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About energy in 40 Flash animations for teachers
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Energy: What it IS, what it is NOT, and How we know
© Walter Scheider 2003

All About Energy:   39 "Flash" animations for teaching about the physics of Energy

from "Energy" workshops at AAPT's 127th Convention 8/1/2003 in Madison, WI, and at CSTA 10/11/2003 in Long Beach, CA

From the links in the script below you can choose to view any of the 40 animated movie snippets that illustrate the points in the script that follows:
(Workshop participants received a CD for more convenient use of these movies in teaching. The CD is available at cost from the Cavendish Press address at the top of this page)

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The bad news is that, in the classical view of physics
  • Energy is not an invisible fluid or anything else substance-like
  • it doesn't flow
  • it's not something that anybody or any object can have
  • nor is it the ability to do work
  • nor can the question,"What is Energy?" be answered.
The good news is that
  • you can go on teaching about energy pretty much as you have been,
  • using familiar models,
  • as long as you don't make any unjustified claims about what energy "really is."
The even better news is that
  • all this is not at all mysterious
  • and the questions,"What do we know about energy?" ...
  • and "How do we know?" ... can be answered quite clearly.

    What follows is an effort to present the origin, meaning, and use of the unified concept called "energy" (encompassing what is normally meant by kinetic energy and potential energy), at the introductory level for the student who may have had a brush with modern physics but whose context for understanding the physical world is classical (Newtonian) physics.
Here are some of the key facts that will be central to our story:
  • The most important facts that we know about energy are contained in the principle of Conservation, which derives from the Work-Energy Theorem, which itself is derived from Newton's Laws of Motion.
  • Because these principles are derived from Newton's Laws of Motion, what they tell us about energy is unambiguous, applies throughout classical mechanics, and is valid in all inertial (i.e. unaccelerated) frames of reference.
  • The principle of Conservation of energy, like Newton's Law, is silent on the zero of location and velocity, and deals only with differences.
  • Because forces act in equal and opposite pairs between pairs of objects (Newton's Third Law), the Energy Conservation principle applies only to closed systems, and does not apply to parts within a system.
  • The definition of the term, energy, originated about 1800. The word, originally a name for the quantity ½mv2, has since been expanded, by general agreement, to include energy-equivalents called, "potential energy." The word, however, can not be freely re-defined.
  • Energy is not itself a quantity independent of the frame of reference in which it is measured, it is not trackable, and no unique calculation of its "flow" in any particular process can be made.

What is energy?
      As Appendix II makes abundantly obvious, teachers, teachers of teachers, and writers of texts, have a lot of trouble when they try to tell their students what energy is all about. They do not have as much freedom to define the territory as they apparently think they do.

      One can disagree about what model or metaphor for "energy" is most suitable for the classroom. It is agreed that a teaching metaphor can, and should, simplify the subject, even at the risk of some inaccuracy. To accomplish this, it is acceptable to strip the model of advanced refinements, but it is important not to mislead students about basic facts.

      Forgotten in much of this discussion is that "energy" itself is a metaphor. Before we can decide how best to simplify it for teaching, we ought to know what we can accurately say about energy before we simplify its presentation in a "model" or "metaphor".

      A metaphor says, "It is like this." And, then comes the question, "Is it really like this?" and "How do we know?"

      The concept of "energy" did not start out as a metaphor. It all began around 1800 when Thomas Young applied that term to the quantity mv2 (he didn't know that it would be better with the factor ½). To physicists of the time, mv2 seemed to be the nearest thing to mv (momentum), as a measure of "motion," that didn't have the nasty habit of totaling zero for two objects that moved like crazy, toward each other.

      The naming of a quantity is an arbitrary act, much like the naming of a star or a baby. A name is neither true nor false, and contains no fact that can be tested experimentally. It becomes accepted by common usage. Only later were some matters of fact revealed: heat turned out to be ½mv2, and energy was found to obey a conservation principle. These, because they are questions of fact, required evidence.

      The discovery that heat is molecular kinetic energy is a story that is well known, and need not be retold here.

      The conservation principle grew out of the question, "How does an object acquire ½mv2?" The answer to that question comes directly from the Laws of Motion proposed by Isaac Newton. These laws, which are the basis of classical mechanics, have been amply verified over 250 years of experimentation and application.

The Frame of Reference problem
      A foreboding of problems with this idea that an object can acquire, and have energy is in the arbitrary zero level of gravitational potential energy. We explain this to students as just one of those things that is obviously there because things can fall just so far. By moving a table away from the path of a falling object, or drilling a hole in the floor, we say we can increase the gravitational potential energy that an object has. Here would be an opportunity, missed by most teachers, to explain that potential energy is not something that an object has, but instead refers to a process in which it undergoes a change.

      Were this the only problem with arbitrary zeroes, we could pass it off as an amusing quirk of gravity. It is, however, only the beginning of more profound difficulties related to frames of reference.

Frames of reference
      What is a frame of reference? Think of it as a piece of graph paper (etched on a transparent glass plate), being used to make measurements such as those in the movie axes that deals with two points in a plane. That piece of graph paper can be placed on top of the two points in a number of different positions and orientations. With each position of the graph paper, there are two pairs of numbers, called coordinates, that, together with the graph paper can be used to tell someone where each of the points is located. With each move of the graph paper, these coordinate numbers will change.

      The coordinate values are not physical quantities. They are marks on a ruler.

      The distance between the two points is a physical quantity, because it is determined in the absence of any graph paper.

      This distance can be calculated from any two pairs of coordinates. Regardless of how the graph paper is laid down, the value of the
square root of (x2+ y2) is the same. The distance between the points is a physical quantity, that is, it is an attribute of those points that is independent of any measurement. even though that distance can be calculated from any one of these sets of measurements.

      Conclusion: It is not wrong to make measurements using the grid of a piece of graph paper. These measurements may enable us to determine physical quantities that are independent of the placement of the graph paper. But it would be wrong to read into the coordinates themselves any physical attribute.

Nature's deafening silence about velocity
      The question, "How does an object acquire ½mv2?" plunges us immediately into one of the more perplexing aspects of trying to devise a metaphor for energy. The question presumes that an object can "have" a certain quantity of ½mv2, which has come to be called, "kinetic energy," (energy of motion).

      To determine how big ½mv2 is, one must know how big v is -- how fast the object is moving. Newton's Law is famously silent on this question. Newton's Law deals only with how velocity changes under the influence of a Force. This silence is the foundation of what is referred to as Galilean, or classical relativity.

      It is not an omission on the part of Isaac Newton that his laws are silent on that question. Newton's Laws are silent because nature is silent on that question. This principle, that the laws of mechanics apply equally in all inertial (non-accelerating) frames of reference, is called Galilean relativity. The principle deals with questions of fact, and is subject to verification.

      You can verify Galilean relativity for yourself. Go up in an airplane, wait until the airplane is cruising at constant velocity, and then, while standing in the aisle, drop, pour, throw, catch, roll any object, play with a yo-yo, walk and run and jump in the airplane's aisle. You will find that, except for the effects of bumpiness in the ride, all of these activities follow the same rules that apply on solid ground.

      For quantities that depend on an object's velocity, there are implications of the arbitrary zero of velocity. Suppose Meredith walks down the aisle of a bus that is moving at constant velocity of 5m/s. Making speed measurements on the bus, we find that Meredith is walking with a speed of 2m/s. But her speed measured by someone at the side of the road would be 7m/s. Neither value is wrong.

      Nature's silence on the zero of velocity means there is no measurement that can tell us what Meredith's velocity is, nor can we tell how much kinetic energy Meredith has. Her kinetic energy is like a coordinate in the scheme of energy, but it is not itself a physical quantity, because it is a function of the velocity of the meter stick used to measure it.

      We can decide to use a frame of reference in which the earth is at rest. This may seem a natural choice, though not necessarily more correct than any other (inertial) frame. A physical quantity must be independent of the choice of frame of reference. No model that correctly represents (kinetic) energy can determine for us how much kinetic energy an object has. Energy is not like "gas in the tank" or "money in a bank account" or "an invisible fluid."

The frame-of-reference problem for changes
      Nature is silent on the question, "What is the object's velocity?" but nature is not silent on the question, "How is an object's velocity changed?" Newton's Law is about that latter question. It says that the application of Force can result in acceleration, which is the rate of change of velocity. Thus, from Newton's Law we should expect to be able to find out how an object's (kinetic) energy is changed, and how an object acquires (kinetic) energy.

      From Newton's Second Law one can derive that the (scalar) product of Force and displacement,
{F · )x}, is exactly the means by which an object acquires additional ½mv2. (To those who named the expression {F · )x} "Work," we owe the endless confusion that this has generated for students. Nevertheless, the nomenclature remains firmly stuck in the scientific vocabulary.)

The Work-Energy Theorem
   F · x = (½mv2)                       [1]

      Because the Work-Energy Theorem is derived from Newton's Law, its scope is all encompassing, valid in all inertial (non-accelerated) frames of reference. This theorem is well known, but is much under-rated. It appears in texts usually as a useful "formula" for a limited class of problems. Not so. It is in fact the basis for understanding what it is about energy that we can say is "conserved."

The Work-Energy Theorem is about "changes in..."
      On each side of [Eq1] is a "delta," ()), meaning the Theorem deals entirely with changes.
On the left, a Force is multiplied by )x, a displacement, or, more generally, a rearrangement. On the right,
)(½mv2) refers to an increase (or decrease) in energy that occurs during this rearrangement.

      Even though we have shown a definite relation [Eq1] that deals with change in (kinetic) energy, there is still a "frame of reference" problem with change in (kinetic) energy. (This is because "velocity" appears as v2 in Eq1.)

      Meredith (meredith01) pulls a wheeled suitcase along the ground. As she pulls it, the speed of the suitcase increases (neglect friction losses).

      Next, (meredith02) we see her do the same thing on a bus that is not moving. This is not different from the previous movement; as she pulls, the speed of the suitcase increases.

      Finally, suppose that the bus is moving -- is already moving and continues to move at constant velocity -- during the time that Meredith pulls the suitcase from the rear to the front of the bus.

      What does the Work-Energy Theorem tell us about the work Meredith does when she pulls her suitcase along the moving bus? Suppose that there are two observers making measurements from which Work done and {½mv2} gained are calculated.

      (1) Observer 1 is on the bus (meredith03). The bus is moving (at constant velocity). Observer 1 makes measurements in the frame of reference of the bus . The suitcase starts at rest, and reaches a speed that can be calculated by the Work Meredith does. A relatively small amount of Work produces a small increase in kinetic energy. The two sides of the Work-Energy Theorem are equal to each other.

      (2) Observer 2 is not on the bus (meredith04), and is observing (measuring) the bus and Meredith in a frame of reference fixed on the ground by the side of the road. Meredith and her suitcase start with non-zero velocity, and non-zero kinetic energy. Not only is the kinetic energy greater now, but also the increase in kinetic energy is now greater [because (72 - 52) is greater than (22 - 02)]. The Work done by Meredith is correspondingly greater as well, because Meredith, carried along by the bus as she walks in the aisle, moves farther. (Note that (2) is not a different process from (1); only the observation is made from a different vantage point.) Both sides of the Work-Energy Theorem [Eq1] are now greater; both sides are greater, but the two sides are still equal to each other.

      The energy that Meredith's Work contributes to the kinetic energy of her suitcase is evidently not a fixed amount, but depends on the vantage point from which the measurements are made.

      This is the "frame-of-reference" problem that occurs when a model of energy is used in which Work and the resulting change of (kinetic) energy are portrayed as a "transfer" of something that is substance-like.

      It is not a failure of the Work-Energy Theorem [Eq1]. As seen on the bus, the Work done and the increase in kinetic energy are both small, and are equal to each other. As seen from the side of the road, the Work done and the increase in kinetic energy are both larger, and are equal to each other. The Work-Energy Theorem correctly describes both cases.

"Flow of energy" and the failure to track
      There is finally the question of "Flow" of energy. Any substance of definite magnitude, like a gallon of gas in the tank, or of definite value, like a dollar in a bank account, that we associate with a conservation principle, has the characteristic of being "trackable." Trackable means there is always an answer to the question, "Where is it now?" or "Where did it go?"

      A quantity of a "substance" may be broken up into smaller segments, or merged with a larger amount, but one can always account for the various portions, and in total they must be equal to the initial amount. The "flow" of these portions must look the same to everyone. If a gallon of gas goes from the reserve tank into the main tank, then everyone who is able to track this flow will observe it going from the reserve tank to the main tank.

      It is, in the end, the fact that energy does not track, that establishes that energy is not a substance-like quantity of any sort.

      Consider, finally, this dramatic example: A space vehicle moving horizontally far above the earth's surface is ready to scuttle its empty main fuel tank. The problem involves the two parts of the spaceship and a spring between them. To make calculations easy, we specify perhaps unlikely parameters: The payload (front) part and the empty tank are of equal mass. They are initially joined by a latch that is released to make the separation. Initially, of course, the front and rear halves move as one.

      When the latch is released, a compressed spring between the front and back part pushes the two halves apart. Suppose we want to portray the Work done by the spring and the changes in kinetic energy of the front and rear parts of the spaceship as transfers or flow of energy.

      Physical intuition suggests that the natural way to solve the problem is in the frame of reference in which the original vehicle is at rest. The center of mass of the two halves will continue to be at rest in this frame of reference after separation. Symmetry (or momentum conservation) dictates that in this frame of reference the two halves will move apart with equal, but opposite, velocity. Each half will have half of the spring's initial potential energy. This apportionment of energy is shown in the movie (rocket01).

      Observers on earth may, however, for very good reason, prefer to solve the problem in earth frame. In this frame both front and back have considerable kinetic energy before separation. In that case, however, the rear half will be slowed down by the spring, so that its kinetic energy decreases (for simplicity choose a spring that decreases the kinetic energy of the rear to zero). The front half gets not only all the spring's initial potential energy, but also the kinetic energy given up by the rear half. The numbers, and the fluid-in-the-tubes model, will look very different, even though the events were exactly the same (rocket02).

      Not only are the quantities of energy different in the two frames of reference, but even the direction of "flow" of energy, if there were such a thing, is quite reversed.

      This is the "failure to track" problem. It is the fatal flaw in the substance model of energy.

      Again, this is not a failure of the Work-Energy Theorem [Eq1]. In the two observations of Meredith on the bus, that "flow" is different, even while the Work-Energy Theorem remains verified.

      In the case of the spaceship, not only the quantities but also the direction of "movement" of energy are different. To sum up, the flow is different in different frames of reference, and is therefore unlike the flow expected of any substance-like "stuff" that might on the surface of it seem to resemble energy.

Systems and Conservation of Energy
      In the examples used so far, we have relied on the Work-Energy Theorem for solutions. All the "Meredith" examples were "one-Force one-Object" examples.

      In the problem of Meredith and the suitcase, we were quite right to recognize that the suitcase had gained energy (it was moving faster), but the question of where it got the energy remains. Surely Meredith deserves some credit; if she had not pulled the suitcase, it would not have increased its speed. Yet, the Work-Energy Theorem, which is the foundation upon which we build, tells us that credit belongs elsewhere.

      According to that Theorem [Eq1] Work is done by Forces, and not by people or things. There is more than semantics involved in that distinction.

      The movies (meredith03 and meredith04) illustrate Work as coming from a tube labeled "-Work". The Work was represented as a source of energy. The Work done represents a certain amount of energy that we imagined was brought into the system from outside, by a Force exerted by Meredith.

      It is not a far reach to describe the Work as a kind of proto-energy, something that could become energy, and to describe its role before it becomes an addition to kinetic energy, as "potentially kinetic energy" -- "potential energy" for short -- somewhat in the way that cake mix could be described as "potential cake."

This appears to bring us to a kind of conservation theorem: First it is potential energy, then through the process called Work, it becomes kinetic energy, and the total amount of energy in both categories remains always the same. In the process, the potential energy is consumed.

      Yes, but not quite.

Newton's Third Law and the Other Force
      Newton's Third Law tells us that there is never just one Force. Forces come in pairs. The pair of Forces is like Forces at the ends of a spring. Neither end of the spring can do anything unless both ends do. The two Forces pull the two objects toward one another, or they push the two objects away from one another. A spring can't pull unless both ends pull, it can't push unless both ends push. Something must be attached on each end. And the potential energy in the spring is consumed or diminished by the Work done by both of the Forces, never by either one alone.

      If the Work done by the spring (or any pair of Forces) is to be associated with the consuming of the potential energy, the Work done by both Forces has to be counted.

      The one-Force one-Object problem neglects that other Force, the one that acts on the other Object. The one-Force one-Object problem can not be solved as a potential energy problem. The Work-Energy Theorem can be used to find the change in kinetic energy of one object as a result of the Work done on that object. In that application the Work done on the one object has to be known. The problem is not solved by a conservation principle.

Potential Energy and Closed Systems
      Only when there are no loose ends -- no loose Third-Law partner Forces that act in undetermined ways on objects outside of what we will call the "System" -- can the total Work done in a particular sequence be described as the result of consumption of Potential Energy of the system. The condition: "No loose ends," defines a Closed system.

      In a closed system, the Work-Energy equation,

            F · )x = )(½ mv2)            [Eq1]

can be summed, for all the objects and all the forces in a system,

            (F · )x)TOTAL = )(½mv2)TOTAL             [2]


            WorkTOTAL   =    )KETOTAL            [2a]

Inside a closed system, Work done can be identified as a decrease of potential energy (abbreviated PE),

             Work  =  -)PE.

Or, summed (totalled), (WE-theorem)

            -)PETOTAL     =    )KETOTAL             [3]

The Principle of Conservation of Energy

For a closed system
   {PE + KE}TOTAL     =     0            [4]

with the variables defined as,
KE = {½mv2}         PE = -{F · x}           [4a]

     Conservation of energy applies to the state of a closed system. The principle is true independent of frame of reference. A closed system is any collection of objects of which none are acted upon by a Force whose Third Law partner Force acts on an object that is not part of the system. It does not apply to individual objects or parts of the system, separately (unless the parts themselves satisfy the condition for a closed system).

      Like the distance between two points in a plane (axes) discussed above, the energy state of a (closed) system is a physical fact that is true in any (inertial) frame of reference, and exists even in the absence of a frame of reference. It is true that, just as the distance between two points can be calculated from measurements of components, so the energy state of a closed system is usually determined by calculation from components in some particular frame of reference. In both cases, the components in themselves do not have physical meaning.

      Notice that all terms in the Energy Conservation law have )'s in front of them: all the terms refer to changes. These changes are brought about by a process in which Work is done.

      It does not follow that, just because one can assign physical value to the change in Potential Energy, one can also assign physical values to Potential Energy itself before and after the Work is done.

    The association, for example, of {)PEGRAV} with {mg)h} does not imply that there are physically significant values PE0 and mgh0 before and values PE1 and mgh1 after the process represented by the )'s.

      Both the Work-Energy Theorem [Eq1] and the conservation statement [Eq4] give meaning only to the increments ()'s) on each side of the equation.

      It is not required that potential energy be reversible, i.e. path independent. Remembering that the origin of the concept of Potential Energy involves no more than its identification with the Work done in a particular re-arrangement process, no restrictions have been placed upon the kind of work done.

      There is a desire to have categories of frequently-encountered configurations of Work that, for convenience, permit us to write formulas for calculating )PE from problem parameters (as in
PEGRAV = mgh). Forces which do Work that is reversible in this sense are called "conservative" Forces. It needs to be understood that this is a convenience; ultimately all Work is done by Forces that are conservative.

How, then, do we make it "work," even when it is wrong?
      We seem, in spite of all we have said above, to be able to illustrate conservation of energy for our students using a "flow of liquid" scheme as if there were a definite, determinable, "amount of flow" of energy, within and outside a system, closed or not, all or in part. What sort of magic enables us to do that?

      We do it by constructing examples and problems that hide the difficulties we have observed (hiding the difficulties that a perceptive student will eventually encounter). Good pedagogy, however, may dictate that we do this. A good teacher brings to the classroom the ability to simplify a subject to make it understandable at the level of preparation and maturity of the majority of students. Some of the points we have raised are subtle, others are of importance mostly in situations that one doesn't encounter in every day life.

      On the other hand it is good to avoid saying things that aren't so. It is probably good teaching to warn our students when we are being allegorical, as when a chemistry teacher portrays an electron as a little point object with the caution not to take that literally, explaining that when they are older and study the matter further they will find out that it isn't quite so.

Some hints on how to simplify without lying
      Here are some ways to skirt some of the more subtle issues of energy:

  • Silently adopt a default frame of reference with any or all of the following characteristics:
    (a) ground level is the implicit vertical zero
    (b) the place where an object begins its motion is the horizontal zero
    (c) zero velocity is when something is at rest with the earth's surface
          By saying nothing, you implicitly define frames of reference for both location and velocity. Having done so, you can deal with "flow of energy" in a particular frame of reference, where it works, as if you were dealing with it in general, where it doesn't. (This will require some explanation in examples in which the default frame of reference is not obvious or natural, as in the rocket example (rocket01/rocket02)).
          What you do here is analogous to what you do when you sketch a problem involving two points in space with an x-y axis implicitly anchored to the corner of the blackboard (axes). It is not wrong to use particular, convenient coordinate axes in numerical problems; it is only wrong to imply that the components are physical absolutes.
  • Take advantage of the fact that when one of the loose ends of a "third law" force pair acts upon a very large object, such as the earth, the Work done by that loose end can generally be neglected. This stems from the fact that in an interaction, (collision, repulsion, attraction) between a small and a large object, the ratio of work done on the large object to that done on the small one is inversely proportional to the ratio of the masses.
          For example, the problem of a falling object can be solved quite well as a one-object-one-force problem, ignoring the Earth both as an object and as the force exerted on it by the falling object, because the Work done on the Earth is negligible. While the two forces of a Third Law pair are equal (and opposite in direction), the Work done by the two forces is not necessarily the same.
  • In examples that are done primarily for development of intuition, as when bar graphs are drawn or a scheme of liquid-containing tubes is used, one may make an estimate of some otherwise undeterminable amount of Work, done by a particular force, based on how much Work must have been done to make everything else in the problem come out as it did.
          For example, the force that does the Work of raising a roller coaster to its initial high point is usually impossible to know, but one can deduce it from the resultant elevation of the coaster from A to B (mg)h) in the movie (rollercoaster4a). Somehow, the machine will come up with this energy, or the coaster won't make it to the top.

Are there different kinds of energy?
      We have rejected the idea of a generic substance called "energy." So, the question of whether this generic stuff is all exactly alike or not doesn't arise. There certainly is, among the things that we recognize as energy, a variety of categories, but these are, like all classifications, matters of choice, based mostly on pragmatic criteria.

      Kinetic energy is {½mv2}. Heat looks and feels different, but it is {½mv2}, only on a scale too small to be seen by the eye.

      Potential energy, we have shown, is to kinetic energy as a check is to cash. A check defines a process by which an IOU yields cash. Potential energy was labeled such when it was observed that Work had the capacity for increasing the supply of kinetic energy. Work is an IOU being cashed.

      Just as checks can be placed in categories by the source (payroll, personal, bank certified, line of credit, etc), or by the issuing institution, or by a variety of other criteria, so can Potential Energy be placed in categories in various ways. Work is done by Forces, and so it is natural that one might want to make categories by the type of Force involved. The four fundamental types of Force that physics tells us exist in this universe are perhaps natural categories (Energy scheme). Other categories may be based on usage, such as "chemical" (which are really electrical forces between atoms and molecules), "contact forces," "wind forces," etc.

It's not all a one way street
      One should not be misled by the derivation of either the Work-Energy Theorem [Eq1] or the Energy Conservation principle [Eq4], into believing that all the processes referred to by the )'s in those equations go from left to right, that it is always Work releasing kinetic energy.

      Work done by gravity (falling water in a hydroelectric power plant) for example, can be used to separate electric charges that are sent into homes in a state of elevated electric potential energy difference (hydroelectric power).

      A heat engine, such as the one in the automobile, can take heat (random kinetic energy on the small scale) and produce automobile motion (directed kinetic energy on the visible scale). Heat can also, of course, be used to generate electric potential energy.

What does energy tell us about how things work?
      There is the story of the physicist in Henry Ford's shop. "It'll never work," he told Ford. "A thing can't push itself - nothing can exert Force on itself. And the road can't do any Work on it, because the road is not moving. You'll never do it without a horse." Ford ignored the physicist.

      Those of us who are accustomed to looking to science, and physics especially, to tell us how things work, may be surprised to find that by and large, energy rules don't tell us how things work. The sequence in the movie (propulsion) explains how a car is propelled. It is pushed neither by itself, nor by the road, but by an elastic Force, like a horizontal spring, compressed between an unevenness in the road and the rubber tread in the tire. Torque from the car engine turns the wheel, compressing a portion of the tangential top-layer of tire, which then pushes forward on the wheel in its attempt to relieve the compression.

      Once you have figured out how it works, you can apply energy concepts. Work is done by the force of the compressed layer of rubber, acting between the road and the car (like the spring in the propulsion model) (road-car propulsion).

      The energy conservation principle tells us about things that can't work, in the thermodynamic sense. The first law of thermodynamics is a special case of the principle of conservation of energy. In a closed system, the best you can do is keep re-circulating the same total amount of energy that you started with.

      By and large, the kind of problems and examples that we give our physics students are ones in which we know the answers and then show how they conform to a Work-Energy exchange that conserves the total energy. Intuition-building is valuable, because the world runs on Work. Having a feel for how we can get Work done for us is a great practical asset. No place is this more true than in our search for fuels.

Bonds: Where the potential energy grows
      Work is done when forces, like springs, attract objects toward one another, or when they push objects apart. There are rare occasions, such as when a positively charged alpha particle is kicked out of a positively charged atomic nucleus during a radioactive decay, when a repulsive Force does Work that we can capture as kinetic energy. Mostly, we get energy from the Work done by forces of attraction. These forces form bonds.

      When two objects attract each other, providing that their motion is not otherwise restricted, they will approach each other in the direction of their respective forces, both forces doing positive Work (bonds00), with the release of kinetic energy. When this occurs, we say a bond has been formed, or made tighter (actually shorter). There are gravitational bonds (waterfall), chemical bonds between atoms (which are really bonds formed by the electromagnetic Force), nuclear bonds between the protons and neutrons in the atomic nucleus (held together by the strong nuclear Force), and many sub-categories among these classifications.

      Just as our food supply grows on farms, so our energy supply grows where bonds are formed. What we know about energy [Eq1 and Eq4] tells us that there is only one way to get energy, and that is to allow Forces to do positive Work. With rare exceptions, the only Forces willing and able to do this for us are Bond-formers.

The Work-Energy Theorem tells us where to look for fuels
      The gravitational force bond between water in Lake Erie and the earth can be tightened (technically speaking, "shortened" is the better word to use here) by allowing the water to fall over the turbines in the hydroelectric power station, to the lower water level in Lake Ontario. In the process the gravitational force does work on the turbines. Rather than going to an increase in kinetic energy, however, this Work is used primarily to increase the electric Potential Energy of the charges in the transmission lines (hydroelectric power).

      Chemical bonds are our most common source of usable energy. It is not the bond that gives us energy, but the process by which the bond is formed. In the combustion of gasoline, for example, carbon (and hydrogen as well) in the gasoline molecule forms a bond with oxygen in the air. It is in the process of forming carbon dioxide, with its bond of carbon to oxygen, that Work is done and heat (kinetic energy) is released. In the process, chemical potential energy is diminished.

      The fuel brings a price (bonds01 beginning) only because the process in which the bond is formed has not yet taken place. When we speak of "paying for the energy in the gasoline," we mean that we pay in the expectation that the fuel can be made to burn (form a bond with oxygen), and provide us with kinetic energy. We say the fuel "has" potential energy, when we mean that the fuel holds the promise of being able to do Work by going through the process of forming the bond.

      When the bond has been formed, there is no longer any promise that the process can yield up more energy (heat). The carbon (and hydrogen) that has burned to Carbon Dioxide (and steam) is no longer valuable (bonds01 later), and is sent to the automobile's exhaust system. It is not the bond that has the capacity to provide energy, but the two parts of the fuel (gas and air) before they form the bonds.

      Similarly, "potential energy" is said to "be stored" in animal muscle in the form of a "high energy bond," between a phosphate group in the molecule of adenosine tri-phosphate (ATP). As in the unburned gas-air, the phosphate in ATP is "high energy" because the bond with adenosine is a more loosely-held bond that actually holds the phosphate away from forming a tighter bond with the surrounding water. When it is released from the ATP, it bonds more tightly as inorganic phosphate ion with water. As before, the value is in the configuration that holds the promise of releasing energy in bond forming (or tightening). Unlike the gas-air fuel, ATP releases not heat but the Work needed to contract muscle.

Energy Without Work?
      The books would have us believe that Nuclear Energy is exempt from the Work-Energy Theorem. There is no Work done there, they say, no bonds formed, only little bits of matter "converted" to Energy.

      Don't believe that for one minute. Consider, where do they shave off the little bits of matter? (uranium01) After the nucleus has released energy, there are just as many of every particle -- proton, neutron, electron -- as there were before. So where did they shave off the "matter?" What "matter?" There are no protons, neutrons or electrons missing.

      The "shaving of little bits of matter" is a misrepresentation of what happens in the nucleus. What happens is that, just as in the formation of chemical bonds, there is Work done in the formation (or shortening) of bonds, only these are bonds held by the nuclear strong Force, rather than by the chemical (electro-magnetic) Force.

      During nuclear fission (splitting of uranium) and during nuclear fusion (binding of hydrogens to make helium), Work is done in the re-packaging of the nuclear particles (neutrons and protons).

      The nuclear forces and the bonds these forces make between the nucleons (nuclear particles) are so strong, that huge amounts of energy are involved. These amounts are so great that they can actually be weighed. The nuclear particles lose not only energy in forming (or shortening) bonds, but they lose the mass-equivalent of that energy that Einstein predicted from his theory of relativity.

      No particles are lost or missing.

      The loss of bond energy and its accompanying mass-equivalent had been known long before people knew about protons and neutrons. Chemists checking out the periodic table had been finding that atoms of the higher elements seemed to weigh less than they would if they were simply made up of integer multiples of hydrogen atoms. They called it the "mass defect," way back in 1880. Most books on nuclear energy contain a graph (nucbondenergy) in which the "mass defect" is graphed as a function of nuclear mass. On the y axis of this graph is the mass equivalent of the total energy that is lost through the formation (in the stars over billions of years) of the various isotopes from the separate protons and neutrons. Now that this "mass defect" has been identified as the mass of the bond energy of the nucleons in the nucleus, the graph is usually labeled, "Binding energy per nucleon."

      In uranium fission, energy is not released as a sliver of mass during the splitting. It is released after the uranium nucleus has been split into its so-called fission products. One such pair of fission products, for example, consists heavy isotopes of Barium and Krypton. These nuclei, because they are smaller, can pack their respective nucleons more compactly. The tightening (shortening) of bonds in these more compact product nuclei releases energy (uranium02).

      Now, back to the question: what got smaller? (We don't mean smaller in size, necessarily, but in mass.) Did some of the protons get smaller? Did some of the neutrons get smaller? Did some of the electrons get smaller?

      In fact, the answer to that question is "yes." But, you will respond, we've been taught that a proton has a definite mass, the mass that is listed in the tables to seven decimal places. Same for the neutron, and the electron. Surprisingly it's not quite like that.

      We'll explain that through an analogy. Electric charges and their interactions in chemical reactions are a good analogy for nuclear reactions, and they are easier to understand. What happens is essentially the same, except the Force is electric in one case and nuclear in the other. Because the nuclear Force is vastly stronger than the electric Force, the details are different. But the principles are similar.

      An electron is surrounded by an electric field. At each point in the surroundings of the electron, the electric field contains energy, with an energy density measured in Joules/cm3, that is well known. Energy and mass are one and the same (Einstein), so that the energy that is in the electric field is also an equivalent amount of mass. That mass is part of the electron. When the electron forms a hydrogen atom by forming a bond with a proton, the field surrounding them is diminished. Both the electron and the proton lose some of their field energy, and that loss represents some diminution in the total mass of the two particles.

      The amount of potential energy lost in chemical reactions such as the forming of the electron-proton bond in the hydrogen atom, or in the formation of water by the neutralization of hydrochloric acid (H+ions) with sodium hydroxide (OH- ions) is small, too small for its mass-equivalent to be measured by weighing it with laboratory instruments. [The neutralization of (dilute) acid and base is an experiment that can be done in the student lab. Temperature rise can give a good estimate of the bond energy of the H-OH bond inside the water molecule.] The energy given off can be said to have come from the electric fields, which have diminished (ions) as the ions have joined. The geometry of the particles and their fields makes calculation of that energy hard.

      Such a calculation is quite readily done, however, for a pair of charged parallel plates the shape of two "cookie sheets" that attract each other with a constant Force. Their Newtonian acceleration as they approach each other gives them kinetic energy that can be shown to be just equal to the original potential energy content of the electric field (cookiesheets). [see also a detailed discussion of this problem in W. Scheider, "A serious but not ponderous book about Nuclear Energy" Cavendish Press Ann Arbor, 2001]

      In contrast, the formation (or tightening) of nuclear bonds through the much stronger attraction of neutrons and protons to each other, releases energy in amounts great enough that its mass-equivalent can be weighed in laboratory scales. For example, the mass of the two protons and two neutrons in a helium nucleus, is measurably less than the total mass listed for these as separate particles in the tables. They have shrunk (in mass) by losing some of the nuclear field energy that formed the bond.

      No little slivers shaved off the nuclei. Just the tightening of bonds.

What kind of Force is Friction?
      One of the strong messages in the Work-Energy Theorem is that (kinetic) energy (including heat) comes and goes from a system by only one route, that is the route of Work (positive or negative) done by a Force.

      Friction does not seem to fit this pattern, whether it be the sliding friction between surfaces or the fluid friction of an object moving through air or water. What kind of force is friction?

      Engineers use a practical model that describes friction as a force (or, really two forces) between two surfaces that rub against each other. It uses a "coefficient of friction," an engineering approximation, to calculate a Force of friction that quite successfully approximates the effect of this rubbing in Force-acceleration problems.

      In fact, however, the friction process is not one in which Work is done by a Force. Instead, friction is a re-distribution of the kinetic energy of the large-scale, visible, rubbing objects, to molecular kinetic energy on the atomic scale (in these same objects), in other words, heat. In millions of tiny collisions between bumps on the two surfaces, molecules are jostled to move with increased average kinetic energy within both surfaces -- hence: heat. During these collisions, the large scale objects are slowed, giving the appearance that negative Work, one on the other, is reducing the motion of the large objects.

      Describing friction as a pair of macroscopic forces has the twin drawbacks that it fails to explain what forces these are, nor does it account for the increase in heat.

      Collisions that create random molecular motion (heat), at the expense of reducing the directed kinetic energy of the large objects, leave the total amount of kinetic energy unchanged. If Work were being done, the total amount of kinetic energy would increase. (There is always a small amount of kinetic energy that goes into permanent deformations in the molecular bonding in the top layers that rub against each other. This would be tallied as an increase in chemical (electric) potential energy, but it is normally negligible.)

Activation: Why are there any fuels left at all?
      When two bond-formers find each other, the bonding Force pulls each toward the other, and... Wham ... they go faster and faster and crash into each other and give up all the kinetic energy they have accumulated on the way, releasing this energy as heat. We say, it is a fuel, and it burns. It can be chemical or nuclear fuel.

      The intriguing question is, why haven't all the bond-formers found each other long ago, and made their bonds? Once the bonds are made, there is no potential energy left; they are no longer fuels. Why do we have any fuels left?

      The answer is "Activation." Carbon in wood and oxygen do not burn just by coming in contact. Why not? Because before the carbon and oxygen can combine to make a chemical bond, they have to smash through a barrier of repulsion at the surface of the molecules that at normal temperatures keeps them apart.

      You know, of course, what has to happen for wood to burn: you have to light it. The heat of the match, or of the kindling, or of an already established fire, causes these molecules to fly at much greater speeds, and if the speed is great enough some of the molecules will break through the barrier and join in a bond. When they form the bond, they release a lot more heat than was required to "light" them, and so once a fire is going it will provide the heat to spread the burning.

      The barrier is called an "energy barrier." Providing the energy to the molecules to be able to overcome the barrier is called "activation." Once a fire is going, the process by which the fire itself is able to activate further burning is described as a "chain reaction."

      A model for activation is shown in (ener-barrier). A ball on a hill would tend to roll down the hill. But if the ball is caught in a pocket in the side of the hill, it will stay there, because the rim of the pocket forms an energy barrier. One can provide the energy to overcome the barrier by lifting the ball out of the pocket, and then releasing it to roll down to the bottom. The overall process releases a lot more energy than was required to activate it.

      If there were no energy barriers, all the balls in the world would have already rolled to the bottom. If there were no energy barrier to combustion of carbon fuels with oxygen, all the carbon fuels would have already burned and become carbon dioxide.

      In order for uranium 235 to undergo fission it must first be activated by a slow neutron. The barrier here consists in the fact that the uranium nucleus is stable and will remain intact, until it is de-stabilized by the absorption of a neutron that turns it into uranium 236, an unstable isotope, which then splits. If there were not the requirement for activation with a neutron, there would be no uranium 235 left; it would all have fissioned millions of years ago.

      The need for activation may seem to be a nuisance, but it is also a great benefit. It saves the fuels of the world until we intervene to cause them to "burn." It gives us control over the burning of our fuels; they will only burn when we are good and ready for them to do so, down to the last split second, as in the internal combustion engine (four cycle internal combustion engine), where the spark plug is the device that lights the gas-air mixture just at the right instant.

Activation by thermal kinetic energy
      In real life, most activation of chemical reactions is not done by specific and directed "lifting over the barrier," as the model of the ball in the hill suggests.

      Most activation is done by control of the temperature of the fuel. "Kinetic theory" is about the random thermal kinetic energy that all particles possess, by virtue of their temperature. This kinetic energy is proportional to T, the absolute (Kelvin) temperature. For a sample containing a large number of particles, the thermal ("thermal" means temperature) kinetic energy is randomly distributed; the average particle kinetic energy is ½kT per degree of freedom of movement, where k is the Boltzman constant. A better model than a ball sitting in the bottom of a pocket, would be a model with a ball bouncing wildly and randomly around. A normal sized ball isn't bouncing wildly, because ½kT is a very small amount of energy for an object the size of a ball.

      But ½kT is not a small amount of kinetic energy for an object the size of a molecule. Atoms and molecules in thermal motion, which would form a bond if they could overcome an energy barrier between them, in fact can bounce around in their little pockets quite fiercely. Since their bouncing around is proportional to the temperature, heating can give them the energy necessary to activate the reaction.

      A wonderful analogy to this process is that of the domino that is set standing vertically on one of its smallest surfaces. It would like to fall over, but to do so it would have to first pivot on one edge, which requires the extra energy of lifting it to a position in which its center of mass is slightly elevated. In (domino) you see Meredith rocking the table back and forth to provide random bouncing energy to the domino.

      As she rocks the table harder and harder (corresponding to higher and higher temperature) the domino rocks as well, and comes quite close to getting up on one of its edges. Eventually it sits on its edge (it has been "pulled out of its energy pocket"), and it falls over.

Rolling dice and chemical equilibrium
      Some chemical reactions are essentially one-way reactions, like the ball on the hill and the domino, in that once they get out of the trap, they go all the way, and do not go back. [In the second part of (domino) Meredith tries to rock the table enough to get the domino to stand up again, and she fails.]

      The more interesting chemical reactions are those that go between two (or more) states that are not energetically very different. In terms of energy barriers, these reactions go between two or more pockets that are about equally deep. They do require activation, usually heat, but once activated they oscillate among the possible states, falling into one or another of the pockets only when the heat is turned down.

      A visual analog of this condition is the position of a cube, a die, with faces numbered 1 to 6. If the die is well made, when it is rolled it will come to rest with all of the six possibilities having equal probability.

      In (dice02) four such dice are pre-aligned in the improbable configuration of four aces, "snake eyes." Suppose that someone picks up the four dice and rolls them vigorously. They will have enough kinetic energy for a while to raise themselves to a position on an edge repeatedly, from which they can tumble down in any number of directions. They will eventually lose the energy to rise to an edge and will fall to the table in the last position they held. The configuration of four such rolled dice that comes up will vary with each roll, but it is unlikely in any one roll that two dice come up with the same number showing.

      "Loaded dice" are dice for cheaters and crooked gamblers. These dice have weights imbedded in them that favor certain numbers to come up more frequently. If such dice are used, there will be one number that is more likely to come up, the number on the opposite face is less likely to come up, and the 4 remaining faces come up with equal likelihood somewhere between. This is shown in the sequence (dice03) where the number 3 pocket is deeper, meaning it is less likely that a die, once in the pocket, will bounce back out. The number 4 pocket is correspondingly shallower, with a die that falls into it being less likely to stay and be there in the end. Out of four such dice rolled, there is an increased likelihood that more than 1 will come up a three, and a smaller likelihood that any will come up a four. With thousands of rolls, there would be probabilities such as 16% each for 1, 2, 5, and 6, 8% for 4, and 24% for 3 (the details depending on how heavily they are weighted).

      All of this is an introduction to what happens in chemical equilibria. In a chemical reaction,

                                                 A    +   B   ->   AB

there are two states for the atoms A and B. Chemists know how to calculate the likelihoods and how these depend on energy parameters, but it is sufficient for us to take, as an example, the case in which {AB} is known to be 75% likely and {A + B} is 25% likely. There are two pockets only, and the energy barrier may be small or great.

Exothermic reactions
It may be that A and B are originally separate, all in the {A + B} pocket. (Suppose that they come out of different bottles and are then mixed.) If the energy barrier is high, they may remain as {A + B} for a long time. If they are then heated, kT, the measure of molecular thermal energy, will increase, and may raise the average kinetic energies of A and B enough so that some of them will be able to jump into the AB pocket. Once there, they may bounce back and forth several times, as long as the temperature remains high. When the temperature is eventually lowered, they will drop into the pocket they were last in. The ones who are in the {A + B} pocket during the cooling will be able to bounce back out longer than those in the AB pocket. At a time when the AB's have sunk to where they no longer have enough energy to get out, those in {A + B} may still be able to jump to join the AB's. The outcome will be that 75% will end up in the AB pocket, and 25% will end up in the {A + B} pocket (exothermic).

      In this example, 75% will have gone from a higher potential energy (they were all originally in the {A + B} pocket) to a lower one. One is accustomed to thinking that chemical reactions go in that direction, toward lower potential energy, and although they did not all go that way, 75% did, and the average potential energy of them all is now lower. The "lost" energy has been released as heat. This is the "normal" way chemical reactions go. Reactions of this type are called exothermic (giving off heat). Reactions that occur at room temperature demonstrate this by heating the beaker. The neutralization of hydrochloric acid with sodium hydroxide (dilute!) is such a reaction; it is a reaction which students can perform in the lab to measure the release of heat, from which they can calculate an estimate of the H-OH bond energy.

Endothermic reactions
      What if all A's and all B's are initially present as AB, the most stable configuration by a ratio of 3 to 1. (The initial 100% preponderance of AB could be the result of a deliberate chemical separation of AB from A's and B's.)

      If the temperature is not elevated, and there is not enough energy on the average for the AB's to pass over the energy barrier, they may all stay as AB's for a long time, even though the equilibrium probability of the AB configuration is only 75%.

      If at that point the temperature is raised so that some AB's will have enough energy to break up, some will become {A + B}.

      As the {A + B} pocket accumulates A's and B's, some will of course bounce out, re-combine, and go back to the {AB} pocket. However, since there are initially so many more AB's, the rate of AB becoming {A + B} will exceed the rate at which the smaller number in the {A + B} pocket will return to {AB}. The net flow will be in the direction of the {A + B} pocket until the population reaches the equilibrium ratio of 75 to 25.

      If the sample is cooled at any time after that, for the same reason as before, the numbers in the cooled down final distribution will be, as before, 75% in {AB} and 25% in {A + B}.

      The interesting thing here is that now the final average potential energy of all the A's B's and AB's is higher than it was to begin with. The net transfer has been of 25% of the initial AB's going to the higher energy configuration, {A + B}. The reaction has taken energy from the surroundings!

      It should be remembered that this did not happen spontaneously. Energy had to be available to cause the AB's to be able to reach the barrier; those that did and came down on the other side took more energy from their environment than they gave back on the other side. Reactions of this sort are called "endothermic" reactions (taking in energy).

What distinguishes exothermic reactions from endothermic reactions is on which side of equilibrium they began. In both cases the samples went from their initial state to the equilibrium state. If they started out below, they had to take in energy to get to the equilibrium; if they started out above, they could give some energy up to get to the equilibrium.

      We are accustomed to reactions that give off energy, because they go to a more stable configuration. In biology especially, where systems go back and forth from one side of equilibrium to the other, it is not unusual for nature to have created conditions that support such cycling. Often energy is provided by catalysts, or enzymes, as they are called in the animal body.

      In the two movies (exothermic) and (endothermic) the equilibrium condition is shown to be the same in both, while the difference is that one begins with all the balls in the upper {A + B} pocket, and the other begins with all the balls in the lower {AB} pocket.

Why condense the steam, when you just have to boil it again?
      The movie, (steam cycle) has one purpose: to provide a qualitative aid to intuition for the practice in power plants to boil steam, let it do its work on the turbine blades, condense the steam, then boil it again, and so around the cycle. There is the appearance of wasted energy in cooling the steam, only to heat it again, with the added burden of having to actively remove the heat (from the condenser).

      This is normally the subject of rather abstract principles of thermodynamics that follow the study of the Carnot cycle, and other material that introductory physics students have not learned.

      Without claiming to be rigorous, we point to the fact that pumping water is energetically much more economical than pumping an equal amount, but hugely larger volume, of steam at the same pressure. The fluid between the water pumps and the turbine blades is essentially, though not exactly, a continuous flow, so that the steam pressure at the turbine and the water pressure at the pump are nearly equal. Even taking account of the difference in blade surface, it is plain that the Work done (Force times distance) is much less at the pump by virtue of the fact that water, and not steam, is being pumped.

      There are many factors that are totally left out in this picture, but the fundamental reason for condensing is accurately presented. The movie can be used to discuss other aspects of the steam cycle.

The winebottle
      The movie, (winebottle), helps answer the question, "How does the table know how hard to push up to support the winebottle?

An article about relativity, potential energy, and mass

      A recent article in The Physics Teacher (E. Hecht, "Is PE Really Real?" TPT 41 486 Nov 2003) deserves some comment. The article suggests that, because all potential energy is energy of Fields (electric fields, quark fields, etc.), and because in special relativity, (E=mc2), energy is mass, it follows that potential energy is mass. He suggests that this proves that potential energy is "real", but that we don't need the term "potential energy" any more, since we now know what it is: mass.
      If potential energy is Field energy (although this doesn't work out easily for quark fields and for gravitation) and Field energy is mass, the question, "What is potential energy?" is answered, assuming that you know what "mass" is.
      Because of mass-energy equivalence, relativistic mechanics can be built entirely on energy. This leaves some options in the definition of mass. Physicists are not agreed on the relation of mass to kinetic energy. Some physicists, especially those who teach non-physicists, use the concept of relativistic mass, which adds the mass of the kinetic energy to the rest mass of a moving object. Some do not accept any definition of mass that makes it frame-dependent. Under the latter option, some energy (kinetic energy) is not mass. What, then, is kinetic energy? In relativity, kinetic energy is that part of the total energy that is not mass (rest mass).
      There are some other problems with mixing a basically Newtonian description of energy with a little bit of relativity. In the classical model that we have tried to describe in these movies, Potential Energy is related to a process in which Work done by a Force (this is what we call potential energy) causes motion energy, i.e., kinetic energy. To replace the concept of "potential energy" with its equivalent "mass" leaves us with an undifferentiated quantity whose mode of action is not clear. It is not explained by the author of the article how mass begets kinetic energy. This question does not come up when there are only interactions, in which things happen without the need for force, or mechanism.
      Some physicists insist that the only right way to do relativity is to go all the way; to write all the equations of relativistic mechanics in terms of four-dimensional variables, from the ground up. This is an excellent idea, and is particularly suited to particle physics, but it does not help us much in the life-sized Force-mass-acceleration-Work-Energy picture as we teach it. There are no longer Forces - there are interactions. Our high school physics students, I believe, are not ready for this.

Appendix II The Dance of the texts
What is energy?

"The general concept of energy is very difficult to define. But to define some particular forms of energy is easy enough"1

"The concept of energy is one of the most important in the world of science. In everyday usage, we think of energy in terms of the cost of fuels... fuels are needed to do a job and those fuels provide us with something we call energy. ... Energy is present in various forms... when energy is transformed from one form to another, the total amount of energy remains the same."2

" is a scalar quantity that is associated with a state of one or more objects. The term state here has its common meaning: it is the condition of an object. In this chapter we shall focus on one form of energy, kinetic energy K, which is associated with the state of motion of an object. The faster the object moves, the greater is its kinetic energy... we define the kinetic energy as K = ½mv2 ."3

"... When we plug the spout, the steam lifts the lid. Here the work is done by the steam. So we reason that the steam in the kettle had the capacity for doing work before it lifted the lid. The steam had energy. Energy is a quantity that has the capacity for doing work, if the conditions are right."4

"... we cannot give a simple but accurate and general definition of energy in only a few words. Energy of various specific types, however, can be defined fairly simply... all the types of energy can be defined consistently with one another and in such a way that the sum of all types, the total energy, is the same after any process occurs as it was before... "For the purposes of this chapter, we can define energy in the traditional way as 'the ability to do work.' This simple definition is not very precise, nor is it really valid for all types of energy..."5

"There is a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact ..."6

And an oldie: "Work is done when a clock is wound and the weights lifted to the upper part of the case. As a result the clock may run for a week... When a projectile is fired, the projectile does considerable work when it is stopped. The clock weight at rest and the projectile in motion are both said to possess energy, because they are capable of doing work, and the measure of energy is the work they can do. We define energy as the capability of doing work." 7
1 "Project Physics" (p 274) F. James Rutherford, Gerald Holton, Fletcher Watson, directors of Harvard Project Physics, by Project Physics, 1970
2 "College Physics" (p 102-3) Raymond A. Serway and Jerry S. Faughn, Saunders College Publishing, Philadelphia, 1989
3 "Fundamentals of Physics" (p131) David Halliday, Robert Resnick, and Jearl Walker, John Wiley & Sons, New York, 1997
4 "Modern Physics" (p 10) John E. Williams, Frederick Trinklein, and H. Clark Metcalfe, Holt, Rinehart, and Winston, 1980
5 "Physics" (p 142) Douglas Giancoli, Prentice Hall, Englewood Cliffs, 1995
6 "Lectures on Physics" Richard Feynman, Robert B. Leighton, and Matthew Sands, (vol 1 p4-1) Addison-Wesley, Reading, Massachusetts, 1963.
7 "General College Physics" Harrison M. Randall, Neil H Williams, and Walter F. Colby, (p 46-7) Harper & Brothers, New York, 1929

Appendix III
Derivation of the Work Energy Theorem

      There are shorter derivations, that use results of kinematics re-arrangements, but we like this derivation because it makes the skeleton of the derivation more transparent.
Note on Notation:
Initial values will have no "modifiers;" initial velocity will be v.
Final values will be written with a prime ('); final velocity will be v'.
The difference between initial and final velocity will be: )v = (v' - v).

      We will derive this important theorem in just the x-direction, and for a single object which undergoes a displacement )x over which a constant Force acts, giving the object constant acceleration. This will simplify the algebra. The result can be easily extended to a system of several objects, to two or three dimensions, to rotational motion, and to motion in which the Force, and hence the acceleration, are not constant.

Click here to view a diagram of the "skeleton" of the derivation.

      This derivation begins with a definition of Work. The development of the energy concept began with attempts to put a value on the "quantity of motion." There are basically two ways people are paid: by the hour and by the piece. Likewise there are two ways in which applying a Force (to produce motion) can be paid. If you pay by the hour you get a quantity,
F)t. If you pursue this idea, replacing EF with "ma "(from Newton's Law: EF = ma), you are led to the law of conservation of momentum.

      If you pay someone for applying a force by the amount of movement produced you get a quantity,
F)x. If you pursue this idea, replacing EF with "ma" you are led to the Work-Energy Theorem.
Let us follow this course, and define Work in this way:


      (This definition is carefully chosen, but, like all definitions is arbitrary, it contains no facts, is not true or false)

      Replace EF with "ma "(from Newton's Law: EF = ma)
(Newton's Law is a Fact. Facts are subject to experimental test and verification)

WTOTAL = ma )x

For 'a' let us put )v/)t (definition of acceleration)

then we have:
WTOTAL= (m)()v/)t)()x)

= (m) [()v)/()t)] ()x)

= (m)()v) [()x)/()t)]

[()x)/()t)] is the average velocity. If the acceleration is constant, or if )t is small, the average velocity is equal to the average of the initial and final velocities:

for [()x)/()t)] put ½(v'+v)

WTOTAL = (m)()v)[½(v'+v)]

()v) = v' - v

WTOTAL = (m) (v' - v)[½(v' + v)]

for any 'a' and any 'b' it is true that: (a - b)(a + b) = a2 - b2

(v' - v)(v' + v) = (v'2 - v2)

and therefore:
WTOTAL = ½(m)(v'2 - v2)

which is the same as:
WTOTAL = ½(m)(v'2) - ½(m)(v2)

This shows that what WTOTAL does to an object is to change the value of ½(m)(v2) from its initial to its final value. The quantity, ½(m)(v2), seems to play a key role in the answer to "What does Work accomplish?" so it may make sense to give it a name. The quantity, ½(m)(v2) is called the Kinetic Energy, 'KE.' The quantity ½(m)(v2) is therefore, by definition, the Kinetic Energy. The naming contains no facts, is not true or false.

With this name, we can now write that:

(check that the units of Kinetic Energy are Joules, as are the units of Work)

      We have proved that if we define Kinetic Energy in this way, then, we obtain the Work-Energy Theorem. In words, this theorem states that the Total Work done on an object (or a system of several objects) produces a change in the object's (or system's) Kinetic Energy that is equal to the Total Work. Each Joule of Work done increases the Kinetic Energy by one Joule.


      It is from the relation between Work and Kinetic Energy that we get our physics definition of what energy is. All other energy expressions that we will encounter later are called "energy" due to their equivalence with Kinetic Energy.

Note: Don't be misled into thinking that because it is called a "theorem," it is somehow uncertain or doubtful. Since it is a consequence of Newton's Law, it is as true as Newton's Law.

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For more on the nuclear energy process, see W. Scheider, "A serious but not ponderous book about Nuclear Energy," Cavendish Press Ann Arbor, 2001. For information

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