What Is Energy Some Preliminary Physics

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Some Definitions

In answer to the question, What is energy? no less a scientist than the late Nobel laureate Richard Feynman said, "In physics today, we have no knowledge of what energy is ... .It is an abstract thing."1 That was in 1963. At a profound epistemological level it is no doubt true to this day. In the same philosophical vein, it is equally true of matter. But for practical purposes that answer is not much help.

Turning to more mundane sources, we find that energy is "the capacity ... to perform work," which is hardly a standalone definition. To be complete, it requires a definition of work. From the same source, the definition of work is "energy transferred to or from a body____it involves an applied force moving a certain distance."2 This circularity is unavoidable: in simple terms, work requires the expenditure of energy, and energy spent performs work.

Let us look more closely at work, the application of a force through a distance. It helps to consider an actual example. To pick up a five-kilogram block of iron from the ground and raise it to a height of two meters is work: it requires energy. Force must be exerted—enough force to overcome the gravitational pull of the earth on the 5 kg block; the force must be applied directly upward, against the pull of gravity, for a distance of 2 m. We can measure this amount of work by multiplying the force times the distance through which it acts; the answer measures both the work done in lifting the block and the energy required to lift it: they are the same. It remains to consider how force is to be measured.

Force is what it takes to accelerate a mass. If your auto has run out of gas and you want to push it along a level road, it takes considerable force to get the movement started—to accelerate the auto from zero speed to walking speed— but hardly any force to make it continue rolling at walking speed; once it is moving steadily, no force is required beyond that necessary to overcome any slight roughness of the road and any friction in the bearings. If there were no roughness and no friction, the force needed to keep the auto moving forward at an unchanging speed would be zero.3

Now let's return to the 5 kg block being lifted from the ground: the force of gravity (the force you are working to overcome) imparts acceleration to anything it acts on, and at the surface of the earth this acceleration, known as gravitational acceleration,4 is 9.81 meters per second per second (briefly, 9.81 m s-2; see page ix for an explanation of the symbols). This means that if you drop an object from a height (as Galileo is said to have done from the Leaning Tower of Pisa), it will fall at an ever increasing speed. It is being accelerated by the force of gravity acting on it. If the object is heavy enough for air resistance to be negligible, it will be falling at a speed of 9.81 meters per second (9.81 m s-1) after one second, twice that, or 19.62 m s-1, after two seconds, 29.43 m s-1 after three seconds, and so on; the speed keeps on increasing steadily. This is true whatever the mass of the object. A measure of the amount of force acting on it is given by multiplying the acceleration by the object's mass.5 The answer is in newtons (abbreviated as N); one newton is the force required to give a mass of one kilogram an acceleration of 1 m s-2.

Therefore, when you hold a 5 kg block you are exerting an upward force of 5 x 9.81 N = 49.05 N. If you stop exerting this force, the block falls to the ground.

An aside is necessary here, to explain the difference between mass and weight. At the surface of the earth, an object's mass and its weight are the same by definition. For example, a 50 kg woman has a mass of 50 kg, and she weighs 50 kg; to use both terms seems mystifying and redundant, or at least it did to schoolchildren in the days before space travel. However, if the woman travels to the moon her mass will not change—it will still be 50 kg—but she will weigh much less, specifically 8.5 kg. The 8.5 kg is the force, confusingly called "weight," that holds her to the moon's surface, where the acceleration due to gravity is only 1.67 m s-2, which is 17 percent of the acceleration on earth.

Now let's return to the topic of work, specifically the work required to raise the 5 kg block vertically through 2 m. This is equivalent to exerting a force of 49.05 N through a distance of 2 m. The answer is force times distance, and the resultant energy, measured in joules, is 49.05 newtons x 2 m = 98.1 joules.

Joules are the units in which both work and energy are measured. Thus one joule is the work done when a force of one newton is applied over a distance of one meter. It is also the energy expended in doing the same thing. Joules will be used throughout this book as a measure of energy. The abbreviation for them is simply J. To compare the energies of, say, earthquakes, rising and falling tides, breaking waves, sunlight falling on a patch of ground, the sunlight trapped by photosynthesis needed to grow a tree, the sound of thunder— whatever it is—one needs a unit for measuring energy, and that unit is the joule.6 It is not, admittedly, a unit familiar from frequent use in everyday life, as is true of kilograms (for measuring mass), meters (for measuring length or distance) and seconds (for measuring time). But once you concentrate your attention on energy, the unit soon becomes familiar: you get used to it.

Energy Conversions

Energy exists in many forms. Electrical energy, electromagnetic energy, chemical energy, heat energy, and nuclear energy are only a few. Moreover, any form of energy is convertible into any other, though not necessarily at a single step. Most of the actions going on in the world involve several energy conversions.

Here is an ecological example. The sun generates its energy by nuclear fusion, which yields enormous amounts of radiant energy (light, heat, and ultraviolet rays); this energy leaves the sun in all directions as electromagnetic energy, a small fraction of which strikes the earth. Suppose some of this solar energy falls on a tract of grassland. The grass uses the solar energy to create sugars by the process of photosynthesis. That is, the chlorophyll in the grass converts electromagnetic energy into chemical energy. The grass grows—en-tailing a whole series of conversions of chemical energy—until some of it is eaten by a jackrabbit.

The jackrabbit leads an active life; to acquire chemical energy to fuel its own activities, reproduction, and growth, it must eat. It must hop hither and thither, biting off blades of grass and chewing them. That is, its limbs and jaws move: chemical energy in the jackrabbit's muscles has been converted into kinetic energy, the energy of movement. Eventually a coyote catches and eats the jackrabbit; this requires a fairly lavish conversion of chemical energy into kinetic energy by the coyote, since the jackrabbit will no doubt resist. Both the animals are warm-blooded, and to keep their temperatures at the physiologically correct level, they must also convert some of their chemical energy into thermal energy. Death finally claims the top predator, the coyote; some of its remains are consumed by scavengers, and what's left decays—it is consumed by decay organisms, chiefly bacteria and fungi. These, though not warm-blooded, still produce heat as a by-product of their activities. In the end the solar energy that was first captured by the grass is finally dissipated as waste heat.

This short story, with many details glossed over—or it would have taken pages and pages—could also have been written as the life history of a joule. Instead of treating it as a tale about a series of different objects—sun, grass, jackrabbit, coyote, bacteria—we could have made it the tale of a single unit of energy, a joule, and the conversions it underwent in a sequence of different settings before ending up, as all energy eventually does, as heat. We return to this ultimate fate of all energy in chapter 3, under the heading entropy.

Change of any kind, anywhere, entails energy conversion of one sort or another. Whenever you see energy being spent in movement—in the flight of a bird, the breaking of a wave, or the flow of a river, for example, it is worth asking how and where the energy originated and how and where it will be dissipated.

Potential Energy

Let's return to the 5 kg block. It was lifted from the ground and placed on a shelf 2 m up (unless you're still standing there holding it). Work was done on it—specifically, 98.1 J of work. It has been given energy, but in spite of that it stolidly sits there, motionless, on the shelf. Where has the energy gone? The answer is that it has become potential energy, or PE for short. If the shelf gives way, the block will fall back to the ground; that is, the PE you gave it by lifting it will be converted back to movement—kinetic energy.

The form of PE possessed by the 5 kg block is known as gravitational PE. Anything poised to fall if something gives way has it—a leaning tree, a boul der on a clifftop, the water behind a dam. But what if the leaning tree is strongly rooted or the boulder is in the middle of a flat plateau, so that neither can truly be called "poised" to fall? Their collapse is not imminent. Does this make a difference to their gravitational PE? Surely the energy is not a mere matter of chance.

No, it's not. Gravitational PE is a relative matter. If one chooses to treat the surface of the earth at mean sea level as the level at which gravitational PE is to be regarded as zero, then anything whatever above that level has measurable PE, whether or not it's poised to fall.7 A person living on a plateau high above sea level might prefer to treat the plateau as the level at which gravitational PE is to be regarded as zero. Then a 5 kg block on a shelf 2 m above the floor in a house on the plateau would have the same gravitational PE as an identical block 2 m above the floor in a house with its floor at sea level.8 But if one chose to use sea level as the reference level for measuring the gravitational PE of both blocks, and if the elevation of the plateau is, say, 250 m, then for the block in the house at the seaside, the gravitational PE would be 98.1 J as before, whereas the PE of the block in the house on the plateau, on its shelf 252 m above sea level, would be

Likewise, a rock below sea level, in Death Valley, say, has negative PE relative to sea level; energy would have to be spent to raise it to sea level.

This demonstrates that measurements of potential energy are arbitrary. The reference level against which gravitational PE is measured is always a matter of choice and must be stated if there could be any doubt.

Energy is stored as PE in a multitude of ways. A stretched spring or an archer's drawn bow stores elastic energy: the stretched spring snaps back to its unstretched length when let go; a stretched bowstring straightens when released, speeding an arrow on its way. In both cases, stored elastic energy has changed to kinetic energy.

Another familiar form of potential energy is chemical PE. An electric battery and a loaf of bread both have it. The conversion from potential to actual produces an electric current in the case of the battery and muscle movement in the case of the bread.

Magnetic PE is stored in magnets, ready to be converted to kinetic energy when a piece of steel is attracted to the magnet.

The list goes on: potential energy in its various manifestations will appear frequently in all that follows.

The Ideal and the Real

In theory (though never in practice), certain actions go on forever. Here are two examples; in both, gravitational PE is equal to zero at the lowest level reached by the moving object.

First, imagine a pendulum suspended from a perfectly frictionless bearing swinging from left to right and back again (fig. 2.1). Its bob (the hanging weight) is suspended by a perfectly inelastic string. Assume that the pendulum has been set up in a perfect vacuum, so that its movements are not affected by air resistance. The pendulum will continue to swing forever without any loss of amplitude. It is intuitively clear that this should happen, even though the conditions prescribed for the experiment are too perfect ever to be attained in practice. What happens to the imaginary pendulum is this: when the bob is at the left extremity of its swing, it is motionless for an instant; that is, it has no kinetic energy (KE). All its energy is potential; more precisely, it is gravitational PE. Then the bob starts to fall because of the force of gravity, but it is constrained by the string to swing to the right; as it swings, its PE is converted to KE. By the time the bob reaches the bottom of its swing, its PE is zero, having all been converted to KE; at this instant its KE, and therefore its speed, has reached a maximum. Nothing stops the bob's continued movement, so it keeps on swinging to the right and begins to ascend, losing KE and gaining PE in the process. The conversion of KE back into PE continues as the bob approaches the right-hand end of its swing. Here the conversion is complete: the bob's KE has decreased to zero so that it is momentarily stationary, and its gravitational PE has increased to a maximum. Then the whole process happens again, from right to left. The total energy remains the same all the time, never dwindling; it is the sum of the KE and the PE, known as the mechanical energy of the pendulum. As an equation, mechanical energy = potential energy + kinetic energy.

In the ideal case, the mechanical energy remains unchanged forever, and the pendulum keeps on swinging.

In real life, with conditions unavoidably less than perfect, this does not happen. Because of friction in the bearings, air resistance, and minute stretching of the string, energy is gradually drained away from the pendulum in the form of imperceptibly slight heating. The mechanical energy slowly declines, and the amplitude of the swings diminishes, until all movement stops. At this stage the pendulum's mechanical energy has all been dissipated and it hangs motionless.

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Figure 2.1. Three positions of a perfect (frictionless) pendulum. (a and c) Here the bob has maximum gravitational PE and zero KE. (b) Here it has maximum KE and zero PE.

For a second theoretical example, imagine a perfectly elastic rubber ball bouncing on a perfectly rigid floor in a perfect vacuum (fig. 2.2). The ball will continue to bounce forever, returning to the same height above the floor at each bounce.9 As with the pendulum, the bouncing ball retains its total mechanical energy, which at every instant is the sum of its PE and its KE. The bouncing ball is slightly more complicated, however. Its PE is gravitational when it is at the top of its bounce and descending floorward and elastic when it recoils from the floor and starts upward. The KE of the ball is at a maximum on its downward journey just as it hits the floor. There the ball is abruptly stopped by the collision with the floor, but its KE is instantly converted to elastic PE and as instantly released, restoring the ball's KE, in an upward direction this time. The renewed KE and the upward speed of the ball are at a maximum just as the ball leaves the floor; they decrease to zero as the ball reaches its highest point.

In the real-life equivalent of this experiment, with an imperfectly elastic ball, an imperfectly rigid floor, and an imperfect vacuum (or none at all), we know that the bounces will steadily become lower and lower until they peter out altogether. That is, the ball's mechanical energy will be dissipated as heat, some of it in the air because of air resistance, and some of it in warming the imperfectly elastic ball and the imperfectly rigid floor; as these compress and expand, shearing within them causes friction.

The foregoing paragraphs have shown, implicitly, that energy results from

Figure 2.2. Five positions of a perfectly elastic bouncing ball. (a and e) Here, at the highest level reached at each bounce, the ball has maximum gravitational PE, zero elastic PE, and zero KE. (b and d) Here it has maximum KE (downward and upward respectively) and zero PE (both gravitational and elastic). (c) Here, where the ball is slightly flattened against the rigid floor, it has maximum elastic PE, zero gravitational PE, and zero KE.

Figure 2.2. Five positions of a perfectly elastic bouncing ball. (a and e) Here, at the highest level reached at each bounce, the ball has maximum gravitational PE, zero elastic PE, and zero KE. (b and d) Here it has maximum KE (downward and upward respectively) and zero PE (both gravitational and elastic). (c) Here, where the ball is slightly flattened against the rigid floor, it has maximum elastic PE, zero gravitational PE, and zero KE.

two kinds of forces. One kind, exemplified by gravity and elasticity, is called a conservative force; its salient feature is that it can be stored—in these examples, as gravitational PE and elastic PE. A system in which the only forces acting are conservative forces never runs down. The other kind of force, exemplified by friction and air resistance, is nonconservative. When nonconservative forces are operating, either alone or in combination with conservative ones, a system inevitably runs down. Nonconservative forces produce heat, and the heat can never spontaneously turn back into another kind of energy.10