Physics
Work and Kinetic Energy
Work is the transfer of energy that occurs when a force is applied to an object and it moves in the direction of the force. Kinetic energy is the energy an object possesses due to its motion. The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.
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12 Key excerpts on "Work and Kinetic Energy"
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
CHAPTER 7 Kinetic energy and work 7.1 Kinetic energy LEARNING OBJECTIVES After reading this module, you should be able to: 7.1.1 apply the relationship between a particle’s kinetic energy, mass, and speed 7.1.2 identify that kinetic energy is a scalar quantity. KEY IDEA Here is the key idea of this module. Below we will use them in the example: • The kinetic energy K associated with the motion of a particle of mass m and speed v, where v is well below the speed of light, is K = 1 2 mv 2 . Why study physics? Whether you are enjoying a jet boat adventure on Milford Sound, New Zealand, or alley-ooping your snowboard at Mount Buller in Victoria, the amount of work done by the jet boat’s motor or your body is being converted to kinetic energy to provide you with the thrills and spills. What is energy? The term energy is so broad that a clear definition is difficult to write. Technically, energy is a scalar quantity associated with the state (or condition) of one or more objects. However, this definition is too vague to be of help to us now. A looser definition might at least get us started. Energy is a number that we associate with a system of one or more objects. If a force changes one of the objects by, say, making it move, then the energy number changes. After countless experiments, scientists and engineers realised that if the scheme by which we assign energy numbers is planned carefully, the numbers can be used to predict the outcomes of experiments and, even more important, to build machines, such as flying machines. This success is based on a wonderful property of our universe: energy can be transformed from one type to another and transferred from one object to another, but the total amount is always the same (energy is conserved). No exception to this principle of energy conservation has ever been found. Money Think of the many types of energy as being numbers representing money in many types of bank accounts.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
More formally, we define work as follows: Work W is energy transferred to or from an object by means of a force acting on the object. Energy transferred to the object is positive work, and energy trans- ferred from the object is negative work. “Work,” then, is transferred energy; “doing work” is the act of transferring the energy. Work has the same units as energy and is a scalar quantity. The term transfer can be misleading. It does not mean that anything mate- rial flows into or out of the object; that is, the transfer is not like a flow of water. Rather, it is like the electronic transfer of money between two bank accounts: The number in one account goes up while the number in the other account goes down, with nothing material passing between the two accounts. Note that we are not concerned here with the common meaning of the word “work,” which implies that any physical or mental labor is work. For example, if you push hard against a wall, you tire because of the continuously repeated muscle contractions that are required, and you are, in the common sense, work- ing. However, such effort does not cause an energy transfer to or from the wall and thus is not work done on the wall as defined here. To avoid confusion in this chapter, we shall use the symbol W only for work and shall represent a weight with its equivalent mg. Work and Kinetic Energy Finding an Expression for Work Let us find an expression for work by considering a bead that can slide along a frictionless wire that is stretched along a horizontal x axis (Fig. 7.2.1). A con- stant force F → , directed at an angle ϕ to the wire, accelerates the bead along the wire. We can relate the force and the acceleration with Newton’s second law, written for components along the x axis: F x = ma x , (7.2.1) where m is the bead’s mass. As the bead moves through a displacement d → , the force changes the bead’s velocity from an initial value v → 0 to some other value v → .
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)
- 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.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
3. Only the component of F → that is along the displacement d → can do work on the object. 4. When two or more forces act on an object, their net work is the sum of the individual works done by the forces, which is also equal to the work that would be done on the object by the net force F → net of those forces. 5. For a particle, a change ΔK in the kinetic energy equals the net work W done on the particle: ΔK = K f − K i = W (work–kinetic energy theorem), in which K i is the initial kinetic energy of the particle and K f is the kinetic energy after the work is done. The equation rearranged gives us K f = K i + W. LEARNING OBJECTIVES 7.2 Work and Kinetic Energy 149 Work If you accelerate an object to a greater speed by applying a force to the object, you increase the kinetic energy K (= 1 _ 2 mv 2 ) of the object. Similarly, if you decelerate the object to a lesser speed by applying a force, you decrease the kinetic energy of the object. We account for these changes in kinetic energy by saying that your force has transferred energy to the object from yourself or from the object to yourself. In such a transfer of energy via a force, work W is said to be done on the object by the force. More formally, we define work as follows: Work W is energy transferred to or from an object by means of a force acting on the object. Energy transferred to the object is positive work, and energy trans- ferred from the object is negative work. “Work,” then, is transferred energy; “doing work” is the act of transferring the energy. Work has the same units as energy and is a scalar quantity. The term transfer can be misleading. It does not mean that anything material flows into or out of the object; that is, the transfer is not like a flow of water. Rather, it is like the electronic transfer of money between two bank accounts: The number in one account goes up while the number in the other account goes down, with nothing material passing between the two accounts.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Energy transferred to the object is positive work, and from the object, negative work. ● The work done on a particle by a constant force F → during displacement d → is W = Fd cos ϕ = F → · d → (work, constant force), in which ϕ is the constant angle between the directions of F → and d → . ● Only the component of F → that is along the displace- ment d → can do work on the object. ● When two or more forces act on an object, their net work is the sum of the individual works done by the forces, which is also equal to the work that would be done on the object by the net force F → net of those forces. ● For a particle, a change ΔK in the kinetic energy equals the net work W done on the particle: ΔK = K f − K i = W (work – kinetic energy theorem), in which K i is the initial kinetic energy of the particle and K f is the kinetic energy after the work is done. The equa- tion rearranged gives us K f = K i + W. Learning Objectives Key Ideas “Work,” then, is transferred energy; “doing work” is the act of transferring the energy. Work has the same units as energy and is a scalar quantity. The term transfer can be misleading. It does not mean that anything mate- rial flows into or out of the object; that is, the transfer is not like a flow of water. Rather, it is like the electronic transfer of money between two bank accounts: The number in one account goes up while the number in the other account goes down, with nothing material passing between the two accounts. Note that we are not concerned here with the common meaning of the word “work,” which implies that any physical or mental labor is work. For example, if you push hard against a wall, you tire because of the continuously repeated muscle contractions that are required, and you are, in the common sense, work- ing. However, such effort does not cause an energy transfer to or from the wall and thus is not work done on the wall as defined here.
eBook - PDFWorkshop Physics Activity Guide Module 2
Mechanics II
- Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Name Section Date UNIT 10: WORK AND ENERGY VIAVAL TOURS / Shutterstock A roller coaster presents a challenge for using Newton’s laws of motion to predict the position of the carts as a function of time. The slope of the track is continually changing; imagine trying to figure out the net force on each cart on a moment-by-moment basis as the carts go uphill, downhill, and even upside down. As opposed to Newton’s laws, the concepts of work and energy can be used to simplify the analysis of such complex motions. We will examine these powerful new concepts in Units 10 and 11. 314 WORKSHOP PHYSICS ACTIVITY GUIDE UNIT 10: WORK AND ENERGY OBJECTIVES 1. To extend the intuitive notion of work as relating to effort into a more formal mathematical definition of physical work. 2. To learn to use the definition of physical work to calculate the work done by a constant force or a force that depends on position. 3. To introduce the concept of power as the rate at which work is done. 4. To define the concept of kinetic energy and its relationship to the net work done on a point mass as embodied in the work-energy principle. 10.1 OVERVIEW Although we have seen that momentum is generally conserved in collisions, different outcomes are still possible. For example, two carts can collide and stick together, or they can bounce off each other after the collision. Two carts can even “explode” apart if you release a compressed spring between them. In this unit we will introduce two new concepts that are useful for studying the interactions just described—work and energy. We start by considering both intuitive and mathematical definitions of the work done on objects.
eBook - PDF- David Agmon, Paul Gluck;;;(Authors)
- 2009(Publication Date)
- WSPC(Publisher)
Since velocity is defined with respect to a particular observer, different observers will ascribe different kinetic energies to the same body, so that E k is not an invariant quantity but depends on the frame of reference. Keeping these facts in mind will prevent many mistakes and pitfalls. For a system consisting of N bodies of masses mi,m 2 ...m N having velocities Vi,V 2 ...Vn , the total kinetic energy is given by the sum i=l ^ 1=1 The above definition of kinetic energy in mechanics considers the body as a single unit, without considering any internal motions of its microscopic constituents. In thermodynamics one also considers the body's internal (or thermal) energy which is the result of such internal motions. We shall encounter processes, such as plastic collisions, in which all the kinetic energy of a body may be converted into internal energy. But it is one of the basic tenets of thermodynamics that the reverse process of the conversion of all internal energy into kinetic energy never occurs spontaneously in nature. The work-energy theorem may be stated as follows: W = AE k = E k f ina i - £ k i n i t i a i (6.6) In words: The work done by the resultant of the external forces acting on a body equals the increase in its kinetic energy. This relation actually justifies the above definitions of W and E k . We see that in the process of doing work energy is transferred between bodies in interaction. The first law of thermodynamics is a generalization of this idea, as discussed in example 15. Furthermore, in a so-called closed system, for which the resultant external force vanishes, the kinetic energy is a conserved quantity. This statement will be later generalized to a many-particle system. We saw above that kinetic energy is not an invariant. By the same token, work too depends on the frame of reference. Nevertheless, the relation between the two quantities as expressed by the work-energy theorem holds in every frame of reference.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
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. 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. † Strictly speaking, the work–energy theorem, as given by Equation 6.3, applies only to a single particle, which occupies a mathematical point in space. A macroscopic object, however, is a collection or system of particles and is spread out over a region of space. Therefore, when a force is applied to a macroscopic object, the point of application of the force may be anywhere on the object. To take into account this and other factors, a discussion of work and energy is required that is beyond the scope of this text. The interested reader may refer to A. B. Arons, The Physics Teacher, October 1989, p. 506. Analyzing Multiple-Concept Problems EXAMPLE 4 The Physics of an Ion Propulsion Drive The space probe Deep Space 1 was launched October 24, 1998, and it used a type of engine called an ion propulsion drive. An ion propulsion drive generates only a weak force (or thrust), but can do so for long periods of time using only small amounts of fuel. Suppose the probe, which has a mass of 474 kg, is traveling at an initial speed of 275 m/s. No forces act on it except the 5.60 × 10 ‒2 -N thrust of its engine. This external force F → is directed parallel to the displacement s → , which has a magnitude of 2.42 × 10 9 m (see Figure 6.6).
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 - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
In physics, when a net force performs work on an object, there is always a result from the effort. The result is a change in the kinetic energy of the object. As we will now see, the relation- ship that relates work to the change in kinetic energy is known as the work–energy theorem. This theorem is obtained by bringing together three basic concepts that we’ve already learned about. First, we’ll apply Newton’s second law of motion, SF 5 ma, which relates the net force SF to the acceleration a of an object. Then, we’ll determine the work done by the net force when the object moves through a certain distance. Finally, we’ll use Equation 2.9, one of the equations of kinematics, to relate the distance and acceleration to the initial and final speeds of the object. The result of this approach will be the work–energy theorem. To gain some insight into the idea of kinetic energy and the work–energy theorem, look at Figure 6.5, where a constant net external force S F B acts on an airplane of mass m. This net force is the vector sum of all the external forces acting on the plane, and, for simplicity, it is assumed to have the same direction as the displacement s B . According to Newton’s second law, the net force produces an acceleration a, given by a 5 SF/m. Check Your Understanding (The answers are given at the end of the book.) 1. Two forces F B 1 and F B 2 are acting on the box shown in the drawing, causing the box to move across the floor. The two force vectors are drawn to scale. Which one of the following statements is correct? (a) F B 2 does more work than F B 1 does. (b) F B 1 does more work than F B 2 does. (c) Both forces do the same amount of work. (d) Neither force does any work. 2. A box is being moved with a velocity v B by a force P B (in the same direction as v B ) along a level horizontal floor. The normal force is F B N , the kinetic frictional force is f B k , and the weight is m g B .
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Work and Kinetic Energy For a particle, a change ΔK in the kinetic energy equals the net work W done on the particle: ΔK = K f − K i = W (work – kinetic energy theorem), (7-10) Review & Summary in which K i is the initial kinetic energy of the particle and K f is the ki- netic energy after the work is done. Equation 7-10 rearranged gives us K f = K i + W. (7-11) Work Done by the Gravitational Force The work W g done by the gravitational force F → g on a particle-like object of mass m as the object moves through a displacement d → is given by W g = mgd cos ϕ, (7-12) in which ϕ is the angle between F → g and d → . Work Done in Lifting and Lowering an Object The work W a done by an applied force as a particle-like object is either lifted or lowered is related to the work W g done by the gravitational force and the change ΔK in the object’s kinetic energy by ΔK = K f − K i = W a + W g . (7-15) If K f = K i , then Eq. 7-15 reduces to W a = −W g , (7-16) which tells us that the applied force transfers as much energy to the object as the gravitational force transfers from it. 169 QUESTIONS F x F 1 –F 1 x 1 x (a) F x F 1 –F 1 x 1 x (b) F x F 1 –F 1 x 1 x (c ) F x F 1 –F 1 x 1 x (d) Figure 7-18 Question 5. Spring Force The force F → s from a spring is F → s = −k d → (Hooke’s law), (7-20) where d → is the displacement of the spring’s free end from its posi- tion when the spring is in its relaxed state (neither compressed nor extended), and k is the spring constant (a measure of the spring’s stiffness). If an x axis lies along the spring, with the origin at the location of the spring’s free end when the spring is in its relaxed state, Eq. 7-20 can be written as F x = −kx (Hooke’s law). (7-21) A spring force is thus a variable force: It varies with the displacement of the spring’s free end.
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