are E t and E t + ∆ t , respectively.

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  Fundamentals of Physics, Extended

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  What Is Physics? One job of physics is to identify the different types of energy in the world, especially those that are of common importance. One general type of energy is potential energy U. Technically, potential energy is energy that can be associated with the configuration (arrangement) of a system of objects that exert forces on one another. C H A P T E R 8 Potential Energy and Conservation of Energy 8-1 POTENTIAL ENERGY Learning Objectives After reading this module, you should be able to . . . ● A force is a conservative force if the net work it does on a particle moving around any closed path, from an initial point and then back to that point, is zero. Equivalently, a force is conservative if the net work it does on a particle moving between two points does not depend on the path taken by the particle. The gravitational force and the spring force are conservative forces; the kinetic frictional force is a nonconservative force. ● Potential energy is energy that is associated with the configuration of a system in which a conservative force acts. When the conservative force does work W on a particle within the system, the change ∆U in the potential energy of the system is ∆U = −W. If the particle moves from point x i to point x f , the change in the potential energy of the system is ΔU = − ∫ x f x i F(x) dx. ● The potential energy associated with a system consist- ing of Earth and a nearby particle is gravitational potential energy. If the particle moves from height y i to height y f , the change in the gravitational potential energy of the particle–Earth system is ∆U = mg( y f − y i ) = mg ∆y. ● If the reference point of the particle is set as y i = 0 and the corresponding gravitational potential energy of the system is set as U i = 0, then the gravitational potential energy U when the particle is at any height y is U( y) = mgy. ● Elastic potential energy is the energy associated with the state of compression or extension of an elastic object.

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  Superstrings and Other Things

  A Guide to Physics, Second Edition

  • Carlos Calle(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  As the girl swings, mechanical energy is continuously transformed FIGURE 5.1 The potential energy of the boulder will be converted into kinetic energy when the boulder rolls down the cliff. Conservation of Energy and Momentum 65 into thermal energy; this results in a decrease in height with each swing. This rub-bing of parts of the system against each other is what we call friction . Likewise, when the child playing in a park slides down a slide some of her kinetic energy is converted into heat or thermal energy. These situations do not violate the principle of conservation of mechanical energy as stated above, however, since thermal energy is not a form of mechanical energy. Could we include nonconservative forces and still have Conservation of Energy? If we consider all forms of energy in a system, we can expand the principle of conser-vation of mechanical energy to a more general principle of Conservation of Energy , which can be stated as follows: Energy is neither created nor destroyed; it only changes from one kind to another. H L H´ FIGURE 5.2 The girl on the swing sways from a maximum height H to a lowest point L . The potential energy at H is transformed into kinetic energy, which reaches a maximum value at L . 66 Superstrings and Other Things: A Guide to Physics, Second Edition THE ENERGY OF MASS I n 1905, Einstein extended the principle of Conservation of Energy still further to include mass. In a beautiful paper written when he was 26 years old—the fifth scien-tific paper that he published that year—Einstein deduced that mass and energy were equivalent. The famous formula E = mc 2 gives the energy equivalent of a mass m ; c is the speed of light. According to Einstein, the mass of an object is a form of energy. Conversely, energy is a form of mass. For example, the combination of one pound of hydrogen with four pounds of oxygen to form water releases enough energy to run a hair dryer for about ten hours.

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  Mechanics of Continuous Media

  • John Botsis et Michel Deville(Author)
  • 2018(Publication Date)
  • PPUR

    (Publisher)

  Chapter 4 Energy 4.1 Introduction Having described the principles of conservation of mass, momentum, and an- gular momentum, we will now introduce the principles related to the thermo- dynamics of continuous media in motion and the Conservation of Energy. We can recall that all deformations in a material produce a thermal effect in the same way that a thermal effect produces a deformation. This is easily observed by heating a metal bar which lengthens under the action of the heat. In this chapter, we will generally work in the spatial or Eulerian represen- tation. The principle of conservation of total energy is first established. It leads to the principle of conservation of internal energy. Then, we will con- sider the conservation of mechanical energy in the Lagragian representation. Later, we will show that from the principle of conservation of total energy, for which objectivity is imposed, we can infer the other conservation laws. Finally, the chapter ends with the introduction of entropy and the second law of ther- modynamics, which is based on the Clausius–Duhem inequality, a measure of the irreversibility of the phenomena associated with the physics of continuous media. Continuous media thermodynamics is covered in detail by the following authors: [15, 17, 18, 22, 58, 68]. 4.2 Conservation of Energy Let ω(t) be the material volume of a continuous medium at the instant t, such that ω(t) ⊆ R, the deformed configuration of the body B. We generalize the concept of kinetic energy by defining it as the integral over the deformed volume ω(t) of half the density, ρ(x, t), multiplied by the square of the local spatial velocity, v(x, t). The kinetic energy of ω(t), which we denote E k (t), is a scalar given by the relation E k (t) = Z ω(t) ρ(x, t) v(x, t) · v(x, t) 2 dv . (4.1) 142 Energy To simplify, the dependence of ω with respect to time will no longer be explic- itly shown in the following.

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  At the Root of Things

  The Subatomic World

  • Palash Baran Pal(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  If however, the motion is random, the momenta of the molecules cancel each other to produce a net zero momentum then the total kinetic energy of the molecules is known as heat energy. This is the only difference. By now, we also know that sound too is a form of kinetic energy. We see the pendulum oscillate in an old fashioned grandfather clock. The pendulum oscillates once in every second. If instead the pendulum had oscillated between 20 and 20,000 times per second, the energy of that vibration (that is the kinetic energy of the motion) induced on the neighbouring air molecules would have been able to stimu-late human auditory senses. Then we would have said that we are hearing a sound. We have progressed even more along the path of uni-fication now. We have very coarse vision, that is the only regret. If we had finer vision, right from the beginning we could have said that material particles can have only two types of energy — kinetic energy and potential energy. And § 1.9: Relativity 33 yes, besides the particles there exist the electromagnetic and the gravitational field. These fields too can have energy. But that’s it. The list ends here. If we keep the consideration of fields aside for a mo-ment, then the law of Conservation of Energy simply means that the total sum of the kinetic and the potential en-ergy of the particles is invariant. This statement made by Helmholtz in 1847 actually included almost all kinds of en-ergy. We said ‘almost’ because now we need to add the energy of the fields. And then we would have a perfect result. 1.9 Relativity By now, the nineteenth century was coming to a close and the twentieth century begun. This was a rather eventful century, right from the beginning. The quantum theory came into existence in 1900 itself — we shall discuss that in the next chapter. In 1905, Einstein proposed the theory of relativity. The main points of this would be outside the scope of our discussion here.

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  Physics, Volume 1

  • Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  The energy inside our sys- tem boundary can change due to either heat transferred to or from the environment or work done by or on the environ- ment. Including the heat, we can write Eq. 13-2 as (13-24) In Eq. 13-24, E total indicates all forms of energy contained within the system boundary: kinetic, potential, internal, and perhaps other forms. For convenience we have dropped the subscript “ext” from W, but we still take it to mean the work done on the system by its external environment. Our E total  Q  W. sign convention for Q is similar to that for work: Q  0 means that heat is transferred to a system and increases its energy, while Q  0 means that heat is transferred from the system and decreases its energy.* Equation 13-24 is the most general statement we can make about Conservation of Energy in a system. In this form it is commonly known as the first law of thermodynamics. Later in this text we will consider more detailed application of this law to a particular thermodynamic system: a gas en- closed in a container. For now we consider how this law ap- plies to some mechanical systems. 1. A block sliding on a horizontal surface. A block is sliding on a flat horizontal table where a frictional force acts. The block has an initial speed v and eventually comes to rest. We first take our system to be the block. Equation 13-24 applied to the block gives (13-25) Here E int, block is the increase in internal energy of the block (which is measured by its rise in temperature), W f is the (negative) frictional work done on the block by the table, and Q is the (negative) heat trans- ferred from the block. We assume that the heat transferred to the air is negligible, and that the only heat transfer is from the hot block to the cooler regions of the table with which it comes into contact. Now applying the first law of thermodynamics to the system of block  table, we find (13-26) K  E int, block  E int, table  0.

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  Discover Physics

  • Benjamin Crowell(Author)
  • 2018(Publication Date)
  • Studium Publishing

    (Publisher)

  1-ml pipette and bulb . . . . . . . . . . . . . . . . . . .   . . . . 1/group 

  magnetic stirrer . . . . . . . . . . . . . . . . . . . . . . . . .  . . . 1/group 

  triple-beam balance . . . . . . . . . . . . . . . . . . . . .  . . . 1/group

  Introduction

  Styles in physics come and go, and once-hallowed principles get modified as more accurate data come along, but some of the most durable features of the science are its conservation laws. A conservation law is a statement that something always remains constant when you add it all up. Most people have a general intuitive idea that the amount of a substance is conserved. That objects do not simply appear or disappear is a conceptual achievement of babies around the age of 9-12 months. Beginning at this age, they will for instance try to retrieve a toy that they have seen being placed under a blanket, rather than just assuming that it no longer exists. Conservation laws in physics have the following general features:

  Physicists trying to find new conservation laws will try to find a measurable, numerical quantity, so that they can check quantitatively whether it is conserved. One needs an operational definition of the quantity, meaning a definition that spells out the operations required to measure it.

  Conservation laws are only true for closed systems. For instance, the amount of water in a bottle will remain constant as long as no water is poured in or out. But if water can get in or out, we say that the bottle is not a closed system, and conservation of matter cannot be applied to it.

  The quantity should be additive. For instance, the amount of energy contained in two gallons of gasoline is twice as much as the amount of energy contained in one gallon; energy is additive. An example of a non-additive quantity is temperature. Two cups of coffee do not have twice as high a temperature as one cup.

  Conservation laws always refer to the total amount of the quantity when you add it all up. If you add it all up at one point in time, and then come back at a later point in time and add it all up, it will be the same. 

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