Physics
Translational Kinetic Energy
Translational kinetic energy refers to the energy possessed by an object due to its motion from one point to another. It is directly proportional to the mass of the object and the square of its velocity. This form of kinetic energy is a fundamental concept in physics and is crucial in understanding the behavior of moving objects.
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9 Key excerpts on "Translational Kinetic Energy"
eBook - PDF- Richard C. Hill, Kirstie Plantenberg(Authors)
- 2013(Publication Date)
- SDC Publications(Publisher)
Therefore, to calculate the translational portion of the kinetic energy we use the velocity of the center of mass of the body. If we choose the center of mass (G) as a reference point, the equation for the kinetic energy simplifies to the expression shown in Equation 8.3-4. Recall that even though different points on a rigid body Conceptual Dynamics Kinetics: Chapter 8 – Rigid Body Work & Energy 8 - 18 may have different linear velocities, the body as a whole has only a single angular velocity since each point on the body sweeps out the same angle in the same amount of time. Kinetic energy for general planar motion with the mass center G as a reference: 2 2 1 1 2 2 G G T mv I (8.3-4) T = kinetic energy m = mass v G = speed of the center of mass I G = mass moment of inertia about an axis passing through G = angular speed (rad/s) Conceptual Example 8.3-4 Consider the following situations where the same bar is translating and/or rotating with the given linear and angular speeds. Rank the kinetic energy the bar in each situation from greatest to least. Greatest _____ Next ______ Next ______ Least ______ 8.4) POTENTIAL ENERGY As with a particle, the potential energy a rigid body possesses is due to its position, regardless of whether it is due to the body's position in a gravitational field or the position of the compression or expansion of a spring. Potential Energy is energy stored due to a body's position. Units of Work SI units: Newton-meter [N-m] Joule [J = 1 N-m] erg [erg = 1 x 10 -7 J] US customary units: foot-pound [ft-lb = 1.3558 J] kilocalorie [kcal = 4187 J] British thermal unit [Btu = 778.16 ft-lb = 1055 J] Conceptual Dynamics Kinetics: Chapter 8 – Rigid Body Work & Energy 8 - 19 8.4.1) GRAVITATIONAL POTENTIAL ENERGY Gravitational potential energy is the energy stored by a body due to its position in a gravitational field. Recall that for a particle, the gravitational potential energy is calculated using V g = mgh.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
This minimum kinetic energy contributes to the system's invariant mass, which is independent of the reference frame. According to classical mechanics (ie ignoring relativistic effects) the kinetic energy of a non-rotating object of mass m traveling at a speed v is mv 2 /2. This will be a good approximation provided v is much less than the speed of light. History and etymology The adjective kinetic has its roots in the Greek word κίνησις (kinesis) meaning motion , which is the same root as in the word cinema, referring to motion pictures. The principle in classical mechanics that E ∝ mv² was first developed by Gottfried Leibniz and Johann Bernoulli, who described kinetic energy as the living force , vis viva . Willem 's Gravesande of the Netherlands provided experimental evidence of this relation-ship. By dropping weights from different heights into a block of clay, 's Gravesande determined that their penetration depth was proportional to the square of their impact speed. Émilie du Châtelet recognized the implications of the experiment and published an explanation. The terms kinetic energy and work in their present scientific meanings date back to the mid-19th century. Early understandings of these ideas can be attributed to Gaspard-Gustave Coriolis, who in 1829 published the paper titled Du Calcul de l'Effet des Machines outlining the mathematics of kinetic energy. William Thomson, later Lord Kelvin, is given the credit for coining the term kinetic energy c. 1849 - 1851. ________________________ WORLD TECHNOLOGIES ________________________ Introduction There are various forms of energy: chemical energy, heat, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, rest energy. These can be categorized in two main classes: potential energy and kinetic energy. Kinetic energy can be best understood by examples that demonstrate how it is tran-sformed to and from other forms of energy.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
This minimum kinetic energy contributes to the system's invariant mass, which is independent of the reference frame. According to classical mechanics (ie ignoring relativistic effects) the kinetic energy of a non-rotating object of mass m traveling at a speed v is mv 2 /2. This will be a good approximation provided v is much less than the speed of light. History and etymology The adjective kinetic has its roots in the Greek word κίνησις (kinesis) meaning motion , which is the same root as in the word cinema, referring to motion pictures. The principle in classical mechanics that E ∝ mv² was first developed by Gottfried Leibniz and Johann Bernoulli, who described kinetic energy as the living force , vis viva . Willem 's Gravesande of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay, 's Gravesande determined that their penetration depth was proportional to the square of their impact speed. Émilie du Châtelet recognized the implications of the experiment and published an explanation. The terms kinetic energy and work in their present scientific meanings date back to the mid-19th century. Early understandings of these ideas can be attributed to Gaspard-Gustave Coriolis, who in 1829 published the paper titled Du Calcul de l'Effet des Machines outlining the mathematics of kinetic energy. William Thomson, later Lord Kelvin, is given the credit for coining the t erm kinetic energy c. 1849 - 1851. ________________________ WORLD TECHNOLOGIES ________________________ Introduction There are various forms of energy: chemical energy, heat, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, rest energy. These can be categorized in two main classes: potential energy and kinetic energy. Kinetic energy can be best understood by examples that demonstrate how it is transformed to and from other forms of energy.
eBook - PDFEngineering Mechanics
Dynamics
- Benson H. Tongue, Daniel T. Kawano(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
(4.1) ◆ Apply potential energy concepts and equations to analysis of translating bodies. (4.2) ◆ Solve dynamics problems involving work, power, and efficiency. (4.3) 4.1 KINETIC ENERGY Learning Objective: Apply kinetic energy concepts and equations to analysis of translating bodies. This section introduces the kinetic energy concepts and equations used in analyzing translating bodies. You can select your preferred presenta- tion style for the material: video or text (or both). Consider a particle P in the x,y plane being acted on by a force F, as shown in Figure 4.1.1. You probably remember from physics that the work expended in moving an object is equal to the force acting on the object times the displacement: dW = F· dr (4.1) 222 Figure 4.1.1 Force acting on a particle y x O P r P / O F P a t h In this case I have written the differential forms dW and dr because I want to look at small changes in work resulting from small changes in the position of particle P. We need to use the vector dot product because it’s only the component of force along the direction of the particle’s displacement that matters. The component normal to the displacement vector can’t do any work because no motion takes place in that direction. Figure 4.1.2 shows what I’m talking about. The particle moves a distance ds. The force F is made up of F t tangent to the path (and therefore points along r) and F n normal to the path. You can see in the figure that F n always points at right angles to the path and therefore can’t do any work on m; there is no displacement associated with the direction in which it points. All of F t, however, points along the path; therefore the work done on m is given by dW = F t ds (4.2) 4.1 KINETIC ENERGY 223 Let s 1 and s 2 represent its positions at times t 1 and t 2 , and let υ 1 and υ 2 represent its corresponding speeds.
No longer available |Learn more- Ping YI, Jun LIU, Feng JIANG(Authors)
- 2022(Publication Date)
- EDP Sciences(Publisher)
As shown in figure 9.16, we establish a fixed reference frame Oxyz and a translating reference frame Cx 0 y 0 z 0 whose origin is attached to and moves with the mass center C of the system of particles. v C is the mass center’s velocity relative to the fixed frame Oxyz, and v i=C is the relative velocity of particle M i with respect to the translating frame Cx 0 y 0 z 0 . From the relative-motion analysis, we have v i ¼ v C þ v i=C Therefore v 2 i ¼ v i v i ¼ ðv C þ v i=C Þðv C þ v i=C Þ ¼ v 2 C þ 2v C v i=C þ v 2 i=C Substituting the above expression into equation (9.19), we can express the kinetic energy of the system as T ¼ X 1 2 m i v 2 i ¼ X 1 2 m i ðv 2 C þ 2v C v i=C þ v 2 i=C Þ ¼ 1 2 X m i v 2 C þ v C X m i v i=C þ X 1 2 m i v 2 i=C FIG. 9.16 – Kinetic energy of a system of particles. Kinetics: Work and Energy 365 In this equation, P m i ¼ m is the total mass of the system. For the mass center C, we have P m i r i ¼ mr C in the fixed frame Oxyz. Similarly, in the translating frame Cx 0 y 0 z 0 , we have P m i r i=C ¼ mr C =C ¼ 0. Taking the time derivative yields P m i v i=C ¼ mv C =C ¼ 0. Then the second term on the right side is zero. The third term represents the kinetic energy of the system due to its relative motion with respect to the translating frame Cx 0 y 0 z 0 and can be denoted as T r . Therefore T ¼ 1 2 mv 2 C þ X 1 2 m i v 2 i=C ¼ 1 2 mv 2 C þ T r ð9:20Þ It means the kinetic energy of a system of particles is the sum of the Translational Kinetic Energy due to the translation with the mass center C, and the relative kinetic energy due to the relative motion with respect to the translating frame Cx 0 y 0 z 0 . 9.2.3 Kinetic Energy of a Rigid Body in Planar Motion For a rigid body undergoing general plane motion shown in figure 9.17, v i=C is the tangential velocity due to the rotation about mass center C, i.e., v i=C ¼ r i=C x.
eBook - PDF- Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
Translational Kinetic Energy (of the center-of-mass motion) is the only kind of energy that ap- pears in these expressions. Other energy terms, including the real work, rotational kinetic energy, potential energy, and internal energy, do not appear. We will refer to Eq. 13-14 or 13-15 as the center-of- mass (COM) energy equation and Eq. 13-2 as the conserva- tion-of-energy (COE) equation. Note that the COM equa- tion is derived directly from Newton’s second law and, although it is a useful formulation, it is not a new and inde- pendent principle. The following examples illustrate the differing and of- ten complementary information that these two equations give. 1. A sliding block. A block slides across a horizontal table with initial velocity and is brought to rest by the v B cm F ext s cm K cm . ( x f x i ) x f x i F ext dx cm K cm, f K cm, i K cm . x f x i F ext dx cm v cm, f v cm, i Mv cm dv cm 1 2 Mv 2 cm, f 1 2 Mv 2 cm, i . frictional force f exerted on it by the tabletop. The center of mass of the block moves through a displacement s cm . Our two energy equations give: COM (Eq. 13-15): (13-16a) COE (Eq. 13-2): (13-16b) The COM equation looks like the work – energy theorem but it is not, because, as we have seen, fs cm is not the mag- nitude of the frictional work. In this and the following ex- amples, we write COE (Eq. 13-2) as so that the COM and COE equations look more similar. 2. Pushing a meter stick. Figure 13-6 shows the result of pushing on a meter stick (initially at rest) that is free to slide on a frictionless horizontal surface. A constant exter- nal force is applied at the 25-cm mark. The point of appli- cation of the force moves through the distance s as the cen- ter of mass of the stick moves through the distance s cm (which is less than s), and the stick acquires a center-of- mass velocity v cm and a rotational velocity .
eBook - PDF- Richard L. Myers(Author)
- 2005(Publication Date)
- Greenwood(Publisher)
D Translational Motion Introduction Chapter 2 presented the historical development of ideas about motion. These ideas were syn- thesized by Newton at the end of the sev- enteenth century in his Principia and have shaped our thinking of motion ever since. Concepts, once questioned and debated, are now commonly accepted, and each of us has a common understanding of the basic prin- ciples of motion. Terms describing motion, such as velocity and acceleration, are used colloquially to describe motion, often with little thought of their precise physical mean- ing. In this chapter the principles of mechan- ics introduced in the previous chapter will be examined in greater detail. The focus in this chapter will be translational, or linear, motion. Translational motion is characterized by movement along a linear path between two points. Scalars and Vectors Before translational motion can be exam- ined, it is important to distinguish between scalar and vector quantities. A scalar quan- tity, or just scalar, is defined by a magnitude and appropriate units. Ten dollars, 3 meters, and 32 °F are examples of scalar quantities. A quantity that is defined by a magnitude and direction with appropriate units is a vector quantity, or vector. Wind velocity is a good example of a vector. When wind velocity is reported, both the magnitude and direction are given, for example, 10 miles per hour out of the north. Vectors require a specified or assumed direction. A vector quantity is incomplete without a direction. The importance of includ- ing the direction for a vector can be illustrated by considering a person standing at the end of a narrow dock. If the person were told to move three steps forward versus three steps directly backward, it would probably make the differ- ence between falling in the water and staying dry. Three steps is a scalar, whereas three steps backward is a vector. Throughout this book, physical quantities will be defined as scalars or vectors.
eBook - ePubOAT Prep Plus 2023-2024
2 Practice Tests + Proven Strategies + Online
- (Author)
- 2023(Publication Date)
- Kaplan Test Prep(Publisher)
In order for momentum to be conserved, the stationary ball must be moving in the same direction after the collision as the incoming ball was before the collision. This is because the incoming ball has negative momentum after the collision relative to its initial momentum. The stationary ball must make up for the lost initial positive momentum by itself acquiring positive momentum. Conservation of kinetic energy requires that the speed of the stationary ball after the collision be less than the speed of the incoming ball because it is more massive. In other words, if the stationary ball moved as fast or faster than the incoming ball, it would have more kinetic energy than the incoming ball, and this would violate conservation of kinetic energy.8.8.33 m/sIn all collision problems, it is important to remember that the vector sum of the momentum before the collision is equal to the vector sum of the momentum after:
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[ CHAPTER 67 ]
THERMODYNAMICS
LEARNING OBJECTIVES
After this chapter, you will be able to:- Define temperature and calculate thermal expansion
- Define heat transfer and calculate specific heat and heat of transformation
- Apply the First and Second Laws of Thermodynamics to a system to solve for unknown variables
Thermodynamics is the study of heat and its effects. Primary to this study are the concepts of temperature, heat, pressure, volume, work, internal energy, and entropy. These ideas will be applied to thermal expansion, heat transfer processes, specific heat, heat of transformation (latent heat), and pressure vs. volume diagrams, including the relationship between work, pressure, and volume. The first law of thermodynamics involving conservation of energy in the presence of heat transfer will be reviewed along with the second law of thermodynamics and the associated concept of entropy.Temperature
Temperature
All bodies possess a property called temperature. In common usage, temperature is the relative measure of how hot or cold something is. In the study of thermodynamics, however, temperature is a measure of the internal energy of the object and must be identified quantitatively on a defined scale. There are three scales used to make these measurements of temperature on a thermometer: the Fahrenheit (°F), the Celsius (°C), and the kelvin
eBook - ePub- (Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
CHAPTER XXII.
ON THE MOLECULAR THEORY OF THE CONSTITUTION OF BODIES.
WE have already shown that heat is a form of energy—that when a body is hot it possesses a store of energy, part at least of which can afterwards be exhibited in the form of visible work.Now energy is known to us in two forms. One of these is Kinetic Energy, the energy of motion. A body in motion has kinetic energy, which it must communicate to some other body during the process of bringing it to rest. This is the fundamental form of energy. When we have acquired the notion of matter in motion, and know what is meant by the energy of that motion, we are unable to conceive that any possible addition to our knowledge could explain the energy of motion, or give us a more perfect knowledge of it than we have already.There is another form of energy which a body may have, which depends, not on its own state, but on its position with respect to other bodies. This is called Potential Energy. The leaden weight of a clock, when it is wound up, has potential energy, which it loses as it descends. It is spent in driving the clock. This energy depends, not on the piece of lead considered in itself, but on the position of the lead with respect to another body—the earth—which attracts it.In a watch, the mainspring, when wound up, has potential energy, which it spends in driving the wheels of the watch. This energy arises from the coiling up of the spring, which alters the relative position of its parts. In both cases, until the clock or watch is set agoing, the existence of potential energy, whether in the clock-weight or in the watch-spring, is not accompanied with any visible motion. We must therefore admit that potential energy can exist in a body or system all whose parts are at rest.
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