Figure 4.2. Force versus time.
Figure 4.2. Force versus time.
time the force is allowed to act. So we have (with only one force acting on the object)
This is shown graphically in Fig. 4.2.
Here again, the force need not be a constant force. The momentum added to the object will be equal to the area under the force versus time curve, whatever shape the curve might display.
Both kinetic energy and momentum are related to the velocity of an object. Kinetic energy is a scalar quantity, whereas momentum is a vector quantity (see Chapter 3, Section C5). Both quantities are determined by the integrated (or cumulative) effect of the action of net force. However, kinetic energy is the integrated effect of net force acting over a distance as contrasted with momentum, which is the integrated effect of a force acting over time.
2. Significance of Kinetic Energy—Work
The product of force times distance is a much more general concept than indicated so far. It is an important quantity and is called work. From the discussion above, work (W) = F X d = K.E., for a force that accelerates an object.1 One must be very careful when using the word work, because it now means something very specific. A physicist would say that if you pushed very hard against a wagon full of bricks, and the wagon did not move, then you did no work on the wagon! Only if the wagon moved as a result of the applied force would you have done some work on it. Note that one can double the amount of work done either by
Actually, it is only the component of the force parallel to the direction of motion that does work. Moreover, it is the net force that results in the acceleration and produces a change in the kinetic energy.
doubling the applied force or by doubling the distance over which the force is applied. In either case, the net effect will be the same—twice the amount of kinetic energy will have been imparted to the object.
Kinetic energy is our first example of one of the forms energy may have. Whenever we do work (in the physicist's sense) on an object, we change the energy of the object.
One can work on an object and not change its kinetic energy, however. For example, the force acting over the distance may just be moving an object slowly up an inclined plane. When the force is removed, the object is left higher up the ramp, but not moving. If work has been done on the object, what kind of energy has been imparted to it? The answer is potential energy because the object, in its higher position, now has the potential to acquire kinetic energy. If the object were nudged off the side of the ramp to make it fall, it would accelerate and gain kinetic energy as it fell. Potential energy is energy associated with an object because of its position or configuration, as will be discussed in the next section.
There are certain common systems not involving collisions, but nevertheless involving movement, for which some aspect is unchanging. Something is being conserved, physicists prefer to say. One such system is a pendulum. A pendulum, if constructed properly, will continue to swing for a very long time. If we watch such a pendulum for only a little while, we can analyze it as if it were not going to stop at all. More precisely, we can measure just how high the pendulum bob rises on every swing and see that it comes back to almost the same height every time. It would not be hard to convince ourselves that if we could just get rid of the effect of air resistance and put some extremely good lubricant on the pivot, the pendulum bob would always return to the exact same height on every swing. This is an interesting observation. What is it that makes nature act this way for a pendulum?
Is there something being conserved in the pendulum's motion that requires it to behave this way? If so, what is being conserved? It is not kinetic energy. The pendulum bob speeds up and slows down, even coming to a stop, momentarily, at the top of each swing. When the bob passes through the bottom position, it is moving quite fast. Hence, because kinetic energy (K.E.) is one-half of mass X (velocity)2, its K.E. is zero at the high points of a swing and a maximum at the low point.
Mathematical analysis shows that the quantity being conserved is the sum of the kinetic energy and the potential energy of the bob. (Potential energy was originally called latent vis viva because it was considered to have the potential of being converted into vis viva.) The potential energy is said to be zero when the bob is at its lowest position and increases directly with the height above that position. It reaches its maximum when the bob is at the top of the swing where the kinetic energy has gone to zero. It works out just right. The total mechanical energy (i.e., the sum of these two kinds of energies) is always the same. This is an example of what we call a conservative system.
Another example of a conservative system is a ball rolling up and down the sides of a valley. If the terrain is very smooth, and one starts a ball at a certain height on a hill on one side of the valley, when released it will roll down into the valley and up the hill on the other side to the same height as it had on the first side. As it moves, it will possess the same combination of potential and kinetic energies as a pendulum bob. The sum, the total mechanical energy, will again always be a constant.
It can be seen why this new form of energy is called potential energy. When the ball is high on one hill, it has a great amount of ' 'potential'' to acquire kinetic energy. The higher up the hill the ball is started, the faster it will be moving, and therefore the more kinetic energy it will have when it reaches the valley. Note that this potential energy is actually gravitational potential energy. It is against the force of gravity that work is done to raise the ball higher, and the force of gravity will accelerate the ball back down into the valley.
Another system that also involves only motion and the gravitational force is the system of a planet moving around the Sun. This is also a conservative system. Kepler's first law of planetary motion (see Chapter 2) states that the planet's orbit is an ellipse, with the Sun at one focus. Thus the planet is sometimes closer, then sometimes farther from the Sun. From Newton's universal law of gravitation, it can be shown that when the planet is farther from the Sun it has more gravitational potential energy (it has further to "fall" into the Sun, so it is "higher"). Kepler's second law of planetary motion states that the radius vector from the Sun to a planet sweeps out equal areas in equal times. This law accounts for the known fact that the planet moves faster when it is closer to the Sun. Again, one can see that things occur in just the right way: The planet moves faster when it is closer to the Sun, so the planet has more kinetic energy when its potential energy is less. If the mathematics is done carefully, it can be seen that the sum of the potential and kinetic energies will remain exactly constant. (Kepler's second law also illustrates another conservation principle—conservation of angular momentum, discussed in Chapter 8).
Conservative systems thus appear to be operating in our universe. These systems always seem to maintain the same value for the total energy when taken as the sum of the kinetic energy plus the potential energy. But these systems are actually only approximately conservative. The pendulum will slowly come to a stop, as will the ball in the valley. The total energy appears to go slowly to zero. The planets are slowly moving out into larger-sized orbits. Is there or isn't there an exact conservation law for these systems?
The answer to the preceding question is that the total energy actually is conserved, but there exists yet a third kind of energy, and the kinetic and potential energies are slowly being converted into this third form. The new form of energy to be considered is heat. The systems have frictional forces (between the pendulum bob and the air and at the pivot for the first example); and friction (as people who have rubbed their hands together to warm them know) generates heat.
The heat so generated is gradually lost to the system. It escapes into the surrounding air and slightly warms it. That all of the lost energy appears as heat, and that heat is another form of energy, was established by nineteenth-century physicists. It is worthwhile to discuss now, in some detail, the evolution of our understanding of heat as a form of energy. In fact, historically, physicists seriously considered that there was a conservation law for total energy only after heat was shown to be a form of energy.
As discussed above, work can produce mechanical energy (i.e., either kinetic energy or potential energy). Work is calculated as force times distance, that is, work = F X d
Kinetic energy is calculated as one-half of the mass times the square of the velocity
Gravitational potential energy is calculated as the weight of an object (W = mg) times the height (i.e., distance) of the object above some reference level,
We can use these mathematical expressions to calculate the amount of work done and kinetic and potential energies present in a simple pendulum. Suppose we have a pendulum bob of mass 1 kg suspended by a thin wire. If we pull the pendulum up (keeping the wire tight) to a height 10 cm above its bottom (lowest) position, then the work done is work = F X d where F is equal to the force of gravity that we must overcome in lifting the bob and is just the weight, W = mg. (Actually, the work done is the product of the distance moved times the component of the force along that distance. In this case, the force is vertical and so only the height matters for the distance moved.) So, work = W • h = mgh
Now 1 kg-m2/sec2 = lN-m (see Chap. 3) which is called 1 joule, and is actually a unit of energy. So the work done is 0.98 joules.
At this point, before we release the pendulum, all the work done went into increasing the potential energy.
Now we release the pendulum and it begins to swing. Using our expression for kinetic energy and assuming that the system is conservative, so that the total energy is the sum of potential energy and kinetic energy, we can calculate the velocity of the pendulum bob as it passes through its lowest point. At this lowest point, the potential energy (mgh) goes to zero since the height (h) goes to zero. Hence all the energy must now be in the form of kinetic energy and this must be equal to the original potential energy, which is equal to the work done.
sec or sec
It is important to realize that this result is obtained by merely recognizing the relationship between work and energy and by assuming the pendulum system is a conservative system.
B. Heat and Motion
Heat has been studied from very early times. Aristotle considered fire to be one of the five basic elements of the universe. Heat was later recognized to be something that flowed from hot objects to colder ones and came to be regarded as a type of fluid. About the time of Galileo this heat fluid was known as phlogiston and was considered to be the soul of matter. Phlogiston was believed to have mass and could be driven out of, or absorbed by, an object when burning.
In the latter part of the eighteenth century the idea that heat was a fluid was further refined by the French chemist Antoine Lavoisier and became known as the caloric theory of heat. The caloric theory was accepted by most scientists as the correct theory of heat by the beginning of the nineteenth century. The fluid of heat was called the caloric fluid and was supposed to be massless, colorless, and conserved in its total amount in the universe. (Lavoisier developed this theory while proving that mass is conserved when the chemical reaction known as oxidation, or burning, takes place.) The caloric fluid, with its special characteristics, could not actually be separated from objects in order to be observed or studied by itself. The caloric fluid theory thus presented a growing abstractness in the explanations of heat phenomena. Several phenomena were accurately understood in terms of the caloric theory, even though it was an incorrect theory.
As an example of a process that was thought to be understood by analysis with the caloric theory of heat, consider the operation of a steam engine. This is a good example both because improved understanding of the nature of heat came from attempts to improve steam engines and because it introduces several important concepts needed for the study of heat. The first working steam engine was constructed about 1712 by Thomas Newcomen, an English blacksmith. Very rapidly, the Newcomen engine was installed as a power source for water pumps in coal mines throughout England. It replaced cumbersome and costly horse-team powered pumps.
A simplified Newcomen steam engine is shown in Fig. 4.3. The fire under the boiler is continually producing steam from the water in the boiler. The operation of the engine is accomplished with essentially four steps. These steps are performed by using the two valves labeled steam valve and water valve. The steps are (1) open the steam valve to let steam into the piston chamber, with the pressure of the steam causing the piston to rise and the pump rod to descend; (2) close the steam valve; (3) open the water valve, which allows cool water to spray into the chamber, condensing the steam, creating a partial vacuum, and causing the piston to come down and the pump rod to be lifted; and (4) close the water valve. The whole cycle is then repeated.
The early ideas regarding the essentials of the steam engine were very crude by today's standards. Although it is called a steam engine, it is the fuel being burned under the boiler that actually provides the power for the engine. Early experimenters were not entirely convinced of this, however. The power source
Figure 4.3. Schematic representation of early steam engines, (a) The Newcomen engine, (b) Steam pressure pushes piston up. (c) Condensing steam pulls down, (d) Watt's improved steam engine with separate condensing chamber.
Figure 4.3. Schematic representation of early steam engines, (a) The Newcomen engine, (b) Steam pressure pushes piston up. (c) Condensing steam pulls down, (d) Watt's improved steam engine with separate condensing chamber.
for the steam engine was considered to be the steam, and the efficiency of the engine was measured in terms of the amount of steam it consumed. Many of these early ideas did improve the steam engine somewhat, especially those of the English inventor James Watt, who patented the first really efficient steam engine in 1769.
In the Newcomen engine, the walls of the cylinder are cooled during the condensation or "down-stroke" step of the cycle. When steam is admitted into the cylinder for the expansion or ' 'up-stroke'' step of the cycle, about two-thirds of
Closed the steam is consumed just to reheat the walls of the cylinder so that the rest of the steam can stay hot enough to exert sufficient pressure to move the piston up. Watt realized that the condensation step was necessary for the cyclic operation of the engine, but he also realized that the actual cooling of the steam could take place in a different location than the hot cylinder. He therefore introduced a separate chamber called the condenser into his engine, to carry out the cooling for the condensation step, as shown in Fig. 4.3d. The condenser was placed between the water valve and the piston chamber, and between steps (2) and (3) above, the steam was sucked into the condenser by a pump (not shown in the diagram) and step (3) was modified appropriately. Also the water from the condenser was returned to the boiler.
Watt also recognized that the necessary condensation step resulted in the loss of some steam, which was capable of pushing the piston further. In order to decrease the relative amount of steam lost in the condensation step, he decided to use higher pressure (and, therefore, higher temperature) steam in the initial stages of the expansion step. Thus even after the steam inlet valve was closed, the steam in the cylinder would still be at a high enough pressure to continue expanding and pushing the piston upward. While it continued expanding, its pressure would drop and it would "cool" off, but it would still be doing work. Only after it had done this additional work would the condensation step begin.
With these and other improvements, Watt's steam engine was so efficient that he was able to give it away rather than sell it directly. All the user of the engine had to pay Watt was the money saved on fuel costs for the first three years of operation of the engine. Watt and his partner Matthew Boulton became wealthy, and the Industrial Revolution in England received a tremendous impetus from a new source of cheap power.
In 1824 a young French military engineer named Sadi Carnot published a short book entitled On the Motive Power of Fire, in which he presented a very penetrating theoretical analysis of the generation of motion by the use of heat. Carnot was considerably ahead of his time, and so his contribution went essentially unnoticed for about 25 years.
In considering Watt's work, Carnot realized that the real source of the power of the steam engine was the heat derived from the fuel, and that the steam engine was simply a very effective means of using that heat to generate motion. He therefore decided to analyze the fundamental way in which heat could generate motion, believing that once this was understood it might be possible to design even more efficient engines than Watt had developed. He made use of the then accepted caloric theory of heat, and arrived at some remarkable conclusions despite using an erroneous theory. In time, after the publication of his book, he realized that the caloric theory might be in error, but unfortunately he died of cholera before he could pursue the matter much further. Nevertheless, Carnot explained the performance of a steam engine in terms of an analogy that is extremely useful for understanding the motive power for heat.
According to the caloric theory, heat is a fluid that flows from hot objects to colder ones. The steam engine is understood to be the heat equivalent of the mechanical waterwheel. The waterwheel derives its power from the flow of water from a higher elevation to a lower one. The temperatures of the objects can be regarded as analogous to the elevations of the pond and the exit steam for the waterwheel. It is commonly known that for a constant amount of water flowing over the waterwheel, the larger the wheel, the more work that can be done (i.e., the farther the water must fall). Hence, Carnot suggested that a steam engine (he said it should more properly be called a heat engine) could do more work with the same amount of heat (caloric fluid) if the heat was made to flow over a larger temperature difference.
One must be somewhat clever to utilize this suggestion. The temperature difference that exists in a steam engine is between the heat of the steam and the temperature of the surroundings. The steam is at the boiling point of water. At first it appears that we cannot control either the high temperature or the low temperature. But Carnot pointed out that if the steam is produced at a higher pressure, the temperature of the steam will be higher than the normal boiling point of water. This is just the same as the principle of a pressure cooker. Thus there is a way to raise the higher temperature and obtain a larger temperature difference. Carnot thus explained the improvement Watt found by using higher pressures. All modern steam engines operate at high pressures and realize an increase in efficiency accordingly.
The caloric theory thus seemed to explain the steam engine very nicely. Some other aspects of heat flow also appeared to be accounted for by the caloric theory. Some concepts and definitions that will be needed again later can be introduced in terms of the caloric theory.
One of the first things noticed about heat flow is that it is always from hot to cold. When two objects at different temperatures are placed in contact with each other, the colder one becomes warmer and the warmer one becomes colder. One never observes that the warmer one gets warmer still and the colder one, colder. The caloric theory accounts for the observed fact that heat always flows from hot to cold by simply stating that temperature differences cause heat flow. Heat flows "down temperature." In terms of the waterwheel analogy, it is just the same as saying that water always flows downhill. This will be discussed further in Chapter 5.
Another characteristic of heat is that it takes a certain amount of heat to raise the temperature of a given object a certain number of degrees. For example, a large object is much harder to "warm up" than a small one; that is, it takes more heat to do it. In terms of the caloric theory, the temperature corresponds to the "height" of the caloric fluid within the object. The amount of heat needed to raise the temperature of an object one degree Celsius is called the heat capacity of the object. Similarly, objects made of different materials are harder or easier to warm up than some others might be. This characteristic of a material is given the name specific heat and its numerical value is determined by the amount of heat necessary to raise one gram of the material by one degree Celsius.
It is important to understand the difference between heat and temperature. Heat refers to the amount, while temperature refers to the degree, or concentration. A bathtub full of slightly warm water can easily contain more heat than a thimbleful of nearly boiling water. The total amount is greater in the former, but the concentration is greater in the latter.
Water has been chosen to be the standard to which other materials are compared and its specific heat is defined to be unity. The amount of heat required to raise the temperature of one gram of water one degree Celsius is called one calorie (one "unit" of caloric fluid). Eighteenth-century physicists believed that materials with different specific heats possessed different abilities to hold caloric fluid. The heat capacity of an object is calculated by multiplying its mass by its specific heat.
Another characteristic of materials that was explained within the framework of the caloric theory of heat is called latent heat. Water provides a good example of this concept. As heat is added to water, its temperature rises relatively quickly until it reaches its boiling point. If a thermometer is placed in the water, it will be seen that the temperature quickly rises to the boiling temperature (212° F, 100° C) and then remains there as the evaporation (boiling) begins. Heat is continually added at the boiling temperature while the water vaporizes or turns into steam. Because there is no temperature rise, it may be asked: Where is the heat going that is being added to the water? The calorists (supporters of the caloric theory) answered that it was going into a hidden, or latent, form in the water. They did not suggest that it was disappearing.
There is no question that some heat is being added during this time. It is simply going into an invisible form and can be completely recovered by cooling the water. This hidden heat is called the latent heat of vaporization. A parallel phenomenon is observed when heating ice to melt it. Additional heat is required after the ice first reaches its melting temperature to actually melt the ice. This hidden heat is called the latent heat of melting. These latent heats have been observed for all materials as they change from solid to liquid or liquid to gas (i.e., change from one state to another) and are different for different materials. These latent heats are determined as the amount of heat required to melt or vaporize one gram of the material.
It was regarded as significant that exactly all of the heat put into latent heat is released when the material is later cooled down. For example, water in the form of steam is more scalding than the same amount of hot liquid water at the same temperature. The latent heat of the steam will be released as it condenses on skin. The point is, however, that something appears to be conserved— that is, not destroyed. This conserved quantity was believed to be the caloric fluid.
There were other phenomena, however, that were not explained sufficiently by the caloric theory; some of these "problem phenomena" led a few physicists to perform investigations that ultimately demonstrated that there is no caloric fluid at all. Perhaps the best known of these "problem phenomena" was heat generated by friction. We can warm our hands just by rubbing them together. There does not appear to be any heat source for the warmth one feels. There is no fire or even a warm object to provide a flow of heat into our cold hands. Where does the heat come from? If somehow heat is being generated where there was none before, then we must be (according to the caloric theory) producing caloric fluid. But caloric fluid was believed to be something that cannot be created or destroyed; it can only flow from one object to another. Where then, does the caloric fluid come from when friction occurs?
The calorists' answer was that a latent heat is released when heat is produced by friction. The claim was that a change of state is involved, just as in changing a liquid to gas, when small particles are ground off an object by friction—during machining, for example. Some calorists said that the material was "damaged" and would "bleed" heat. These answers were never entirely satisfactory. Any small pieces ground off by friction appear to be just tiny amounts of the original material. Eventually it was shown that such grindings have the same characteristics, including the same specific heat, as the original material, and they certainly had not undergone a change of state.
One of the first people to be impressed by this difficulty with the caloric theory was Count Rumford.2 In a famous experiment performed in 1798, he measured the amount of heat developed from friction during the boring of a cannon at the Bavarian royal arsenal in Munich. He was greatly impressed with the large amount of heat produced and the small amount of metal shavings that resulted. Count Rumford was convinced that the tremendous amount of heat being generated could not be from some latent heat. He measured the specific heat of the shavings and found it to be the same as the original metal. Count Rumford proceeded to demonstrate that, in fact, one could produce any amount of heat one desired, without ever reducing the total amount of metal. The metal was simply being slowly reduced to shavings with no change of state involved. He even proceeded to show that heat could be generated without producing any shavings at all. Count Rumford seriously questioned whether the calorists' interpretation could be correct and suggested that the heat being produced was new heat.
Actually, the calorists readily accepted the results of Rumford's experiments and felt they provided important clues regarding the nature of the caloric fluid. It was said that these experiments showed that caloric fluid must be composed of small, nearly massless particles whose total number in an object is vastly greater than the number ever released by friction. This interpretation is very close to our modern understanding of electricity in terms of atomic electrons. In many ways, Rumford's experiments were simply taken to further the understanding of the caloric fluid.
Besides friction, a second class of phenomena that caused difficulties for the caloric theory was the expansion and contraction of gases. In a little-known (then or now) experiment in 1807, the French physicist Gay-Lussac measured the temperature of a gas allowed to expand into an evacuated chamber (i.e., one with all the air removed). He measured the temperature of the gas both in its original chamber and in the chamber that had originally been evacuated. The temperatures were found to be the same and equal to the original temperature of the gas. The caloric theory predicted that half of the original caloric fluid should be in each chamber (if the two chambers were of equal size). Because temperature was believed to be determined by the concentration of caloric fluid, this should have resulted in the temperature being considerably lower than it was originally. This
2Rumford, originally named Benjamin Thompson, was a former American colonist and Tory who fled to England during the American Revolution. Interestingly enough, Rumford, who helped overthrow the caloric theory, married the widow of Lavoisier, who first proposed the caloric theory.
experiment and others similar to it simply were not understood by the calorists and, in fact, were not understood by anyone for over 30 years.
The final blow to the caloric theory was provided by the careful experiments of British physicist James Joule, in the 1840s. It was well-known that heat could do work and this was understood by the calorists with their waterwheel analogy. Heat did work as it flowed down-temperature just as water does when it flows downhill. But can work create heat? This was one of the most important questions to arise in the entire development of our present-day understanding of heat and energy. The caloric theory answer was a very definite no. This would be creating new caloric fluid; it would be like creating water by mechanical means, and could not be done—except that is just what Joule's experiments demonstrated.
Joule, encouraged by the work of Count Rumford and others, began to study whether mechanical work could produce heat. His experimental apparatus consisted of a paddle wheel inside a cylinder of water (see Fig. 4.4). The vanes placed inside the cylinder just barely allowed the paddles to pass and kept the water from swirling with the paddles. Joule wanted to see if one could raise the temperature of the water just by rotating the paddle. The vanes ensured that the rotating paddle could only "agitate" the water, and microscopic constituents (i.e., molecules) would be set into faster motions. Additionally, if the temperature of the water rose, it must have been because of the mechanical work expended in turning the paddle wheel. Thus if Joule measured a temperature rise in the water, he would have direct evidence that work can produce heat. Finally, one certainly cannot talk about damaging or bruising water to make it "bleed" heat.
Although the temperature rise was rather small, and Joule had to construct his own very sensitive thermometer, he observed a definite increase in the tempera-
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