Technology & Engineering
Work Energy Equation
The work-energy equation states that the work done on an object is equal to the change in its kinetic energy. This principle is based on the concept that work transfers energy to an object, resulting in a change in its motion. The equation is a fundamental tool in analyzing the relationship between work, energy, and the motion of objects.
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12 Key excerpts on "Work Energy Equation"
eBook - PDF- Donald F. Elger, Barbara A. LeBret, Clayton T. Crowe, John A. Robertson(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
228 CHAPTER 7 • THE ENERGY EQUATION 7.1 Technical Vocabulary: Work, Energy, and Power Conservation of energy is perhaps the single most useful equation in all of engineering. The key to applying this equation is to have solid knowledge of the foundational concepts of energy, work, and power. In addition to reviewing these topics, this section also defines pumps and turbines. Energy Energy is the property of a system that characterizes the amount of work that this system can do on its environment. In simple terms, if matter (i.e., the system) can be used to lift a weight, then that matter has energy. Examples • Water behind a dam has energy because the water can be directed through a pipe (i.e., a penstock), then used to rotate a wheel (i.e., a water turbine) that lifts a weight. Of course, this work can also rotate the shaft of an electrical generator, which is used to produce electrical power. • Wind has energy because the wind can pass across a set of blades (e.g., a windmill), rotate the blades, and lift a weight that is attached to a rotating shaft. This shaft can also do work to rotate the shaft of an electrical generator. • Gasoline has energy because it can be placed into a cylinder (e.g., a gas engine), burned, and expanded to move a piston in a cylinder. This moving cylinder can then be connected to a mechanism that is used to lift a weight. The SI unit of energy, the joule, is the energy associated with a force of one newton acting through a distance of one meter. For example, if a person with a weight of 700 newtons travels up a 10-meter flight of stairs, then their gravitational potential energy has changed by ΔPE = (700 N)(10 m) = 700 N∙m = 700 J. In traditional units, the unit of energy, the foot-pound force (lbf) is defined as energy associated with a force of 1.0 lbf moving through a distance of 1.0 foot. Another way to define a unit of energy is describe the heating of water.
eBook - PDFThe Engineering Dynamics Course Companion, Part 1
ParticlesKinematics and Kinetics
- Edward Diehl(Author)
- 2022(Publication Date)
- Springer(Publisher)
where KE 1 D Kinetic Energy at state 1 PE 1 D Potential Energy at state 1 U 1!2 D External Work acting on system between states 1 and 2 KE 2 D Kinetic Energy at state 2 PE 2 D Potential Energy at state 2 The Work-Energy equation might seem familiar to students who have taken or are taking Fluid Mechanics or Thermodynamics. You can think of this as the solid mechanics form of energy accounting. It’s good to use the following grouping to organize how you think of energy accounting: KE 1 C PE 1 „ ƒ‚ … Initial Energy C U 1!2 „ƒ‚… Outside Work Happens D KE 2 C PE 2 „ ƒ‚ … Final Energy : Note that this equation can be used in multiple locations in between, not just initial and final. We can break problems apart into stages, and this can be a very useful solution strategy in many problems. 8.2. WORK (U) 131 F F ∆x ∆y Figure 8.2: Newtdog works by applying forces in the direction of motion (©E. Diehl). Energy methods are powerful tools to solve many kinds of engineering problems, often as an alternative to other solution techniques that become excessively complicated. We’ll see that the kinetic energy change is closely related to N2L since it’s derived from it. We’ll start the derivation/explanation with the definition of work. 8.2 WORK (U) What is the definition of “work”? You might answer with an equation, but perhaps a layman’s answer like “effort to move stuff ” or “getting stuff done” is more descriptive. As a technical definition we might phrase this as “work is force through a distance.” There is emphasis on the word “through” because this is a necessary part of the concept. Work requires both force and movement, and only when the force and movement align is work done. In Figure 8.2, Newtdog is pushing the loaded wheel barrow up the hill. He’s specifically pushing the handles in the direction of motion. The force he’s exerting on the wheel barrow multiplied by the distance it moves in the direction of the force is the work he is applying.
eBook - PDF- Claus Borgnakke, Richard E. Sonntag(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
One can therefore characterize all cycles as energy conversion devices; the energy is conserved, but it comes out in a form different from that of the input. Further discussion of such cycles is provided in Chapter 5, and the details of the cycles are presented in Chapters 9 and 10. Before we can apply the energy equation or the first law of thermodynamics, we need to elaborate on the work and heat transfer terms as well as the internal energy. 3.3 THE DEFINITION OF WORK The classical definition of work is mechanical work defined as a force F acting through a displacement x, so incrementally W = F dx and the finite work becomes 1 W 2 = ∫ 2 1 F dx (3.13) To evaluate the work, it is necessary to know the force F as a function of x. In this section, we show examples with physical arrangements that lead to simple evaluations of the force, so the integration is straightforward. Real systems can be very complex, and some mathe- matical examples will be shown without a mechanical explanation. Work is energy in transfer and thus crosses the control volume boundary. In addition to mechanical work done by a single point force, work can be done by a rotating shaft, as in a car’s transmission system; by electrical power, as from a battery or a power outlet; or it can be chemical work, to mention a few possibilities. Figure 3.4 shows a simple system FIGURE 3.4 Example of work crossing the boundary of a system. System boundary – + Weight Pulley Battery Motor A C B F .................................................................................................................................................................... ............. THE DEFINITION OF WORK 63 of a battery, a motor, and a pulley. Depending upon the choice of control volume, the work crossing the surface, as in sections A, B, or C, can be electrical through the wires, mechanical by a rotating shaft out of the motor, or a force from the rope on the pulley.
eBook - PDF- Richard L. Myers(Author)
- 2005(Publication Date)
- Greenwood(Publisher)
Work Inherent in the definition of energy are the terms work and heat. Work and heat are transfer properties and can be considered processes that transfer energy across a sys- tem's boundary. Work can be transferred to the system across the system boundary from the surroundings or from the system to the surroundings. Work is done when a force is applied to an object, causing it to move. You can push on a wall all day but no work is done on the wall because it doesn't move. Work, W, is the product of force, F, and the magnitude of displacement or distance over which the force acts, W - Fd. In the simple case where a constant force is applied to an object and the force and motion are parallel and in the same direction, work is calculated by multiplying the resultant force applied to the object by how far it moves. Often the force applied to the object is applied at an angle. In this case only the component of force in the direction of motion is used to determine work, and so the equation for work becomes W - Fd cos 8, where 0 is the angle between the force and direction of motion (Figure 5.1). When the force and motion point in the same direction, work is defined as positive, while work is negative when the force and displacement point in 70 Work, Energy, and Heat opposite directions. An example of the latter takes place with friction. As a person pushes an object across the floor, the person does positive work on the object, but the force of friction does negative work. In the previ- ous example, the force applied to the object was considered constant. When the force is variable, it is required to consider the force exerted over small increments and use the methods of integral calculus to determine the work done on the object. Heat and Temperature Heat is another means by which energy is transferred between a system and its sur- roundings. Heat is defined as a transfer of thermal energy from a body of higher tem- perature to a body of lower temperature.
eBook - PDF- Donald F. Elger, Barbara A. LeBret, Clayton T. Crowe, John A. Roberson(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
* The definition that we like best is this: † Work is any interaction at the boundary of a system that is not heat transfer or the transfer of matter. For example, when electrical power is supplied to a motor, the electric current is classified as a work term. Power Power, which expresses a rate of work or energy, is defined by P ≡ quantity of work (or energy) ________________________ interval of time = lim Δt→0 ΔW _ Δt = W ˙ (7.2) Equation (7.2) is defined at an instant in time because power can vary with time. To calculate power, engineers use several different equations. For rectilinear motion, such as a car or bicy- cle, the amount of work is the product of force and displacement: ΔW = FΔx. Then, power can be found using P = lim Δt→0 FΔx _ Δt = FV (7.3a) where V is the velocity of the moving body. When a shaft is rotating (Fig. 7.1b), the amount of work is given by the product of torque and angular displacement, ΔW = TΔθ. In this case, the power equation is P = lim Δt→0 TΔθ _ Δt = Tω (7.3b) where ω is the angular speed. The SI units of angular speed are rad/s. Because power has units of energy per time, the SI unit is a joule/second, which is called a watt. Common units for power are the watt (W), horsepower (hp), and the ft-lbf/s. Some typical values of power include the following: *This generalized kind of work is sometimes called thermodynamic work to distinguish it from mechanical work. In this text, we use the label work to represent all types of work, including mechanical work. † This definition comes from chemical engineering professor and Nobel Prize winner John Fenn in his book Engines, Energy, and Entropy: A Thermodynamics Primer, p. 5. Chapter 7: The Energy Equation 250 • An incandescent lightbulb can use 60–100 J/s of energy. • A well-conditioned athlete can sustain a power output of about 300 J/s for an hour. This is about four-tenths of a horsepower. One horsepower is the approximate power that a draft horse can supply.
eBook - PDF- Donald F. Elger, Barbara A. LeBret, Clayton T. Crowe, John A. Roberson(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
The Energy Equation CHAPTER ROAD MAP This chapter describes how conservation of energy can be applied to a flowing fluid. The resulting equation, called the energy equation, can be applied to solve many problems; see Fig. 7.1 for an example. CHAPTERSEVEN LEARNING OUTCOMES WORK AND ENERGY (§7.1) ● Define energy, work, and power. ● Define a pump and a turbine. ● Classify energy into categories. ● Know common units. CONSERVATION OF ENERGY FOR A CLOSED SYSTEM (§7.2) ● Know the main ideas about conservation of en- ergy for a closed system. ● Apply the equation(s) to solve problems and answer questions. THE ENERGY EQUATION (§7.3) ● Know the most important ideas about the energy equation. ● Calculate α. ● Define flow work and shaft work. ● Define head and know the various types of head. ● Apply the energy equation to solve problems. THE POWER EQUATION (§7.4) ● Know the concepts associated with each of the power equations. ● Solve problems that involve the power equation. EFFICIENCY (§7.4) ● Define mechanical efficiency. ● Solve problems that involve efficiency of compo- nents such as pumps and turbines. THE SUDDEN EXPANSION (§7.7) ● Calculate the head loss for a sudden expansion. THE EGL/HGL (§7.8) ● Explain the main ideas about the EGL and HGL. ● Sketch the EGL and HGL. ● Solve problems that involve the EGL and HGL. FIGURE 7.1 The energy equation can be applied to hydroelectric power generation. In addition, the energy equation can be applied to thousands of other applications. It is one of the most useful equations in fluid mechanics. Penstock Flow Generator Power lines Power house Turbine 184 Technical Vocabulary: Work, Energy, and Power 185 7.1 Technical Vocabulary: Work, Energy, and Power Conservation of energy is perhaps the single most useful equation in all of engineering. The key to applying this equation is to have solid knowledge of the foundational concepts of energy, work, and power.
eBook - PDFApplied Mathematics
Made Simple
- Patrick Murphy(Author)
- 2014(Publication Date)
- Butterworth-Heinemann(Publisher)
Work, Energy, and Power 217 CHAPTER TWELVE WORK, E N E R G Y A N D P O W E R (1) Work In applied mathematics, work is associated with the movement of bodies from one position to another. Since the movement of a body from rest re-quires force, and the change in position is represented by displacement, we shall define work by relating it to force and displacement. If you help to push-start a car with a flat battery on a winter's morning you will readily agree on three things : pushing the car 20 m will be only half the 'work' of pushing it 40 m ; directing your push in the line of motion is better than pushing in any other direction ; finally, the sooner you can find a down-ward slope the better. If a young child helps you push she will not do the same amount of work as you, even though she will move just as far. The differ-ence between your work contribution and hers will have something to do with the magnitude of the force applied to the car. We therefore conclude that a definition of work involving force and distance looks promising. In order to keep the early ideas as simple as possible we shall consider first, the work done by a force acting on a particle. DEFINITION : / / a constant force of magnitude Fis applied to a particle which then moves a distance s along the line of action of F, the product Fs is called work and is said to be a measure of the work done by the force on the particle. To measure anything we must have a standard unit. The search for a unit need only be brief because we have already obtained units for F and s. Since the standard units are newtons for F and metres for s, it follows that Fs must be measured in newton metres. Unfortunately this is the same measurement as for moment of a force, so it would be helpful to have some means of distinguishing between the two. We therefore refer to the newton metre, when it is a unit of work, as a joule (pronounced 'jewel').
eBook - ePub- Nihal E Wijeysundera(Author)
- 2016(Publication Date)
- WSPC(Publisher)
chapter 1 on the attributes of properties, we conclude that the kinetic energy is a property of the body.To evaluate the work done on the body, using the left hand side of Eq. (3.2) , we need to know how the force F varies with the displacement x . Therefore, the work done by the force is, in general, path-dependent. However, for certain forces like the gravitational force, the work done in moving the body from one point to another is independent of the path followed. Such a force is called a conservative force .The vector form of the expression for work done can be written aswhere F and are the force and position vectors respectively. The work, W however, is a scalar quantity. Note that the common units of work may be expressed as: [W ] = Nm (Newton-meter) = J (Joule).3.2Work Interactions in Thermodynamics
The interactions of a thermodynamic system with its surroundings occur at the system boundary. These interactions are usually the result of pressure forces, electrical currents, surface tension forces and others. For a concise formulation of the laws of thermodynamics we categorize all boundary interactions under two broad headings. These are called work interactions and heat interactions .3.2.1Criterion for a work interaction
It is possible to develop a criterion to determine unambiguously the nature of the interaction at a system boundary. A given boundary interaction is a work interaction if its sole effect on the surroundings could be resolved into the raising or lowering of a weight. In other words we should be able conceive of a device which when coupled to the given system (i) raises or lowers a weight in the surroundings using the same interaction, and (ii) leaves no other permanent effect on the surroundings. The concept is best illustrated using the piston-cylinder arrangement that has a gas, and an electrical heating element connected to a battery, as shown in Fig. 3.2(a) .Consider the gas as our system. As the gas expands due to heating, the piston will move up pushing the outside ambient air. There are two different boundary interaction that could be clearly identified. These are (i) the work done by the gas in raising the piston with the weight while pushing the outside air and (ii) the flow of electrical energy across the sections of the wire where they cross the system boundary.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Kinetic energy, like work, is a scalar quantity. These are not surprising observations, because work and kinetic energy are closely related, as is clear from the following statement of the work–energy theorem. THEWORK– ENERGYTHEOREM WhenanetexternalforcedoesworkWonanobject,thekineticenergyofthe objectchangesfromitsinitialvalueofKE 0 toafinalvalueofKE f ,thedifference betweenthetwovaluesbeingequaltothework: W= KE f − KE 0 = 1 _ 2 m υ f 2 − 1 _ 2 m υ 0 2 (6.3) Work done by net ext. force *For extra emphasis, the final speed is now represented by the symbol υ f , rather than υ. INTERACTIVE FIGURE 6.5 A constant net external force Σ → Facts over a displacement → s and does work on the plane. As a result of the work done, the plane’s kinetic energy changes. s v 0 v f Final kinetic energy = m f 2 _ 1 2 1 2 Initial kinetic energy = m 0 2 _ ΣF ΣF υ υ 6.2 The Work–Energy Theorem and Kinetic Energy 163 The work–energy theorem may be derived for any direction of the force relative to the displacement, not just the situation in InteractiveFigure6.5. In fact, the force may even vary from point to point along a path that is curved rather than straight, and the theorem remains valid. According to the work–energy theorem, a moving object has kinetic energy, because work was done to accelerate the object from rest to a speed υ f . † Conversely, an object with kinetic energy can perform work, if it is allowed to push or pull on another object. Example 4 illustrates the work–energy theorem and considers a single force that does work to change the kinetic energy of a space probe. Description Symbol Value Comment ExplicitData Mass m 474 kg Initial speed υ 0 275 m/s Magnitude of force F 5.60 × 10 −2 N Magnitude of displacement s 2.42 × 10 9 m ImplicitData Angle between force → Fand displacement → s θ 0° The force is parallel to the displacement.
eBook - PDF- Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
F B CHAPTER 11 CHAPTER 11 ENERGY 1: WORK AND KINETIC ENERGY W e have seen how Newton’s laws are useful in un- derstanding and analyzing a wide variety of problems in mechanics. In this and the following two chapters we consider a different approach based on one of the truly fundamental and universal concepts in physics: energy. There are many kinds of energy. In this chapter we consider one particular form — kinetic energy, the energy associated with a body because of its motion. We also introduce the concept of work, which is re- lated to kinetic energy through the work – energy theorem. This theorem, derived from Newton’s laws, pro- vides new and different insight into the behavior of mechanical systems. In Chapter 12 we introduce a sec- ond kind of energy — potential energy — and begin developing a conservation law for energy. In Chapter 13 we discuss energy in a more comprehensive way and generalize the law of conservation of energy, which is one of the most useful laws of physics. application moves through some distance, and one way to define the energy of a system is a measure of its capacity to do work. In the case of the wheelchair rider, he does work because he exerts a force as the wheelchair moves forward through some distance. For him to do work, he must ex- pend some of his supply of energy — that is, the chemical energy stored in his muscle fibers — which can be replen- ished from his body’s store of energy through resting and which ultimately comes from the food he eats. The energy stored in a system may take many forms: for example, chemical, electrical, gravitational, or mechanical. In this chapter we study the relationship between work and one particular type of energy — the energy of motion of a body, which we call kinetic energy. 11- 2 WORK DONE BY A CONSTANT FORCE Figure 11-2a shows a block of mass m being lifted through a vertical distance h by a winch that is turned by a motor.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
force { Final KE { Initial KE This expression is the work–energy theorem. Its left side is the work W done by the net external force, whereas its right side involves the difference between two terms, each of which has the form 1 2 (mass)(speed) 2 . The quantity 1 2 (mass)(speed) 2 is called kinetic energy (KE) and plays a significant role in physics, as we will see in this chapter and later on in other chapters as well. DEFINITION OF KINETIC ENERGY The kinetic energy KE of an object with mass m and speed υ is given by KE = 1 2 mυ 2 (6.2) SI Unit of Kinetic Energy: joule (J) The SI unit of kinetic energy is the same as the unit for work, the joule. Kinetic energy, like work, is a scalar quantity. These are not surprising observations, because work and kinetic energy are closely related, as is clear from the following statement of the work–energy theorem. THE WORK–ENERGY THEOREM When a net external force does work W on an object, the kinetic energy of the object changes from its initial value of KE 0 to a final value of KE f , the difference between the two values being equal to the work: W = KE f − KE 0 = 1 2 mυ f 2 − 1 2 mυ 0 2 (6.3) } s v 0 v f Final kinetic energy = m f 2 _ 1 2 1 2 Initial kinetic energy = m 0 2 _ ΣF ΣF υ υ INTERACTIVE FIGURE 6.5 A constant net external force Σ F → acts over a displacement s → and does work on the plane. As a result of the work done, the plane’s kinetic energy changes. *For extra emphasis, the final speed is now represented by the symbol υ f , rather than υ. 6.2 The Work–Energy Theorem and Kinetic Energy 149 The work–energy theorem may be derived for any direction of the force relative to the displace- ment, not just the situation in Interactive Figure 6.5. In fact, the force may even vary from point to point along a path that is curved rather than straight, and the theorem remains valid.
No longer available |Learn more- Irving Granet, Maurice Bluestein(Authors)
- 2014(Publication Date)
- CRC Press(Publisher)
As stated earlier, work is a transitory effect and is neither a property of a system nor stored in a system. There is one process, however, that does permit the evalua-tion of the work done, because the path is uniquely defined. This process is frictionless and quasi-static, and we shall find it useful in subsequent discussions. To describe this process, we first define equilibrium state in the manner given by Hatsopoulos and Keenan (1961): A state is an equilibrium state if no finite rate of change can occur without a finite change, temporary or permanent, in the state of the environment. The term permanent change of state refers to one that is not canceled out before completion of the process. Therefore, the frictionless quasi-static process can be identified as a succession of equilibrium states. Involved in this definition is the concept of a process carried out infinitely slowly, so that it is in equilibrium at all times. The utility of the frictionless, quasi-static process lies in our ability to evaluate the work terms involved in it, because its path is uniquely defined. Energy or work done per unit time is called power or the rate of energy change. Energy, work, and power units as well as their conversions are detailed in Table 1.8. 2.4 Internal Energy To this point, we have considered the energy in a system that arises from the work done on the system. However, it was noted in Chapter 1 and earlier in this chapter that a body possesses energy by virtue of the motion of the molecules of the body. In addition, it pos-sesses energy due to the internal attractive and repulsive forces between particles. These forces become the mechanism for energy storage whenever particles become separated, such as when a liquid evaporates or the body is subjected to a deformation by an external energy source. Also, energy may be stored in the rotation and vibration of the molecules.
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