Friction and Drag
Friction is regarded with disfavor by most people except, possibly, the manufacturers of lubricants. Whenever something sticks that should slide, friction is to blame. Friction is an indispensable force, however: without it you could not walk or write; you could not make an auto move forward—the clutch would never stop slipping—and if you could the brakes would fail completely. Bedclothes would slide off the moment you got into bed. Friction's services are virtually endless, and they are all taken for granted.
Friction is equally indispensable in the natural world; a walk in the country provides unlimited examples. Take birds' nests: most are held together by friction and would fall apart without it. It is friction that allows a bear to flip a salmon from a stream, a cormorant to alight on a sloping rock, and a bighorn sheep to clamber over steep terrain. Again the list is endless.
One other force is as important as friction in impeding motion: it is drag, or more precisely viscous drag. The term in-
cludes both air resistance and water resistance, and it slows everything that moves through air and water. Drag is often treated as a form of friction, but the two are fundamentally different. Consider a whale swimming through the water: the water does not slip past the whale's flanks, in spite of appearances. At the surface where the skin of the whale and the water make contact, they stick firmly together because of a strong attraction between the molecules of a solid and a liquid. This surprising effect is known as the no-slip condition.1 It prompts the question, How does a whale move effortlessly through water if slippage does not take place?
The answer is that all the slippage takes place by shearing movements within exceedingly thin layers of water encasing the whale (see fig. 3.1). This viscous shearing brings about a progressive increase in the velocity of the water relative to the whale, from zero right at the interface between whale and water. Molecules of water stick to each other and resist the shearing to some extent—hence drag. But water sticks to a solid more tenaciously than it sticks to itself; this accounts for the no-slip phenomenon and for the fact that drag rather than true friction (the resistance to sliding between two solids) impedes the motion between solids and fluids. The no-slip condition applies to every motion between a fluid and a solid: for example, the flow of a river over its bed, the flow of the wind past a crag, and the flow of air past a flying bird. It also ap plies to relative motion between a liquid and a gas, for example, a raindrop falling through the air.
The property friction and drag have in common is that both are noncon-servative forces (see the final paragraph of chapter 2); that is, they cannot be stored as potential energy of one kind or another for later retrieval. Rather, they "go to waste" and produce "useless" heat. These common phrases conceal some fundamental facts of physics, as we shall see in what follows.
Unlike a force such as gravity, which causes an object to accelerate, friction does the opposite: it resists motion and thus generates heat. The classic example of generating heat by friction is starting a fire by spinning a hardwood stick with its pointed end pressed against a wooden block. A clearer manifestation of mechanical energy being turned into heat would be hard to find.
In the 1840s the great British physicist James Prescott Joule carried out experiments to find how much work has to be done to produce a given amount of heat. At that time heat was measured in calories; one calorie is the amount of heat needed to raise the temperature of one milliliter of water by one degree Celsius. More precisely, it is the amount of heat needed to raise the temperature of 1 ml of water from 14.5°C to 15.5°C; this allows for the fact that the heat required varies slightly depending on the starting temperature.
Joule's most famous experiment was remarkably straightforward. He arranged for paddles submerged in a container of water to rotate under the action of a falling weight; knowing the weight, and the distance it fell, he could calculate the work done by the moving paddles. The viscous drag of the paddles stirring the water caused it to heat up, and the change in temperature was recorded. It was then possible to state how much work produced how much heat. Refined modern measurements show that, using joules (J) as the unit for work, 1 calorie (cal) is produced by 4.186 J of work. This is known as the mechanical equivalent of heat.2 Equivalently, 1 J = 0.2389 cal.
The calorie is not yet obsolete as an energy unit, as any dieter knows. The unit listed as a Calorie (with a capital C) on food packages is equal to a thousand calories, that is, one kilocalorie. The energy in a slice of whole-wheat bread, for instance, is said to be 71 Cal, or about 300,000 J; this does not mean that eating a slice will fuel that much useful work—the efficiency of conversion must be taken into account.
Imagine something—anything—gaining heat; it could be the air in a room when the sun shines in, the water in a kettle put on a hot stove, or a wooden block heated by friction when rubbed briskly with a stick. In every case, a gain of heat brings a rise in temperature of the object heated. This is so obvious that it seldom calls for comment. The problem that arises in a scientific mind, however, is this: What has changed within the heated object to make its temperature greater? The question is meaningless until we attach a meaning to temperature.
As nearly everybody knows, the molecules any object consists of, be it a roomful of air, a kettleful of water, a wooden block, or anything else, are in constant random motion. They are never still. And when an object's temperature rises, all its molecules speed up.
For a gas, the link between its temperature and the velocities of its molecules is surprisingly straightforward. The temperature of a mass of gas depends wholly on the average kinetic energy of its molecules.3
The kinetic energy (KE) of a body of mass m moving with velocity v is given by the formula KE = V2 mv2. All the molecules in a gas collide with each other repeatedly; at each collision, the two colliding molecules bounce off each other in new directions and at altered speeds. Consequently each molecule's kinetic energy changes every time it collides, but this doesn't affect the rule, which relates to the average KE of all the molecules. Furthermore, the average KE (averaged over time) is the same for every molecule whatever its weight; therefore lightweight molecules must move (on average) faster than heavy ones. For example, if a given molecule is one-fourth as heavy as another, its average velocity must be twice as great.
Another way of wording the rule is to say, "When we measure the temperature of a gas, we are measuring the average . . . kinetic energy of its mol-ecules."4 This is the meaning of the word "temperature." It also makes clear what the absolute zero of temperature is. It is the temperature at which all the molecules have zero energy because they are motionless. This does not imply that subatomic particles are also motionless. Even at absolute zero, which is -273°C, they continue to oscillate, perpetually.
Absolute zero is used as the zero of the absolute temperature scale, also called the thermodynamic temperature scale; each division of the scale, a kelvin (abbreviated as K), is of the same magnitude as a degree Celsius. Thus the temperature of freezing water (0°C) is 273 K, and that of boiling water
(100°C) is 373 K. Blood temperature in a healthy human (37°C) is 273 + 37 = 310 K, and so on.5 Note that the units are called "kelvins," not "degrees Kelvin."
That the molecules of a gas at a temperature above 0 K are forever randomly moving, colliding, and changing direction makes one wonder: What is their mean free path? In other words, how far does a molecule travel, on average, between one collision and the next? The answer depends on the amount of space available to the molecules, which depends in turn on the density of the gas. It is not dependent on the temperature of the gas, or equivalently, on the molecules' velocities: slow-moving molecules will take longer to travel from one collision to the next, but the distance traversed is the same. The density of the air is greatest at sea level and decreases rapidly at higher and higher altitudes.6 At sea level, the mean free path of a molecule of air is 0.1 |im (micrometer); at an altitude of 100 km above sea level, it is 0.16 m; and at an altitude of 300 km, it is 20 km. This means that the mean free path of an air molecule 300 km up is 2 x 1011 times greater than at the surface.
At an altitude of 300 km, in the farthest fringes of the atmosphere, the air temperature is about 1,500°C. How would this feel if we could experience it? It would unquestionably seem bitterly cold, because the familiar relationship between true, measured temperature and the subjective sensation of temperature holds only if the density (or pressure) of the air is what we are accustomed to. Recall that the temperature of the air is a measure of the average kinetic energy of each molecule, regardless of the number of molecules in a given volume. Now imagine two parcels of air, one at ground level and the other 300 km up, and suppose they are at the same temperature. It is easy to see that the total energy in the parcel at ground level far exceeds the total energy in the high-altitude parcel, because the former contains so many more molecules than the latter; it is the total energy, not the energy per molecule, that determines how the air "feels," either warm or cold. Thus the statement that at an altitude of 300 km the air temperature is 1,500°C, while true, gives no idea of how the air at that height feels: temperatures in air at unfamiliar densities cannot be imagined because we have never had the opportunity to become accustomed to them.
As we have noted already, unless an object is at a temperature of 0 K, all its molecules are in constant random motion; the object has internal energy. This is not to say that heat and internal energy are the same thing, however. They are not.
The distinction between them is best appreciated by considering what happens when you heat a kettle of water on a hot plate. Heat passes from the hot plate into the water and increases the water's internal energy. But that is not all it does; in addition, the heat boils the water, and the steam produced rattles the kettle's lid—the heat has done work on the lid. To repeat: the added heat has done more than merely raise the water's temperature; it has also done mechanical work. This can be written concisely as an equation:
Here Q represents the added heat, U the increase in internal energy of the water, and W the work done.7
We can learn more by considering the classic piston engine driven by steam. In outline, the piston engine of an old steam locomotive works like this: water is heated in a boiler with a coal fire under it, producing steam under pressure; the steam expands into hollow cylinders, forcing out sliding pistons within the cylinders. The movement of the pistons is converted by camshafts and linkages into the rotary motion of the wheels. Indirectly, therefore, heat from the burning coal turns the locomotive wheels. Only a fraction of the heat supplied is turned into mechanical energy, however; most of the rest is lost in the steam escaping into the atmosphere; note that the heat put in from the firebox is at a much higher temperature than the comparatively cool steam discharged from the funnel. All the same, the cool steam carries away a proportion of the heat supplied by the firebox.
This is the crux of the matter: high-temperature heat yields a mixture of mechanical energy and low-temperature heat; the latter is wasted energy, but waste cannot be avoided. The thermal efficiency of any heat engine is defined as work done + heat absorbed. Both parts of the fraction are measured in joules. Thermal efficiency is always less than one.
The maximum efficiency theoretically possible is given by the formula
(Th-Tc)/TH , where TH and TC are, respectively, the temperatures of the hot (input) steam and the cool (output) steam, measured in kelvins.8 It is easy to see that this fraction could reach one only if Tc were absolute zero, an unattainably low temperature.
Thus no engine can be 100 percent efficient. Note that this is not a conse quence of friction; even if friction could be reduced to zero (impossible in practice), the maximum efficiency of any heat engine would still be less than one, because of the very nature of thermal energy, expressed in the famous second law of thermodynamics.According to the law, "It is not possible to change heat completely into work, with no other change taking place."9 In brief, there are no perfect engines. Yet another way of expressing it is to say that the random motion of the molecules in a hot substance can never change, completely and spontaneously, into ordered, macroscopic motion.10 The wasted heat that cannot be made to change into mechanical energy and do work is the form of energy known as entropy. The meaning of this famous term has been the topic of whole books. Here we can give it only a section.
The law of the conservation of energy tells us that energy can be neither created nor destroyed. As we have emphasized repeatedly up to this point, energy put into a system is always, without exception, passed on in the same or another form: it never disappears.
At the same time, it is never true that all the energy supplied to a system can be made to do useful work. Some is always dissipated as unavailable heat, at too low a temperature to serve as the heat source for a heat engine. This heat is entropy; it could also be called useless energy.
We have now come across two entirely different obstacles to so-called perpetual motion. First, recall the swinging pendulum and the bouncing ball described at the end of chapter 2; if their energy came only from conservative forces, their motion would be perpetual. But in real life, nonconservative forces—friction and drag—are always acting as well, and the motions of the two devices are inevitably brought to a stop. The energy they lose in slowing down is converted into "useless" heat, that is, entropy.
Second, as we have seen, some of the energy produced by heat engines is always useless heat (entropy again).This follows from the fact that a heat engine cannot, by the second law of thermodynamics, ever be 100 percent efficient.
These two points lead to the inescapable conclusion that although the total energy of the universe remains forever the same, the fraction of it that is entropy forever increases.
Another way of saying the same thing is, first, to contrast "ordered energy," such as the kinetic energy of a moving macroscopic object, with "disordered energy," namely thermal energy, the disordered, random motion of individual molecules. The law just given then becomes: Ordered energy is always ultimately transformed, spontaneously, into disordered energy.11 The converse is not true—disordered (thermal) energy is never spontaneously transformed entirely into ordered energy.
To put the matter in a nutshell, the universe is running down. Everybody ought to know this nowadays, but we are still sometimes exhorted to "conserve energy," as if we could do anything else. What needs to be conserved, of course, is potential energy, especially that stored as chemical energy in fossil fuels. Entropy does not need conserving; it is increasing all too fast. Governments should be urging us to conserve fuels and slow down, as much as possible, the transformation of their energy into entropy.
As we have just seen, two separate facts make perpetual motion machines impossible. Therefore the two supposed forms of perpetual motion machines are both nonexistent.12
The first kind of mythical machine is exemplified by a water wheel that powers itself. The water in one of the buckets that has reached the top of the wheel tips its contents into an empty bucket below it, driving the wheel onward unceasingly. This device must have been independently invented by generations of mechanically minded children. But it can never work in practice because rotation of the wheel is resisted by friction, and it could keep on rotating only by creating new energy—which from the law of the conservation of energy is impossible.
The second kind of mythical perpetual motion machine is a heat engine working with 100 percent efficiency. This is impossible because it would entail the complete conversion of heat into work, violating the second law of thermodynamics.
Accepting the inevitable—that all energy will ultimately be converted into entropy—it is time to consider what is happening, and will continue to happen for a very long time, here on earth. The earth is continuously supplied with external energy from the sun, and it also generates internal energy of its own. These are the topics to be considered in the rest of this book.
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