Physics
Conservation of Energy and Momentum
The conservation of energy and momentum is a fundamental principle in physics stating that the total amount of energy and momentum in a closed system remains constant over time, provided no external forces act on it. This principle is crucial for understanding and predicting the behavior of physical systems, from simple collisions to complex interactions in the universe.
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12 Key excerpts on "Conservation of Energy and Momentum"
eBook - PDFSuperstrings and Other Things
A Guide to Physics, Second Edition
- Carlos Calle(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
Conservation of Energy and Momentum 75 the momentum of each cart, and adding them together, we discover that the total momentum of the carts is conserved. Representing the momenta of the two carts after the collision with primes, we have p 1 + p 2 = p ′ 1 + p ′ 2 or p before = p after This is a very important law in physics. It is called the principle of conservation of momentum , and can be stated as follows: If no net external force acts on a system, the total momentum of the system is conserved . This law is general and universal, which means that it applies to any system of bodies anywhere in the universe on which no external forces are acting. It holds regardless of the type of force that the bodies exert on each other. The principle of conservation of momentum was clearly stated first by the Dutch physicist Christiaan Huygens, one of the most gifted of Newton’s contemporaries. Employing kinematical analyses and the methods of ancient geometers rather than the modern analytical methods of dynamics (at the time known only to Newton), Huygens extended the work of John Wallis and Robert Boyle who, responding to a challenge by the Royal Society to investigate the behavior of colliding bodies, sug-gested in 1668 that the product mv was conserved in collisions. ELASTIC AND INELASTIC COLLISIONS You might have seen displayed in novelty shops a small device consisting of five or six shiny steel balls attached by means of two threads to two parallel, horizontal rods (see Figure 5.5). The device is often called Newton’s cradle . When one of the two (a) (b) FIGURE 5.5 (a) After the swinging ball collides with the first stationary ball, the momen-tum of the swinging ball is transmitted to this stationary ball which collides with the next, until finally the last ball swings up to nearly the same height. Conservation of momentum alone does not explain completely the motion of these balls. (b) An actual Newton’s cradle.
eBook - PDFQuestioning the Universe
Concepts in Physics
- Ahren Sadoff(Author)
- 2008(Publication Date)
- Chapman and Hall/CRC(Publisher)
61 8 Conservation Laws Conservation laws are extremely important in physics. They are relevant to a wide variety of physical phenomena. They are very powerful and hence very useful. And interestingly, they follow directly from the symmetry principles we discussed in the last chapter. We can see this by just considering what it means if something is conserved. If something is conserved, it does not increase or decrease; it does not change; it does not vary. In other words, it remains invariant . So, we can give the following definition to conservation: Something is conserved if we can do something to it such that afterwards it remains invariant. Of course, this sounds almost identical to our definition of symmetry. It is no acci-dent. Emmy Noether, a woman mathematician, proved that symmetry principles lead directly to conservation laws. For instance, space translation symmetry leads to the conservation of momentum, and time translation symmetry leads to the conserva-tion of energy. We will discuss both of these conservation laws in this chapter. Using these as examples, we will be able to see why conservation laws are so powerful. 8.1 CONSERVATION OF MOMENTUM The word momentum is often used in everyday speech to imply that something is in motion. But in physics, momentum has a very well-defined meaning. The momen-tum of a single object with a mass, m, is given by the equation p = m v (8.1) Momentum is a vector quantity since it is directly proportional to the velocity vec-tor. The units of momentum are [Kg][M]/[s]. Many quantities in physics have units named after some famous physicist. For instance, the unit of force is the Newton for Isaac Newton. But no one has been honored for momentum. If there is a group of objects and we are interested in the momentum of the entire system, then we just add up the momentum of the individual objects to get the momentum of the system.
eBook - PDF- 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.
eBook - PDF- R. Douglas Gregory(Author)
- 2006(Publication Date)
- Cambridge University Press(Publisher)
Part Two MULTI-PARTICLE SYSTEMS AND CONSERVATION PRINCIPLES CHAPTERS IN PART TWO Chapter 9 The energy principle Chapter 10 The linear momentum principle Chapter 11 The angular momentum principle Chapter Nine The energy principle and energy conservation KEY FEATURES The key features of this chapter are the energy principle for a multi-particle system, the poten- tial energies arising from external and internal forces, and energy conservation. This is the first of three chapters in which we study the mechanics of multi-particle systems. This is an important development which greatly increases the range of problems that we can solve. In particular, multi-particle mechanics is needed to solve problems involving the rotation of rigid bodies. The chapter begins by obtaining the energy principle for a multi-particle system. This is the first of the three great principles of multi-particle mechanics ∗ that apply to every mechanical system without restriction. We then show that, under appropriate conditions, the total energy of the system is conserved. We apply this energy conservation principle to a wide variety of systems. When the system has just one degree of freedom, the energy conservation equation is sufficient to determine the whole motion. 9.1 CONFIGURATIONS AND DEGREES OF FREEDOM A multi-particle system S may consist of any number of particles P 1 , P 2 , . . ., P N , with masses m 1 , m 2 . . ., m N respectively. † A possible ‘position’ of the system is called a configuration. More precisely, if the particles P 1 , P 2 , . . ., P N of a system have position vectors r 1 , r 2 , . . ., r N , then any geometrically possible set of values for the posi- tion vectors { r i } is a configuration of the system. If the system is unconstrained, then each particle can take up any position in space (independently of the others) and all choices of the { r i } are possible.
eBook - PDF- J. N. Reddy(Author)
- 2007(Publication Date)
- Cambridge University Press(Publisher)
5 Conservation of Mass, Momenta, and Energy Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena. Leonard Euler Nothing is too wonderful to be true if it be consistent with the laws of nature. Michael Faraday 5.1 Introduction Virtually every phenomenon in nature, whether mechanical, biological, chemical, geological, or geophysical, can be described in terms of mathematical relations among various quantities of interest. Most mathematical models of physical phe- nomena are based on fundamental scientific laws of physics that are extracted from centuries of research on the behavior of mechanical systems subjected to the ac- tion of natural forces. What is most exciting is that the laws of physics, which are also termed principles of mechanics, govern biological systems as well (because of mass and energy transports). However, biological systems may require addi- tional laws, yet to be discovered, from biology and chemistry to complete their description. This chapter is devoted to the study of fundamental laws of physics as applied to mechanical systems. The laws of physics are expressed in analytical form with the aid of the concepts and quantities introduced in previous chapters. The laws or principles of physics that we study here are (1) the principle of conservation of mass, (2) the principle of conservation of linear momentum, (3) the principle of conserva- tion of angular momentum, and (4) the principle of conservation of energy. These laws allow us to write mathematical relationships – algebraic, differential, or integral type – of physical quantities such as displacements, velocities, temperature, stresses, and strains in mechanical systems. The solution of these equations represents the response of the system, which aids the design and manufacturing of the system.
eBook - PDFAt the Root of Things
The Subatomic World
- Palash Baran Pal(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
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. We shall only refer to some of the results. It was seen from the theory of relativity that the con-servation of momentum and the conservation of energy are not two different concepts but are different ways of looking at the same thing. Even though we can definitely smell an-other unifying notion, there is no need to worry about that here. In fact, we can completely forget about it. Instead, let us see what else we have learnt from the theory of relativity. Consider, for example, two colliding particles. There is no external force. We shall expect the momentum to be 34 Ch. 1: Conservation laws conserved. We know how to calculate the momentum of a material particle. Momentum is a vector, its direction is the same as that of the velocity, and its magnitude is equal to the product of the magnitude of the velocity with mass. Einstein showed that a vector is indeed being conserved and its direction is also along the direction of the velocity. But if the magnitude of the velocity is v then the magnitude of this conserved vector p is given by, p = mv radicalbig 1 -( v 2 /c 2 ) . So, if we still want to define momentum as the product of mass and velocity we shall have to say that the mass is now given by, M = m radicalbig 1 -( v 2 /c 2 ) . The mass of a material particle is fixed but it increases with the velocity. Meaning, the mass depends on the velocity. This is why sometimes it is referred to as the kinetic mass , which is denoted by M . And what is m then? It is the rest mass , because this is the mass of the particle when its veloc-ity is zero.
eBook - PDF- David Agmon, Paul Gluck;;;(Authors)
- 2009(Publication Date)
- WSPC(Publisher)
282 Classical and Relativistic Mechanics 9.3 Momentum conservation in many-body systems So far we have dealt with momentum conservation for a single particle. We now extend our treatment to a system of many particles: We start with two and then generalize to N particles. Consider an isolated system consisting of two particles on which no 12 * w ~~~~* F n external forces act. The particles may interact n}l among themselves (with gravitational, electromagnetic or other forces). Denote by F2 the force on particle 2 exerted by particle 1, and by F 2 the force on particle 1 exerted by particle 2. By Newton's third law F 1 2 +F 2 1 =0 (9.6) During a short time interval At particle 1 exerts an impulse J X2 = F i2 At on particle 2, causing a momentum change Ap 2 = F l2 At. Similarly, particle 2 exerts an impulse J 2 = F 2X At on particle 1, causing a momentum change Api = F 2i At. The momentum of each particle changes, but the total vector momentum of the two particles remains unchanged, Ap tot =Ap l +Ap 2 =(F l2 +F 2l )At = 0 Equivalently, (Pi+P 2 )f=(Pi+P 2 )i Generalization to a closed system of N particles is straightforward: (9.7) (9.8) (9.9) Ptot=Y,Pk= const k= Note that a system defined as closed with respect to an inertial frame will certainly not be closed relative to an accelerating frame. An observer in the latter will discern an inertial (or d'Alembert) force whose direction is opposite to that of the acceleration. For him momentum will not be conserved in this direction. An impressive example of momentum conservation is a shell at rest in space suddenly exploding into a myriad fragments. The initial momentum being zero, the vector sum of the momenta of all the fragments must vanish at all times subsequent to the explosion. We conclude with a comment about Newton's third law. It is not valid for the electric interaction of charged particles accelerated in an electromagnetic field, F 2 ± -F 2X .
eBook - PDFPrinciples of Continuum Mechanics
Conservation and Balance Laws with Applications
- J. N. Reddy(Author)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
5 CONSERVATION OF MASS AND BALANCE OF MOMENTA AND ENERGY It is the mark of an educated mind to be able to entertain a thought without accepting it. Aristotle An error does not become truth by reason of multiplied propagation, nor does truth become error because nobody sees it. Mahatma Gandhi 5.1 Introduction 5.1.1 Preliminary Comments Most phenomena in nature, whether mechanical, biological, chemical, geologi-cal, or geophysical can be described, based on the goal of the study, in terms of mathematical relations among various quantities of interest. Such relationships are called mathematical models and are based on fundamental scientific laws of physics that are extracted from centuries of research on the behavior of mechan-ical systems subjected to the action of external stimuli. What is most exciting is that the laws of physics also govern biological systems because of mass and energy transports. However, biological systems may require additional laws, yet to be discovered, from biology and chemistry to complete their description. This chapter is devoted to the study of fundamental laws of physics as applied to mechanical systems. The laws of physics are expressed in analytical form with the aid of the concepts and quantities introduced in previous chapters. The laws or principles of physics that we study here are: (1) the principle of conservation of mass, (2) the principle of balance of linear momentum, (3) the principle of bal-ance of angular momentum, and (4) the principle of balance of energy. These laws allow us to write mathematical relationships – algebraic, differential, or integral type – of physical quantities of interest such as displacement, velocity, temper-ature, stress, and strain in mechanical systems. The solution of these equations represents the response of the system, which aids the design and manufacturing of the system.
eBook - PDFThermodynamics
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.
eBook - PDFTransport Processes in Chemically Reacting Flow Systems
Butterworths Series in Chemical Engineering
- Daniel E. Rosner, Howard Brenner(Authors)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
2 Governing Conservation Principles INTRODUCTION Approach Chemically reacting fluids, including gas mixtures, can be quantitatively understood in terms of two types of laws applied to matter treated as a continuum: a. Conservation laws, which summarize the experience of the last three centuries on the behavior of all forms of matter; b. Constitutive laws, which quantitatively describe the behavior of certain subclasses of fluids (e.g., perfect gas mixtures, elastic solids, etc.). The conservation principles, to which we first direct our attention, ensure that, for any fluid, in any state of motion, the following quantities are either conserved or, more generally, balanced: Mass Momentum Energy Entropy 1 I Total mixture mass Individual chemical species (e.g., C H 4 , 0 2 , C 0 2 , etc.) mass Individual chemical element (e.g., C, O, S, N, etc.) mass [Total linear momentum of the mixture (a vector) [Total angular momentum of the mixture (a vector) [Total energy (thermodynamic + kinetic) (First Law of | Thermodynamics) [Kinetic (mechanical) energy Total entropy of mixture (describing the consequences of transformations between forms of energy that are not thermodynamically equivalent (Second Law) and identifying all sources of irreversibility and the nature of entropy transport by diffusion). 1 From the Greek: εντρωπη, meaning evolution. 27 28 Introduction Each of these quantities is associated with a field density, i.e., a spatial concentration of that quantity, related, in turn, to the local material mass density, as shown in Table 2-1. Note that while most of these field densities are scalars (defined by one number at each point/instant), pv is a vector field (defined by three numbers at each point/instant). These conservation principles apply not only to any fluid, but to any region of space (subject to the continuum restriction). Thus, we state them below in three important forms, applicable to a control volume (CV) that is: in.
eBook - PDFWhy Toast Lands Jelly-Side Down
Zen and the Art of Physics Demonstrations
- Robert Ehrlich(Author)
- 2020(Publication Date)
- Princeton University Press(Publisher)
Chapter 5 Conservation of Momentum and Energy 5.1 Momentum conservation on a ruler Demonstration Two balls, initially at rest, which push off against each other must recoil with speeds having the inverse ratio of their masses. Equipment A grooved plastic ruler, an index card, and three smooth metal or glass balls having a diameter of one inch (2.54 cm), which have masses approximately in the ratio of 1 to 3 to 3. For example, you could use two stainless steel balls and one aluminum ball. If you use a glass ball (marble) as one of the three, be sure that it is sufficiently round and rolls smoothly in the groove of the ruler. Discussion Place two of the balls in contact in the groove of the ruler, which should be placed on a horizontal surface such as an overhead projector. In general, overhead projectors are not exactly level, but there will always be some orientation of the ruler on the OHP so that a ball placed in its groove will not roll. Now place a folded index card sandwiched between the balls to serve as a low-force constant spring, and squeeze the card closed by finger pressure on the balls. The advantage of using an index card to push the balls apart instead of a spring is that the card exerts a gentle force over a large distance, and is therefore less sensitive to nonsimultaneous finger releases— just be sure that you don't have sticky fingers! When you suddenly remove your fingers from the balls, the unfolding card will gently drive the balls apart with equal and opposite momenta. Therefore, if one ball is x times more massive than the other, its recoil speed should be the fraction 1/x of the lighter ball's speed. A simple way to verify the preceding prediction is to initially place the balls so that their contact point is 82 Cnnser ation of located at a point on the ruler where the lighter ball ,, , . has x times the distance to travel to reach its end of Momentum and ., . , ., , . , „ T . . r . _ the ruler than the heavier ball.
- Robert Ehrlich(Author)
- 2020(Publication Date)
- Princeton University Press(Publisher)
Section E Energy and Linear Momentum Conservation E.1. Inelastic collisions between two balls v /~^ Demonstration When one Velcro-covered ball is swung through a 90° arc into a second ball, conservation of momentum requires that the center of mass of the two balls reach one-fourth the ini-tial height of the first ball. Equipment Two balls the size of tennis balls or larger; some Velcro with adhesive backing, obtainable at a crafts or fabric store; and string. Comment The two balls should be entirely covered with Velcro, and tied to the two ends of a string held at its middle. When one ball is released from a 90° angle and allowed to strike the other, momentum conservation requires that the two balls together have half the velocity of the first ball just before impact. Energy conservation then requires that the center of mass of the two balls rise to a height of one-quarter the first ball's height above the lowest part of the swing. In order to verify this prediction, the half-length of the string, L, should be large compared to the size of the balls. But if you use a string longer than about 2 or 3 feet it will be difficult to achieve head-on collisions. You could, however, suspend each ball trapeze-style using two strings attached to a bar in order to facilitate head-on collisions. Rather than directly observing if the balls rise to a height L/4 above the lowest part of the swing, it may be easier for you to observe if they swing to the correct final angle A, where A = cos^O^ = 41°, measured from the vertical. This can be tested most easily by swinging the balls in front of a blackboard or over-head-projector screen on which you have marked a line making a 41° angle with the vertical. 51
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