Work and Energy

Energy is a very important concept that is heavily used in everyday life. Everything around us, including ourselves, needs energy to function. For example, electricity provides home appliances with the energy they require, food gives us energy to survive, and the sun provides earth with the energy needed for the existence of life! Experiments show that energy is a scalar quantity related to the state of an object. Energy may exist in various forms: mechanical, chemical, gravitational, electromagnetic, nuclear, and thermal. Furthermore, energy cannot be created or destroyed; it can only be transformed from one form to another. In other words, if energy were to be exchanged between objects inside a system, then the total amount of energy (the sum of all forms of energy) in the system will remain constant.


INTRODUCTION
Energy is much in the news lately. The term "energy" usually refers to the inherent ability of a material system, such as a person, a flashlight battery, or rocket fuel, to bring about changes in its environment or in itself. Some common sources of energy are the fuel used to heat hot water, the gasoline that propels a car, the dammed water that drives the turbine in a hydroelectric plant, and the spinning yo-yo that can climb up its own string. Inanimate energy sources are of central importance in raising the standard of living of mankind above the subsistence level.
The physicist distinguishes among several types of energy, including kinetic energy (associated with a flying arrow or other moving object), elastic energy (associated with stretched or compressed strings), chemical energy (associated with fuel-oxygen systems or a storage battery), thermal energy (associated with the sun and other objects that are hotter than their surroundings), and nuclear energy. Applications of the energy concept in the science of mechanics, which you are studying now, usually concentrate on kinetic energy, potential energy (to be introduced in the module Conservation of Ener and work (the transfer of energy by the action of a force. Sometimes the phrase "mechanical energy" is used to refer to the forms of energy of importance in mechanics.
When you get on your bicycle, you have undoubtedly noticed that it takes a good deal of effort to get yourself moving rapidly. If you exert yourself very strenuously, you can reach a given speed after a short distance; or you can take it easy and pedal over a longer distance to reach the same speed. In some sense it always takes the same amount of "wor k" to reach a given speed -either a large exertion for a short distance or a small exertion for a long distance. You may also have noticed that if you are carrying a passenger on your bike, then it takes more "wor k" to reach the same speed.
It turns out that these intuitive relationships among the "wor k" done on a system, its mass, and changes in its speed can be sharpened into a precise statement, called the work-energy theorem. (One caution,though: the technical definition of work needed for this precise statement is different from its everyday usage and physiologic meanings; e.g., you do no work on a heavy box \ by merely holding it still.) As you will begin to see in the present module, this relationship between' work and mechanical energy gives you a new and powerful tool for the solution of many problems, a tool that is often easier to use than a direct application of Newton's second law.

GENERAL COMMENTS
Your text deals exclusively with the energy concept in the context of Newtonian mechanics. Actually, energy is much broader than that, and your understanding will be helped if you are aware of other scientific uses of the energy concept.* The text's definitions of work and kinetic energy are chosen to be useful in mechanical theory, and therefore may sound very abstract to you. For Objective 2 don't be put off by the reference to impulse at the beginning of Section 8.4. The same result follows directly from Newton's law as stated in Eq. (5.1) in Section 5.1. You may be surprised by the integral over the variable v on page 116. The purely mathematical manipulation is an application of the abstract definition of the definite integral to the sum that has to be evaluated. The kinetic energy K = (~)mv2 is defined in the text after Eq. (8.5). It is a scalar quantity that may be found for a system of several particles by merely adding the kinetic energies of the individual particles.
Objective 3 is treated only briefly on p. 117; you will gain your understanding of the work-energy theorem by working on problems or by looking at another text. Quest.
The latter is applicable in the limit as ~r approaches zero. By integrating both sides you obtain Eq. (6-10). As explained in the Introduction, the work concept is especially useful when forces in a problem vary as functions of position, because the integration over position in Eq. (6-10) can easily take this variation into account. The integral in Eq. (6-10)is called a line integral because the integration extends along the line (path) of the particle's motion. In this course you will only be asked to evaluate the work for motion along a straight line.
A very important point to remember in applications of Eqs. (6-2) (constant force) or (6-10) (variable force) is that the force may be the resultant force acting on a particle, or it may be a particular force of interaction of the particle with another object (pull of a rope, friction, force of gravity). The two very good examples in Section 6-2 both happen to have zero resultant force (zero acceleration), and that will not always be true (Examples 3 and 4).
Note that kinetic energy K = (~)mv2 is a scalar quantity, not a vector, so that the relative directions of the velocities of many particles in a system are immaterial as far as the total kinetic energy is concerned.
Objective 3 is stated in Eq. (6-14) and illustrated by Examples 3 and 4. Applications of the work-energy theorem are very diverse and can involve the use of Newton's laws to help find the forces that must be known if the work is to be calculated. Here you must again remember to distinguish between the resultant force on a moving body (this accelerates the body and affects its kinetic energy) and the individual forces of interaction with various separate objects that may do positive or negative work on the moving body. Free-body diagrams will help you keep these in mind.
Objective 4 is treated in Section 6-7, and one example with a constant force is given. If the force is not constant, Eq. (6-15) applies. Together with Eq. (1) in this study guide, you can show that the first equation in Example 5 is true in general: If a car or other vehicle has an engine that delivers constant power (constant P), and the velocity changes, then the resultant force F acting on it must vary; if the resultant force is constant and the velocity varies, then the power P must vary. Keep all the possibilities in mind, and look closely at the study guide Problems G and H keyed to this objective. We suggest that you begin reading Section 7-2, which deals with Objective 1. Note that the physicist's technical definition of work is contrasted carefully with the everyday notion. We consider the expression "work done by force" to be better physics idiom than lithe work of a force II used to introduce Eq. (7-3).
The integral in Eq. (7-3) is called a line integral because the integration extends along the (possibly) curved line (path) of the particle's motion. In this course you will only be asked to evaluate the work for motion along a straight line. 7-1 to 7-6, 7-15, 7-21 K 7-7 to 7-9 L, M 7-16 to 7-19(a), For Objective 2, return to Section 7-1, where the kinetic-energy formula is derived. You may have been surprised to see the integral over the variable v on p. 94. This purely mathematical manipulation applies the definition of the definite integral as the limit of a certain sum, provided that the limits of integration correspond on the two sides of the equation, as pointed out in the text just before Eq. (7-1). Continue with Section 7-3. For a system consisting of several particles, the total kinetic energy is the sum of the individual particles' kinetic energies.
Objective 3 requires you to put together Eqs. (7-2), (7-3), and the unnumbered equation in the box just before Eq. (7-2): For Objective 4, turn ahead to Sections 7-9 and 7-10. The important result is presented in Eq. (7-17). The power P is, of course, constant when both vector force and velocity are constant. When a car or other vehicle accelerates with an engine that delivers constant power, however, then the velocity changes and so does the force (Problem H). If the force is constant and leads to acceleration of the moving object, then the power must vary according to Eq. {7 -17) (Prob 1 em G).
*Seven lines from the bottom on p. 134, the first equation should read v fl = v i2 ' not v fi = v i2 .

4(WS 2}
The limitation of infinitesimal displacements is removed in Section 9-3 where there is a careful derivation of the work: -+ done on a particle by the resultant force F. Two very useful theo~ems*follow, concerning the work done by each individual force of interaction (rl' r2' ... ) and the work done by each rectangular component (F x ' F y ' F z ) of the force.
Regarding the statement after Eq. (9-7a), consider this remark: The relation between work and kinetic energy holds regardless of the path followed by the particle, but the numerical values of the work done and kinetic energy change will usually differ for different paths. Study the informative examples at the end of the section carefully. Sections 9-4 and 9-5 add more details concerning Objective 3. You may skip the first subsection "Work Energy and Impulse Momentum" in Section 9-4.
For Objective 4, turn to Section 9-6. The important result is presented in Eq. 9-13, which is illustrated by Problems G and H in this study guide. The power P, according to the equation, is constant when both vector force and velocity are constant. When a car or other vehicle is accelerated by an engine that delivers constant power, however, the velocity changes and so does the force. If the force is constant and leads to acceleration of the moving object, then the power must vary according to Eq. (9-13).
STUDY GUIDE: Work and Energy 5 PROBLEM SET WITH SOLUTIONS A(l). The forces F l , F 2 , and F3 act on a particle of mass M. The forces vary with the particle's position.
(a) State the work done by force F2 as the particle moves along curve C from position r l to position r 2 .
(b) State the total work done on the particle when it moves subject to the three forces as described above. Solution (b) W = b B(l). A particle moves along the y axis from Yl to Y2 according to the force law given below. Find the work done on the particle and Ql the particle in each case. The constant Yo equals 3 m.   (f) Because of the gravitational force, not all the power is available to increase K, but some is used to raise the elevator.
H(4) . The following questions deal with a sports car of mass 1500 kg (including the driver and a companion). Its maximum power is 7.2 x 10 4 W: There are three forces: gravity, friction, and the normal force due to the inclined plane. Since the displacement is given, the net force, acceleration, and speed do not need to be calculated. Answers: ~ (b) 890 J, -520y, 0; (c) vi = 6.0 m/s. N M(3). To come to rest, the block has to lose its kinetic energy. Since the spring is at rest at the end, when the block is at rest, any work done to give the spring kinetic energy will be returned.
..- 2. Find the work done on the block by the applied force and by friction after setting up the work integral. 3. What is the kinetic energy at the moment the force ceases acting? 4. What is the power expended by the applied force at the moment before it ceases acting? 5. State the work-energy theorem (as applied to a particle) in the form of an equation and briefly identify the meaning of each symbol used. A 20 OOO-hp (1.5 x 10 7 W) railroad locomotive accelerates a 10 OOO-ton (1.00 x 10 7 kg) train from a speed of 10.0 m/s to 30.0 m/s at full power along a straight track. 4. How long a time interval is required for this process, neglecting friction? 5. Find the kinetic energy and the speed of the train as functions of time during the interval. 6. Find the force accelerating the train as function of time during the interval.
5. Introduce time variable. Use initial speed. Solve for v(t).