# Thermodynamics

## The First Law of Thermodynamics and Some Simple Processes

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the processes of a simple heat engine.
2. Explain the differences among the simple thermodynamic processes—isobaric, isochoric, isothermal, and adiabatic.
3. Calculate total work done in a cyclical thermodynamic process.
One of the most important things we can do with heat transfer is to use it to do work for us. Such a device is called a heat engine. Car engines and steam turbines that generate electricity are examples of heat engines.  shows schematically how the first law of thermodynamics applies to the typical heat engine.
The illustrations above show one of the ways in which heat transfer does work. Fuel combustion produces heat transfer to a gas in a cylinder, increasing the pressure of the gas and thereby the force it exerts on a movable piston. The gas does work on the outside world, as this force moves the piston through some distance. Heat transfer to the gas cylinder results in work being done. To repeat this process, the piston needs to be returned to its starting point. Heat transfer now occurs from the gas to the surroundings so that its pressure decreases, and a force is exerted by the surroundings to push the piston back through some distance. Variations of this process are employed daily in hundreds of millions of heat engines. We will examine heat engines in detail in the next section. In this section, we consider some of the simpler underlying processes on which heat engines are based.

### PV  Diagrams and their Relationship to Work Done on or by a Gas

A process by which a gas does work on a piston at constant pressure is called an isobaric process. Since the pressure is constant, the force exerted is constant and the work done is given as
See the symbols as shown in . Now , and so
Because the volume of a cylinder is its cross-sectional area  times its length , we see that , the change in volume; thus,
Note that if  is positive, then  is positive, meaning that work is done by the gas on the outside world.
(Note that the pressure involved in this work that we’ve called

      is the pressure of the gas inside the tank.  If we call the pressure outside the tank

     , an expanding gas would be working against the external pressure; the work done would therefore be

       (isobaric process). Many texts use this definition of work, and not the definition based on internal pressure, as the basis of the First Law of Thermodynamics.  This definition reverses the sign conventions for work, and results in a statement of the first law that becomes

     .)
It is not surprising that , since we have already noted in our treatment of fluids that pressure is a type of potential energy per unit volume and that pressure in fact has units of energy divided by volume. We also noted in our discussion of the ideal gas law that  has units of energy. In this case, some of the energy associated with pressure becomes work.
 shows a graph of pressure versus volume (that is, a  diagram for an isobaric process. You can see in the figure that the work done is the area under the graph. This property of  diagrams is very useful and broadly applicable: the work done on or by a system in going from one state to another equals the area under the curve on a .
We can see where this leads by considering (a), which shows a more general process in which both pressure and volume change. The area under the curve is closely approximated by dividing it into strips, each having an average constant pressure . The work done is  for each strip, and the total work done is the sum of the . Thus the total work done is the total area under the curve. If the path is reversed, as in (b), then work is done on the system. The area under the curve in that case is negative, because  is negative.
 diagrams clearly illustrate that the work done depends on the path taken and not just the endpoints. This path dependence is seen in (a), where more work is done in going from A to C by the path via point B than by the path via point D. The vertical paths, where volume is constant, are called isochoric processes. Since volume is constant, , and no work is done in an isochoric process. Now, if the system follows the cyclical path ABCDA, as in (b), then the total work done is the area inside the loop. The negative area below path CD subtracts, leaving only the area inside the rectangle. In fact, the work done in any cyclical process (one that returns to its starting point) is the area inside the loop it forms on a  diagram, as (c) illustrates for a general cyclical process. Note that the loop must be traversed in the clockwise direction for work to be positive—that is, for there to be a net work output.

### Reversible Processes

Both isothermal and adiabatic processes such as shown in  are reversible in principle. A reversible process is one in which both the system and its environment can return to exactly the states they were in by following the reverse path. The reverse isothermal and adiabatic paths are BA and CA, respectively. Real macroscopic processes are never exactly reversible. In the previous examples, our system is a gas (like that in ), and its environment is the piston, cylinder, and the rest of the universe. If there are any energy-dissipating mechanisms, such as friction or turbulence, then heat transfer to the environment occurs for either direction of the piston. So, for example, if the path BA is followed and there is friction, then the gas will be returned to its original state but the environment will not—it will have been heated in both directions. Reversibility requires the direction of heat transfer to reverse for the reverse path. Since dissipative mechanisms cannot be completely eliminated, real processes cannot be reversible.
There must be reasons that real macroscopic processes cannot be reversible. We can imagine them going in reverse. For example, heat transfer occurs spontaneously from hot to cold and never spontaneously the reverse. Yet it would not violate the first law of thermodynamics for this to happen. In fact, all spontaneous processes, such as bubbles bursting, never go in reverse. There is a second thermodynamic law that forbids them from going in reverse. When we study this law, we will learn something about nature and also find that such a law limits the efficiency of heat engines. We will find that heat engines with the greatest possible theoretical efficiency would have to use reversible processes, and even they cannot convert all heat transfer into doing work.  summarizes the simpler thermodynamic processes and their definitions.

### Test Prep for AP Courses

### Section Summary

1. One of the important implications of the first law of thermodynamics is that machines can be harnessed to do work that humans previously did by hand or by external energy supplies such as running water or the heat of the Sun. A machine that uses heat transfer to do work is known as a heat engine.
2. There are several simple processes, used by heat engines, that flow from the first law of thermodynamics. Among them are the isobaric, isochoric, isothermal and adiabatic processes.
3. These processes differ from one another based on how they affect pressure, volume, temperature, and heat transfer.
4. If the work done is performed on the outside environment, work () will be a positive value. If the work done is done to the heat engine system, work () will be a negative value.
5. Some thermodynamic processes, including isothermal and adiabatic processes, are reversible in theory; that is, both the thermodynamic system and the environment can be returned to their initial states. However, because of loss of energy owing to the second law of thermodynamics, complete reversibility does not work in practice.

### Conceptual Questions

### Problem Exercises


# Thermodynamics

## Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency

### Learning Objectives

By the end of this section, you will be able to:
1. State the expressions of the second law of thermodynamics.
2. Calculate the efficiency and carbon dioxide emission of a coal-fired electricity plant, using second law characteristics.
3. Describe and define the Otto cycle.
The second law of thermodynamics deals with the direction taken by spontaneous processes. Many processes occur spontaneously in one direction only—that is, they are irreversible, under a given set of conditions. Although irreversibility is seen in day-to-day life—a broken glass does not resume its original state, for instance—complete irreversibility is a statistical statement that cannot be seen during the lifetime of the universe. More precisely, an irreversible process is one that depends on path. If the process can go in only one direction, then the reverse path differs fundamentally and the process cannot be reversible. For example, as noted in the previous section, heat involves the transfer of energy from higher to lower temperature. A cold object in contact with a hot one never gets colder, transferring heat to the hot object and making it hotter. Furthermore, mechanical energy, such as kinetic energy, can be completely converted to thermal energy by friction, but the reverse is impossible. A hot stationary object never spontaneously cools off and starts moving. Yet another example is the expansion of a puff of gas introduced into one corner of a vacuum chamber. The gas expands to fill the chamber, but it never regroups in the corner. The random motion of the gas molecules could take them all back to the corner, but this is never observed to happen. (See .)
The fact that certain processes never occur suggests that there is a law forbidding them to occur. The first law of thermodynamics would allow them to occur—none of those processes violate conservation of energy. The law that forbids these processes is called the second law of thermodynamics. We shall see that the second law can be stated in many ways that may seem different, but which in fact are equivalent. Like all natural laws, the second law of thermodynamics gives insights into nature, and its several statements imply that it is broadly applicable, fundamentally affecting many apparently disparate processes.
The already familiar direction of heat transfer from hot to cold is the basis of our first version of the second law of thermodynamics.
Another way of stating this: It is impossible for any process to have as its sole result heat transfer from a cooler to a hotter object.

### Heat Engines

Now let us consider a device that uses heat transfer to do work. As noted in the previous section, such a device is called a heat engine, and one is shown schematically in (b). Gasoline and diesel engines, jet engines, and steam turbines are all heat engines that do work by using part of the heat transfer from some source. Heat transfer from the hot object (or hot reservoir) is denoted as , while heat transfer into the cold object (or cold reservoir) is , and the work done by the engine is . The temperatures of the hot and cold reservoirs are  and , respectively.
Because the hot reservoir is heated externally, which is energy intensive, it is important that the work is done as efficiently as possible. In fact, we would like  to equal , and for there to be no heat transfer to the environment (). Unfortunately, this is impossible. The second law of thermodynamics also states, with regard to using heat transfer to do work (the second expression of the second law):
A cyclical process brings a system, such as the gas in a cylinder, back to its original state at the end of every cycle. Most heat engines, such as reciprocating piston engines and rotating turbines, use cyclical processes. The second law, just stated in its second form, clearly states that such engines cannot have perfect conversion of heat transfer into work done. Before going into the underlying reasons for the limits on converting heat transfer into work, we need to explore the relationships among , , and , and to define the efficiency of a cyclical heat engine. As noted, a cyclical process brings the system back to its original condition at the end of every cycle. Such a system’s internal energy  is the same at the beginning and end of every cycle—that is, . The first law of thermodynamics states that
where  is the net heat transfer during the cycle () and  is the net work done by the system. Since  for a complete cycle, we have
so that
Thus the net work done by the system equals the net heat transfer into the system, or
just as shown schematically in (b). The problem is that in all processes, there is some heat transfer  to the environment—and usually a very significant amount at that.
In the conversion of energy to work, we are always faced with the problem of getting less out than we put in. We define conversion efficiency  to be the ratio of useful work output to the energy input (or, in other words, the ratio of what we get to what we spend). In that spirit, we define the efficiency of a heat engine to be its net work output  divided by heat transfer to the engine ; that is,
Since  in a cyclical process, we can also express this as
making it clear that an efficiency of 1, or 100%, is possible only if there is no heat transfer to the environment (). Note that all s are positive. The direction of heat transfer is indicated by a plus or minus sign. For example,  is out of the system and so is preceded by a minus sign.
With the information given in , we can find characteristics such as the efficiency of a heat engine without any knowledge of how the heat engine operates, but looking further into the mechanism of the engine will give us greater insight.  illustrates the operation of the common four-stroke gasoline engine. The four steps shown complete this heat engine’s cycle, bringing the gasoline-air mixture back to its original condition.
The Otto cycle shown in (a) is used in four-stroke internal combustion engines, although in fact the true Otto cycle paths do not correspond exactly to the strokes of the engine.
The adiabatic process AB corresponds to the nearly adiabatic compression stroke of the gasoline engine. In both cases, work is done on the system (the gas mixture in the cylinder), increasing its temperature and pressure. Along path BC of the Otto cycle, heat transfer  into the gas occurs at constant volume, causing a further increase in pressure and temperature. This process corresponds to burning fuel in an internal combustion engine, and takes place so rapidly that the volume is nearly constant. Path CD in the Otto cycle is an adiabatic expansion that does work on the outside world, just as the power stroke of an internal combustion engine does in its nearly adiabatic expansion. The work done by the system along path CD is greater than the work done on the system along path AB, because the pressure is greater, and so there is a net work output. Along path DA in the Otto cycle, heat transfer  from the gas at constant volume reduces its temperature and pressure, returning it to its original state. In an internal combustion engine, this process corresponds to the exhaust of hot gases and the intake of an air-gasoline mixture at a considerably lower temperature. In both cases, heat transfer into the environment occurs along this final path.
The net work done by a cyclical process is the area inside the closed path on a  diagram, such as that inside path ABCDA in . Note that in every imaginable cyclical process, it is absolutely necessary for heat transfer from the system to occur in order to get a net work output. In the Otto cycle, heat transfer occurs along path DA. If no heat transfer occurs, then the return path is the same, and the net work output is zero. The lower the temperature on the path AB, the less work has to be done to compress the gas. The area inside the closed path is then greater, and so the engine does more work and is thus more efficient. Similarly, the higher the temperature along path CD, the more work output there is. (See .) So efficiency is related to the temperatures of the hot and cold reservoirs. In the next section, we shall see what the absolute limit to the efficiency of a heat engine is, and how it is related to temperature.

### Section Summary

1. The two expressions of the second law of thermodynamics are: (i) Heat transfer occurs spontaneously from higher- to lower-temperature bodies but never spontaneously in the reverse direction; and (ii) It is impossible in any system for heat transfer from a reservoir to completely convert to work in a cyclical process in which the system returns to its initial state.
2. Irreversible processes depend on path and do not return to their original state. Cyclical processes are processes that return to their original state at the end of every cycle.
3. In a cyclical process, such as a heat engine, the net work done by the system equals the net heat transfer into the system, or

   , where

   is the heat transfer from the hot object (hot reservoir), and

   is the heat transfer into the cold object (cold reservoir).
4. Efficiency can be expressed as

,

 the ratio of work output divided by the amount of energy input.
5. The four-stroke gasoline engine is often explained in terms of the Otto cycle, which is a repeating sequence of processes that convert heat into work.

### Conceptual Questions

### Problem Exercises


# Thermodynamics

## Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated

### Learning Objectives

By the end of this section, you will be able to:
1. Identify a Carnot cycle.
2. Calculate maximum theoretical efficiency of a nuclear reactor.
3. Explain how dissipative processes affect the ideal Carnot engine.
We know from the second law of thermodynamics that a heat engine cannot be 100% efficient, since there must always be some heat transfer  to the environment, which is often called waste heat. How efficient, then, can a heat engine be? This question was answered at a theoretical level in 1824 by a young French engineer, Sadi Carnot (1796–1832), in his study of the then-emerging heat engine technology crucial to the Industrial Revolution. He devised a theoretical cycle, now called the Carnot cycle, which is the most efficient cyclical process possible. The second law of thermodynamics can be restated in terms of the Carnot cycle, and so what Carnot actually discovered was this fundamental law. Any heat engine employing the Carnot cycle is called a Carnot engine.
What is crucial to the Carnot cycle—and, in fact, defines it—is that only reversible processes are used. Irreversible processes involve dissipative factors, such as friction and turbulence. This increases heat transfer  to the environment and reduces the efficiency of the engine. Obviously, then, reversible processes are superior.
 shows the  diagram for a Carnot cycle. The cycle comprises two isothermal and two adiabatic processes. Recall that both isothermal and adiabatic processes are, in principle, reversible.
Carnot also determined the efficiency of a perfect heat engine—that is, a Carnot engine. It is always true that the efficiency of a cyclical heat engine is given by:
What Carnot found was that for a perfect heat engine, the ratio  equals the ratio of the absolute temperatures of the heat reservoirs. That is,  for a Carnot engine, so that the maximum or Carnot efficiency  is given by
where  and  are in kelvins (or any other absolute temperature scale). No real heat engine can do as well as the Carnot efficiency. But the ideal Carnot engine, like the drinking bird above, while a fascinating novelty, has zero power. This makes it unrealistic for any applications.
Carnot’s interesting result implies that 100% efficiency would be possible only if  —that is, only if the cold reservoir were at absolute zero, a practical and theoretical impossibility. But the physical implication is this—the only way to have all heat transfer go into doing work is to remove all thermal energy, and this requires a cold reservoir at absolute zero.
It is also apparent that the greatest efficiencies are obtained when the ratio  is as small as possible. Just as discussed for the Otto cycle in the previous section, this means that efficiency is greatest for the highest possible temperature of the hot reservoir and lowest possible temperature of the cold reservoir. (This setup increases the area inside the closed loop on the  diagram; also, it seems reasonable that the greater the temperature difference, the easier it is to divert the heat transfer to work.) The actual reservoir temperatures of a heat engine are usually related to the type of heat source and the temperature of the environment into which heat transfer occurs. Consider the following example.
Since all real processes are irreversible, the actual efficiency of a heat engine can never be as great as that of a Carnot engine, as illustrated in (a). Even with the best heat engine possible, there are always dissipative processes in peripheral equipment, such as electrical transformers or car transmissions. These further reduce the overall efficiency by converting some of the engine’s work output back into heat transfer, as shown in (b).

### Section Summary

1. The Carnot cycle is a theoretical cycle that is the most efficient cyclical process possible. Any engine using the Carnot cycle, which uses only reversible processes (adiabatic and isothermal), is known as a Carnot engine.
2. Any engine that uses the Carnot cycle enjoys the maximum theoretical efficiency.
3. While Carnot engines are ideal engines, in reality, no engine achieves Carnot’s theoretical maximum efficiency, since dissipative processes, such as friction, play a role. Carnot cycles without heat loss may be possible at absolute zero, but this has never been seen in nature.

### Conceptual Questions

### Problem Exercises


# Thermodynamics

## Applications of Thermodynamics: Heat Pumps and Refrigerators

### Learning Objectives

By the end of this section, you will be able to:
1. Describe the use of heat engines in heat pumps and refrigerators.
2. Demonstrate how a heat pump works to warm an interior space.
3. Explain the differences between heat pumps and refrigerators.
4. Calculate a heat pump’s coefficient of performance.
Heat pumps, air conditioners, and refrigerators utilize heat transfer from cold to hot. They are heat engines run backward. We say backward, rather than reverse, because except for Carnot engines, all heat engines, though they can be run backward, cannot truly be reversed. Heat transfer occurs from a cold reservoir  and into a hot one. This requires work input , which is also converted to heat transfer. Thus the heat transfer to the hot reservoir is . (Note that , , and  are positive, with their directions indicated on schematics rather than by sign.) A heat pump’s mission is for heat transfer  to occur into a warm environment, such as a home in the winter. The mission of air conditioners and refrigerators is for heat transfer  to occur from a cool environment, such as chilling a room or keeping food at lower temperatures than the environment. (Actually, a heat pump can be used both to heat and cool a space. It is essentially an air conditioner and a heating unit all in one. In this section we will concentrate on its heating mode.)

### Heat Pumps

The great advantage of using a heat pump to keep your home warm, rather than just burning fuel, is that a heat pump supplies . Heat transfer is from the outside air, even at a temperature below freezing, to the indoor space. You only pay for , and you get an additional heat transfer of  from the outside at no cost; in many cases, at least twice as much energy is transferred to the heated space as is used to run the heat pump. When you burn fuel to keep warm, you pay for all of it. The disadvantage is that the work input (required by the second law of thermodynamics) is sometimes more expensive than simply burning fuel, especially if the work is done by electrical energy.
The basic components of a heat pump in its heating mode are shown in . A working fluid such as a non-CFC refrigerant is used. In the outdoor coils (the evaporator), heat transfer  occurs to the working fluid from the cold outdoor air, turning it into a gas.
The electrically driven compressor (work input ) raises the temperature and pressure of the gas and forces it into the condenser coils that are inside the heated space. Because the temperature of the gas is higher than the temperature inside the room, heat transfer to the room occurs and the gas condenses to a liquid. The liquid then flows back through a pressure-reducing valve to the outdoor evaporator coils, being cooled through expansion. (In a cooling cycle, the evaporator and condenser coils exchange roles and the flow direction of the fluid is reversed.)
The quality of a heat pump is judged by how much heat transfer  occurs into the warm space compared with how much work input  is required. In the spirit of taking the ratio of what you get to what you spend, we define a heat pump’s coefficient of performance () to be
Since the efficiency of a heat engine is , we see that , an important and interesting fact. First, since the efficiency of any heat engine is less than 1, it means that  is always greater than 1—that is, a heat pump always has more heat transfer  than work put into it. Second, it means that heat pumps work best when temperature differences are small. The efficiency of a perfect, or Carnot, engine is ; thus, the smaller the temperature difference, the smaller the efficiency and the greater the  (because ). In other words, heat pumps do not work as well in very cold climates as they do in more moderate climates.
Friction and other irreversible processes reduce heat engine efficiency, but they do not benefit the operation of a heat pump—instead, they reduce the work input by converting part of it to heat transfer back into the cold reservoir before it gets into the heat pump.
Real heat pumps do not perform quite as well as the ideal one in the previous example; their values of  range from about 2 to 4. This range means that the heat transfer  from the heat pumps is 2 to 4 times as great as the work  put into them. Their economical feasibility is still limited, however, since  is usually supplied by electrical energy that costs more per joule than heat transfer by burning fuels like natural gas. Furthermore, the initial cost of a heat pump is greater than that of many furnaces, so that a heat pump must last longer for its cost to be recovered. Heat pumps are most likely to be economically superior where winter temperatures are mild, electricity is relatively cheap, and other fuels are relatively expensive. Also, since they can cool as well as heat a space, they have advantages where cooling in summer months is also desired. Thus some of the best locations for heat pumps are in warm summer climates with cool winters.  shows a heat pump, called a “reverse cycle” or “split-system cooler”  in some countries.

### Air Conditioners and Refrigerators

Air conditioners and refrigerators are designed to cool something down in a warm environment. As with heat pumps, work input is required for heat transfer from cold to hot, and this is expensive. The quality of air conditioners and refrigerators is judged by how much heat transfer  occurs from a cold environment compared with how much work input  is required. What is considered the benefit in a heat pump is considered waste heat in a refrigerator. We thus define the coefficient of performance  of an air conditioner or refrigerator to be
Noting again that , we can see that an air conditioner will have a lower coefficient of performance than a heat pump, because  and  is greater than . In this module’s Problems and Exercises, you will show that
for a heat engine used as either an air conditioner or a heat pump operating between the same two temperatures. Real air conditioners and refrigerators typically do remarkably well, having values of  ranging from 2 to 6. These numbers are better than the  values for the heat pumps mentioned above, because the temperature differences are smaller, but they are less than those for Carnot engines operating between the same two temperatures.
A type of  rating system called the “energy efficiency rating” (

    ) has been developed. This rating is an example where non-SI units are still used and relevant to consumers. To make it easier for the consumer, Australia, Canada, New Zealand, and the U.S. use an Energy Star Rating out of 5 stars—the more stars, the more energy efficient the appliance.

           are expressed in mixed units of British thermal units (Btu) per hour of heating or cooling divided by the power input in watts. Room air conditioners are readily available with

           ranging from 6 to 12. Although not the same as the  just described, these

           are good for comparison purposes—the greater the

          , the cheaper an air conditioner is to operate (but the higher its purchase price is likely to be).
The

       of an air conditioner or refrigerator can be expressed as
where

      is the amount of heat transfer from a cold environment in British thermal units,

  is time in hours,

 is the work input in joules, and

      is time in seconds.

### Section Summary

1. An artifact of the second law of thermodynamics is the ability to heat an interior space using a heat pump. Heat pumps compress cold ambient air and, in so doing, heat it to room temperature without violation of conservation principles.
2. To calculate the heat pump’s coefficient of performance, use the equation .
3. A refrigerator is a heat pump; it takes warm ambient air and expands it to chill it.

### Conceptual Questions

### Problem Exercises


# Thermodynamics

## Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy

### Learning Objectives

By the end of this section, you will be able to:
1.  Define entropy and calculate the increase of entropy in a system with reversible and irreversible processes.
2. Explain the expected fate of the universe in entropic terms.
3. Calculate the increasing disorder of a system.
There is yet another way of expressing the second law of thermodynamics. This version relates to a concept called entropy. By examining it, we shall see that the directions associated with the second law—heat transfer from hot to cold, for example—are related to the tendency in nature for systems to become disordered and for less energy to be available for use as work. The entropy of a system can in fact be shown to be a measure of its disorder and of the unavailability of energy to do work.
We can see how entropy is defined by recalling our discussion of the Carnot engine. We noted that for a Carnot cycle, and hence for any reversible processes, . Rearranging terms yields
for any reversible process.  and  are absolute values of the heat transfer at temperatures  and , respectively. This ratio of  is defined to be the change in entropy  for a reversible process,
where  is the heat transfer, which is positive for heat transfer into and negative for heat transfer out of, and  is the absolute temperature at which the reversible process takes place. The SI unit for entropy is joules per kelvin (J/K). If temperature changes during the process, then it is usually a good approximation (for small changes in temperature) to take  to be the average temperature, avoiding the need to use integral calculus to find .
The definition of  is strictly valid only for reversible processes, such as used in a Carnot engine. However, we can find  precisely even for real, irreversible processes. The reason is that the entropy  of a system, like internal energy , depends only on the state of the system and not how it reached that condition. Entropy is a property of state. Thus the change in entropy  of a system between state 1 and state 2 is the same no matter how the change occurs. We just need to find or imagine a reversible process that takes us from state 1 to state 2 and calculate  for that process. That will be the change in entropy for any process going from state 1 to state 2. (See .)
Now let us take a look at the change in entropy of a Carnot engine and its heat reservoirs for one full cycle. The hot reservoir has a loss of entropy , because heat transfer occurs out of it (remember that when heat transfers out, then  has a negative sign). The cold reservoir has a gain of entropy , because heat transfer occurs into it. (We assume the reservoirs are sufficiently large that their temperatures are constant.) So the total change in entropy is
Thus, since we know that  for a Carnot engine,
This result, which has general validity, means that the total change in entropy for a system in any reversible process is zero.
The entropy of various parts of the system may change, but the total change is zero. Furthermore, the system does not affect the entropy of its surroundings, since heat transfer between them does not occur. Thus the reversible process changes neither the total entropy of the system nor the entropy of its surroundings. Sometimes this is stated as follows: Reversible processes do not affect the total entropy of the universe. Real processes are not reversible, though, and they do change total entropy. We can, however, use hypothetical reversible processes to determine the value of entropy in real, irreversible processes. The following example illustrates this point.
It is reasonable that entropy increases for heat transfer from hot to cold. Since the change in entropy is , there is a larger change at lower temperatures. The decrease in entropy of the hot object is therefore less than the increase in entropy of the cold object, producing an overall increase, just as in the previous example. This result is very general:
There is an increase in entropy for any system undergoing an irreversible process.
With respect to entropy, there are only two possibilities: entropy is constant for a reversible process, and it increases for an irreversible process. There is a fourth version of the second law of thermodynamics stated in terms of entropy:
The total entropy of a system either increases or remains constant in any process; it never decreases.
For example, heat transfer cannot occur spontaneously from cold to hot, because entropy would decrease.
Entropy is very different from energy. Entropy is not conserved but increases in all real processes. Reversible processes (such as in Carnot engines) are the processes in which the most heat transfer to work takes place and are also the ones that keep entropy constant. Thus we are led to make a connection between entropy and the availability of energy to do work.

### Entropy and the Unavailability of Energy to Do Work

What does a change in entropy mean, and why should we be interested in it? One reason is that entropy is directly related to the fact that not all heat transfer can be converted into work. The next example gives some indication of how an increase in entropy results in less heat transfer into work.
When entropy increases, a certain amount of energy becomes permanently unavailable to do work. The energy is not lost, but its character is changed, so that some of it can never be converted to doing work—that is, to an organized force acting through a distance. For instance, in the previous example, 933 J less work was done after an increase in entropy of 9.33 J/K occurred in the 4000 J heat transfer from the 600 K reservoir to the 250 K reservoir. It can be shown that the amount of energy that becomes unavailable for work is
where  is the lowest temperature utilized. In the previous example,
as found.

### Heat Death of the Universe: An Overdose of Entropy

In the early, energetic universe, all matter and energy were easily interchangeable and identical in nature. Gravity played a vital role in the young universe. Although it may have seemed disorderly, and therefore, superficially entropic, in fact, there was enormous potential energy available to do work—all the future energy in the universe.
As the universe matured, temperature differences arose, which created more opportunity for work. Stars are hotter than planets, for example, which are warmer than icy asteroids, which are warmer still than the vacuum of the space between them.
Most of these are cooling down from their usually violent births, at which time they were provided with energy of their own—nuclear energy in the case of stars, volcanic energy on Earth and other planets, and so on. Without additional energy input, however, their days are numbered.
As entropy increases, less and less energy in the universe is available to do work. On Earth, we still have great stores of energy such as fossil and nuclear fuels; large-scale temperature differences, which can provide wind energy; geothermal energies due to differences in temperature in Earth’s layers; and tidal energies owing to our abundance of liquid water. As these are used, a certain fraction of the energy they contain can never be converted into doing work. Eventually, all fuels will be exhausted, all temperatures will equalize, and it will be impossible for heat engines to function, or for work to be done.
Entropy increases in a closed system, such as the universe. But in parts of the universe, for instance, in the Solar system, it is not a locally closed system. Energy flows from the Sun to the planets, replenishing Earth’s stores of energy. The Sun will continue to supply us with energy for about another five billion years. We will enjoy direct solar energy, as well as side effects of solar energy, such as wind power and biomass energy from photosynthetic plants. The energy from the Sun will keep our water at the liquid state, and the Moon’s gravitational pull will continue to provide tidal energy. But Earth’s geothermal energy will slowly run down and won’t be replenished.
But in terms of the universe, and the very long-term, very large-scale picture, the entropy of the universe is increasing, and so the availability of energy to do work is constantly decreasing. Eventually, when all stars have died, all forms of potential energy have been utilized, and all temperatures have equalized (depending on the mass of the universe, either at a very high temperature following a universal contraction, or a very low one, just before all activity ceases) there will be no possibility of doing work.
Either way, the universe is destined for thermodynamic equilibrium—maximum entropy. This is often called the heat death of the universe, and will mean the end of all activity. However, whether the universe contracts and heats up, or continues to expand and cools down, the end is not near. Calculations of black holes suggest that entropy can easily continue for at least  years.

### Order to Disorder

Entropy is related not only to the unavailability of energy to do work—it is also a measure of disorder. This notion was initially postulated by Ludwig Boltzmann in the 1800s. For example, melting a block of ice means taking a highly structured and orderly system of water molecules and converting it into a disorderly liquid in which molecules have no fixed positions. (See .) There is a large increase in entropy in the process, as seen in the following example.
In another easily imagined example, suppose we mix equal masses of water originally at two different temperatures, say  and . The result is water at an intermediate temperature of . Three outcomes have resulted: entropy has increased, some energy has become unavailable to do work, and the system has become less orderly. Let us think about each of these results.
First, entropy has increased for the same reason that it did in the example above. Mixing the two bodies of water has the same effect as heat transfer from the hot one and the same heat transfer into the cold one. The mixing decreases the entropy of the hot water but increases the entropy of the cold water by a greater amount, producing an overall increase in entropy.
Second, once the two masses of water are mixed, there is only one temperature—you cannot run a heat engine with them. The energy that could have been used to run a heat engine is now unavailable to do work.
Third, the mixture is less orderly, or to use another term, less structured. Rather than having two masses at different temperatures and with different distributions of molecular speeds, we now have a single mass with a uniform temperature.
These three results—entropy, unavailability of energy, and disorder—are not only related but are in fact essentially equivalent.

### Life, Evolution, and the Second Law of Thermodynamics

Some people misunderstand the second law of thermodynamics, stated in terms of entropy, to say that the process of the evolution of life violates this law. Over time, complex organisms evolved from much simpler ancestors, representing a large decrease in entropy of the Earth’s biosphere. It is a fact that living organisms have evolved to be highly structured, and much lower in entropy than the substances from which they grow. But it is always possible for the entropy of one part of the universe to decrease, provided the total change in entropy of the universe increases. In equation form, we can write this as
Thus  can be negative as long as  is positive and greater in magnitude.
How is it possible for a system to decrease its entropy? Energy transfer is necessary. If I pick up marbles that are scattered about the room and put them into a cup, my work has decreased the entropy of that system. If I gather iron ore from the ground and convert it into steel and build a bridge, my work has decreased the entropy of that system. Energy coming from the Sun can decrease the entropy of local systems on Earth—that is,  is negative. But the overall entropy of the rest of the universe increases by a greater amount—that is,  is positive and greater in magnitude. Thus, , and the second law of thermodynamics is not violated.
Every time a plant stores some solar energy in the form of chemical potential energy, or an updraft of warm air lifts a soaring bird, the Earth can be viewed as a heat engine operating between a hot reservoir supplied by the Sun and a cold reservoir supplied by dark outer space—a heat engine of high complexity, causing local decreases in entropy as it uses part of the heat transfer from the Sun into deep space. There is a large total increase in entropy resulting from this massive heat transfer. A small part of this heat transfer is stored in structured systems on Earth, producing much smaller local decreases in entropy. (See .)

### Test Prep for AP Courses

### Section Summary

1. Entropy is the loss of energy available to do work.
2. Another form of the second law of thermodynamics states that the total entropy of a system either increases or remains constant; it never decreases.
3. Entropy is zero in a reversible process; it increases in an irreversible process.
4. The ultimate fate of the universe is likely to be thermodynamic equilibrium, where the universal temperature is constant and no energy is available to do work.
5. Entropy is also associated with the tendency toward disorder in a closed system.

### Conceptual Questions

### Problem Exercises


# Thermodynamics

## Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

### Learning Objectives

By the end of this section, you will be able to:
1. Identify probabilities in entropy.
2. Analyze statistical probabilities in entropic systems.
The various ways of formulating the second law of thermodynamics tell what happens rather than why it happens. Why should heat transfer occur only from hot to cold? Why should energy become ever less available to do work? Why should the universe become increasingly disorderly? The answer is that it is a matter of overwhelming probability. Disorder is simply vastly more likely than order.
When you watch an emerging rain storm begin to wet the ground, you will notice that the drops fall in a disorganized manner both in time and in space. Some fall close together, some far apart, but they never fall in straight, orderly rows. It is not impossible for rain to fall in an orderly pattern, just highly unlikely, because there are many more disorderly ways than orderly ones. To illustrate this fact, we will examine some random processes, starting with coin tosses.

### Coin Tosses

What are the possible outcomes of tossing 5 coins? Each coin can land either heads or tails. On the large scale, we are concerned only with the total heads and tails and not with the order in which heads and tails appear. The following possibilities exist:
These are what we call macrostates. A macrostate is an overall property of a system. It does not specify the details of the system, such as the order in which heads and tails occur or which coins are heads or tails.
Using this nomenclature, a system of 5 coins has the 6 possible macrostates just listed. Some macrostates are more likely to occur than others. For instance, there is only one way to get 5 heads, but there are several ways to get 3 heads and 2 tails, making the latter macrostate more probable.  lists of all the ways in which 5 coins can be tossed, taking into account the order in which heads and tails occur. Each sequence is called a microstate—a detailed description of every element of a system.
The macrostate of 3 heads and 2 tails can be achieved in 10 ways and is thus 10 times more probable than the one having 5 heads. Not surprisingly, it is equally probable to have the reverse, 2 heads and 3 tails. Similarly, it is equally probable to get 5 tails as it is to get 5 heads. Note that all of these conclusions are based on the crucial assumption that each microstate is equally probable. With coin tosses, this requires that the coins not be asymmetric in a way that favors one side over the other, as with loaded dice. With any system, the assumption that all microstates are equally probable must be valid, or the analysis will be erroneous.
The two most orderly possibilities are 5 heads or 5 tails. (They are more structured than the others.) They are also the least likely, only 2 out of 32 possibilities. The most disorderly possibilities are 3 heads and 2 tails and its reverse. (They are the least structured.) The most disorderly possibilities are also the most likely, with 20 out of 32 possibilities for the 3 heads and 2 tails and its reverse. If we start with an orderly array like 5 heads and toss the coins, it is very likely that we will get a less orderly array as a result, since 30 out of the 32 possibilities are less orderly. So even if you start with an orderly state, there is a strong tendency to go from order to disorder, from low entropy to high entropy. The reverse can happen, but it is unlikely.
This result becomes dramatic for larger systems. Consider what happens if you have 100 coins instead of just 5. The most orderly arrangements (most structured) are 100 heads or 100 tails. The least orderly (least structured) is that of 50 heads and 50 tails. There is only 1 way (1 microstate) to get the most orderly arrangement of 100 heads. There are 100 ways (100 microstates) to get the next most orderly arrangement of 99 heads and 1 tail (also 100 to get its reverse). And there are  ways to get 50 heads and 50 tails, the least orderly arrangement.  is an abbreviated list of the various macrostates and the number of microstates for each macrostate. The total number of microstates—the total number of different ways 100 coins can be tossed—is an impressively large . Now, if we start with an orderly macrostate like 100 heads and toss the coins, there is a virtual certainty that we will get a less orderly macrostate. If we keep tossing the coins, it is possible, but exceedingly unlikely, that we will ever get back to the most orderly macrostate. If you tossed the coins once each second, you could expect to get either 100 heads or 100 tails once in  years! This period is 1 trillion () times longer than the age of the universe, and so the chances are essentially zero. In contrast, there is an 8% chance of getting 50 heads, a 73% chance of getting from 45 to 55 heads, and a 96% chance of getting from 40 to 60 heads. Disorder is highly likely.

### Disorder in a Gas

The fantastic growth in the odds favoring disorder that we see in going from 5 to 100 coins continues as the number of entities in the system increases. Let us now imagine applying this approach to perhaps a small sample of gas. Because counting microstates and macrostates involves statistics, this is called statistical analysis. The macrostates of a gas correspond to its macroscopic properties, such as volume, temperature, and pressure; and its microstates correspond to the detailed description of the positions and velocities of its atoms. Even a small amount of gas has a huge number of atoms:  of an ideal gas at 1.0 atm and  has  atoms. So each macrostate has an immense number of microstates. In plain language, this means that there are an immense number of ways in which the atoms in a gas can be arranged, while still having the same pressure, temperature, and so on.
The most likely conditions (or macrostates) for a gas are those we see all the time—a random distribution of atoms in space with a Maxwell-Boltzmann distribution of speeds in random directions, as predicted by kinetic theory. This is the most disorderly and least structured condition we can imagine. In contrast, one type of very orderly and structured macrostate has all of the atoms in one corner of a container with identical velocities. There are very few ways to accomplish this (very few microstates corresponding to it), and so it is exceedingly unlikely ever to occur. (See (b).) Indeed, it is so unlikely that we have a law saying that it is impossible, which has never been observed to be violated—the second law of thermodynamics.
The disordered condition is one of high entropy, and the ordered one has low entropy. With a transfer of energy from another system, we could force all of the atoms into one corner and have a local decrease in entropy, but at the cost of an overall increase in entropy of the universe. If the atoms start out in one corner, they will quickly disperse and become uniformly distributed and will never return to the orderly original state ((b)). Entropy will increase. With such a large sample of atoms, it is possible—but unimaginably unlikely—for entropy to decrease. Disorder is vastly more likely than order.
The arguments that disorder and high entropy are the most probable states are quite convincing. The great Austrian physicist Ludwig Boltzmann (1844–1906)—who, along with Maxwell, made so many contributions to kinetic theory—proved that the entropy of a system in a given state (a macrostate) can be written as
where  is Boltzmann’s constant, and  is the natural logarithm of the number of microstates  corresponding to the given macrostate.  is proportional to the probability that the macrostate will occur. Thus entropy is directly related to the probability of a state—the more likely the state, the greater its entropy. Boltzmann proved that this expression for  is equivalent to the definition , which we have used extensively.
Thus the second law of thermodynamics is explained on a very basic level: entropy either remains the same or increases in every process. This phenomenon is due to the extraordinarily small probability of a decrease, based on the extraordinarily larger number of microstates in systems with greater entropy. Entropy can decrease, but for any macroscopic system, this outcome is so unlikely that it will never be observed.

### Test Prep for AP Courses

### Section Summary

1. Disorder is far more likely than order, which can be seen statistically.
2. The entropy of a system in a given state (a macrostate) can be written as
where

  is Boltzmann’s constant, and  is the natural logarithm of the number of microstates   corresponding to the given macrostate.

### Conceptual Questions

### Problem Exercises


# Oscillatory Motion and Waves

## Connection for AP® Courses

In this chapter, students are introduced to oscillation, the regular variation in the position of a system about a central point accompanied by transfer of energy and momentum, and to waves. A child’s swing, a pendulum, a spring, and a vibrating string are all examples of oscillations. This chapter will address simple harmonic motion and periods of vibration, aspects of oscillation that produce waves, a common phenomenon in everyday life. Waves carry energy from one place to another.” This chapter will show how harmonic oscillations produce waves that transport energy across space and through time. The information and examples presented support Big Ideas 1, 2, and 3 of the AP® Physics Curriculum Framework.
The chapter opens by discussing the forces that govern oscillations and waves. It goes on to discuss important concepts such as simple harmonic motion, uniform harmonic motion, and damped harmonic motion. You will also learn about energy in simple harmonic motion and how it changes from kinetic to potential, and how the total sum, which would be the mechanical energy of the oscillator, remains constant or conserved at all times. The chapter also discusses characteristics of waves, such as their frequency, period of oscillation, and the forms in which they can exist, i.e., transverse or longitudinal. The chapter ends by discussing what happens when two or more waves overlap and how the amplitude of the resultant wave changes, leading to the phenomena of superposition and interference.
The concepts in this chapter support:
Big Idea 3 The interactions of an object with other objects can be described by forces.
Enduring Understanding 3.B Classically, the acceleration of an object interacting with other objects can be predicted by using
  .
Essential Knowledge 3.B.3 Restoring forces can result in oscillatory motion. When a linear restoring force is exerted on an object displaced from an equilibrium position, the object will undergo a special type of motion called simple harmonic motion. Examples should include gravitational force exerted by the Earth on a simple pendulum and a mass-spring oscillator.
Big Idea 4 Interactions between systems can result in changes in those systems.
Enduring Understanding 4.C Interactions with other objects or systems can change the total energy of a system.
Essential Knowledge 4.C.1 The energy of a system includes its kinetic energy, potential energy, and microscopic internal energy. Examples should include gravitational potential energy, elastic potential energy, and kinetic energy.
Essential Knowledge 4.C.2 Mechanical energy (the sum of kinetic and potential energy) is transferred into or out of a system when an external force is exerted on a system such that a component of the force is parallel to its displacement. The process through which the energy is transferred is called work.
Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws.
Enduring Understanding 5.B The energy of a system is conserved.
Essential Knowledge 5.B.2 A system with internal structure can have internal energy, and changes in a system’s internal structure can result in changes in internal energy. [Physics 1: includes mass-spring oscillators and simple pendulums. Physics 2: includes charged object in electric ﬁelds and examining changes in internal energy with changes in conﬁguration.]
Big Idea 6 Waves can transfer energy and momentum from one location to another without the permanent transfer of mass and serve as a mathematical model for the description of other phenomena.
Enduring Understanding 6.A A wave is a traveling disturbance that transfers energy and momentum.
Essential Knowledge 6.A.1 Waves can propagate via different oscillation modes such as transverse and longitudinal.
Essential Knowledge 6.A.2 For propagation, mechanical waves require a medium, while electromagnetic waves do not require a physical medium. Examples should include light traveling through a vacuum and sound not traveling through a vacuum.
Essential Knowledge 6.A.3 The amplitude is the maximum displacement of a wave from its equilibrium value.
Essential Knowledge 6.A.4 Classically, the energy carried by a wave depends on and increases with amplitude. Examples should include sound waves.
Enduring Understanding 6.B A periodic wave is one that repeats as a function of both time and position and can be described by its amplitude, frequency, wavelength, speed, and energy.
Essential Knowledge 6.B.1 The period is the repeat time of the wave. The frequency is the number of repetitions over a period of time.
Essential Knowledge 6.B.2 The wavelength is the repeat distance of the wave.
Essential Knowledge 6.B.3 A simple wave can be described by an equation involving one sine or cosine function involving the wavelength, amplitude, and frequency of the wave.
Essential Knowledge 6.B.4 The wavelength is the ratio of speed over frequency.
Enduring Understanding 6.C Only waves exhibit interference and diffraction.
Essential Knowledge 6.C.1 When two waves cross, they travel through each other; they do not bounce off each other. Where the waves overlap, the resulting displacement can be determined by adding the displacements of the two waves. This is called superposition.
Enduring Understanding 6.D Interference and superposition lead to standing waves and beats.
Essential Knowledge 6.D.1 Two or more wave pulses can interact in such a way as to produce amplitude variations in the resultant wave. When two pulses cross, they travel through each other; they do not bounce off each other. Where the pulses overlap, the resulting displacement can be determined by adding the displacements of the two pulses. This is called superposition.
Essential Knowledge 6.D.2 Two or more traveling waves can interact in such a way as to produce amplitude variations in the resultant wave.
Essential Knowledge 6.D.3 Standing waves are the result of the addition of incident and reﬂected waves that are conﬁned to a region and have nodes and antinodes. Examples should include waves on a ﬁxed length of string, and sound waves in both closed and open tubes.
Essential Knowledge 6.D.4 The possible wavelengths of a standing wave are determined by the size of the region to which it is conﬁned.
Essential Knowledge 6.D.5 Beats arise from the addition of waves of slightly different frequency.


# Oscillatory Motion and Waves

## Hooke’s Law: Stress and Strain Revisited

### Learning Objectives

By the end of this section, you will be able to:
1. Explain Newton’s third law of motion with respect to stress and deformation.
2. Describe the restoration of force and displacement.
3. Calculate the energy in Hooke’s Law of deformation, and the stored energy in a spring.
Newton’s first law implies that an object oscillating back and forth is experiencing forces. Without force, the object would move in a straight line at a constant speed rather than oscillate. Consider, for example, plucking a plastic ruler to the left as shown in . The deformation of the ruler creates a force in the opposite direction, known as a restoring force. Once released, the restoring force causes the ruler to move back toward its stable equilibrium position, where the net force on it is zero. However, by the time the ruler gets there, it gains momentum and continues to move to the right, producing the opposite deformation. It is then forced to the left, back through equilibrium, and the process is repeated until dissipative forces dampen the motion. These forces remove mechanical energy from the system, gradually reducing the motion until the ruler comes to rest.
The simplest oscillations occur when the restoring force is directly proportional to displacement. When stress and strain were covered in Newton’s Third Law of Motion, the name was given to this relationship between force and displacement was Hooke’s law:
Here,  is the restoring force,  is the displacement from equilibrium or deformation, and  is a constant related to the difficulty in deforming the system. The minus sign indicates the restoring force is in the direction opposite to the displacement.
The force constant  is related to the rigidity (or stiffness) of a system—the larger the force constant, the greater the restoring force, and the stiffer the system. The units of  are newtons per meter (N/m). For example,  is directly related to Young’s modulus when we stretch a string.  shows a graph of the absolute value of the restoring force versus the displacement for a system that can be described by Hooke’s law—a simple spring in this case. The slope of the graph equals the force constant  in newtons per meter. A common physics laboratory exercise is to measure restoring forces created by springs, determine if they follow Hooke’s law, and calculate their force constants if they do.

### Energy in Hooke’s Law of Deformation

In order to produce a deformation, work must be done. That is, a force must be exerted through a distance, whether you pluck a guitar string or compress a car spring. If the only result is deformation, and no work goes into thermal, sound, or kinetic energy, then all the work is initially stored in the deformed object as some form of potential energy. The potential energy stored in a spring is . Here, we generalize the idea to elastic potential energy for a deformation of any system that can be described by Hooke’s law. Hence,
where  is the elastic potential energy stored in any deformed system that obeys Hooke’s law and has a displacement  from equilibrium and a force constant .
It is possible to find the work done in deforming a system in order to find the energy stored. This work is performed by an applied force . The applied force is exactly opposite to the restoring force (action-reaction), and so .  shows a graph of the applied force versus deformation  for a system that can be described by Hooke’s law. Work done on the system is force multiplied by distance, which equals the area under the curve or (Method A in the figure). Another way to determine the work is to note that the force increases linearly from 0 to , so that the average force is , the distance moved is , and thus  (Method B in the figure).

### Test Prep for AP Courses

### Section Summary

1. An oscillation is a back and forth motion of an object between two points of deformation.
2. An oscillation may create a wave, which is a disturbance that propagates from where it was created.
3. The simplest type of oscillations and waves are related to systems that can be described by Hooke’s law:

    where

4. Elastic potential energy  stored in the deformation of a system that can be described by Hooke’s law is given by

### Conceptual Questions

### Problems & Exercises


# Oscillatory Motion and Waves

## Period and Frequency in Oscillations

### Learning Objectives

By the end of this section, you will be able to:
1. Observe the vibrations of a guitar string.
2. Determine the frequency of oscillations.
When you pluck a guitar string, the resulting sound has a steady tone and lasts a long time. Each successive vibration of the string takes the same time as the previous one. We define periodic motion to be a motion that repeats itself at regular time intervals, such as exhibited by the guitar string or by an object on a spring moving up and down. The time to complete one oscillation remains constant and is called the period . Its units are usually seconds, but may be any convenient unit of time. The word period refers to the time for some event whether repetitive or not; but we shall be primarily interested in periodic motion, which is by definition repetitive. A concept closely related to period is the frequency of an event. For example, if you get a paycheck twice a month, the frequency of payment is two per month and the period between checks is half a month. Frequency  is defined to be the number of events per unit time. For periodic motion, frequency is the number of oscillations per unit time. The relationship between frequency and period is
The SI unit for frequency is the cycle per second, which is defined to be a hertz (Hz):
A cycle is one complete oscillation. Note that a vibration can be a single or multiple event, whereas oscillations are usually repetitive for a significant number of cycles.

### Test Prep for AP Courses

### Section Summary

1. Periodic motion is a repetitious oscillation.
2. The time for one oscillation is the period .
3. The number of oscillations per unit time is the frequency .
4. These quantities are related by

### Problems & Exercises


# Oscillatory Motion and Waves

## Simple Harmonic Motion: A Special Periodic Motion

### Learning Objectives

By the end of this section, you will be able to:
1. Describe a simple harmonic oscillator.
2. Explain the link between simple harmonic motion and waves.
The oscillations of a system in which the net force can be described by Hooke’s law are of special importance, because they are very common. They are also the simplest oscillatory systems. Simple Harmonic Motion (SHM) is the name given to oscillatory motion for a system where the net force can be described by Hooke’s law, and such a system is called a simple harmonic oscillator. If the net force can be described by Hooke’s law and there is no  damping (by friction or other non-conservative forces), then a simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position, as shown for an object on a spring in . The maximum displacement from equilibrium is called the amplitude . The units for amplitude and displacement are the same, but depend on the type of oscillation. For the object on the spring, the units of amplitude and displacement are meters; whereas for sound oscillations, they have units of pressure (and other types of oscillations have yet other units). Because amplitude is the maximum displacement, it is related to the energy in the oscillation.

What is so significant about simple harmonic motion? One special thing is that the period  and frequency  of a simple harmonic oscillator are independent of amplitude. The string of a guitar, for example, will oscillate with the same frequency whether plucked gently or hard. Because the period is constant, a simple harmonic oscillator can be used as a clock.
Two important factors do affect the period of a simple harmonic oscillator. The period is related to how stiff the system is. A very stiff object has a large force constant , which causes the system to have a smaller period. For example, you can adjust a diving board’s stiffness—the stiffer it is, the faster it vibrates, and the shorter its period. Period also depends on the mass of the oscillating system. The more massive the system is, the longer the period. For example, a heavy person on a diving board bounces up and down more slowly than a light one.
In fact, the mass  and the force constant  are the only factors that affect the period and frequency of simple harmonic motion.

### The Link between Simple Harmonic Motion and Waves

If a time-exposure photograph of the bouncing car were taken as it drove by, the headlight would make a wavelike streak, as shown in . Similarly,  shows an object bouncing on a spring as it leaves a wavelike "trace" of its position on a moving strip of paper. Both waves are sine functions. All simple harmonic motion is intimately related to sine and cosine waves.
The displacement as a function of time t in any simple harmonic motion—that is, one in which the net restoring force can be described by Hooke’s law, is given by
where  is amplitude. At , the initial position is , and the displacement oscillates back and forth with a period . (When , we get  again because .). Furthermore, from this expression for , the velocity  as a function of time is given by:
where . The object has zero velocity at maximum displacement—for example,  when , and at that time . The minus sign in the first equation for  gives the correct direction for the velocity. Just after the start of the motion, for instance, the velocity is negative because the system is moving back toward the equilibrium point. Finally, we can get an expression for acceleration using Newton’s second law. [Then we have  and , the quantities needed for kinematics and a description of simple harmonic motion.] According to Newton’s second law, the acceleration is . So,  is also a cosine function:
Hence,  is directly proportional to and in the opposite direction to .
 shows the simple harmonic motion of an object on a spring and presents graphs of  and  versus time.
The most important point here is that these equations are mathematically straightforward and are valid for all simple harmonic motion. They are very useful in visualizing waves associated with simple harmonic motion, including visualizing how waves add with one another.

### Test Prep for AP Courses

### Section Summary

1. Simple harmonic motion is oscillatory motion for a system that can be described only by Hooke’s law. Such a system is also called a simple harmonic oscillator.
2. Maximum displacement is the amplitude . The period  and frequency  of a simple harmonic oscillator are given by

3. Displacement in simple harmonic motion as a function of time is given by

4. The velocity is given by

, where

.
5. The acceleration is found to be

### Conceptual Questions

### Problems & Exercises


# Oscillatory Motion and Waves

## The Simple Pendulum

### Learning Objectives

By the end of this section, you will be able to:
1. Measure acceleration due to gravity.
Pendulums are in common usage. Some have crucial uses, such as in clocks; some are for fun, such as a child’s swing; and some are just there, such as the sinker on a fishing line. For small displacements, a pendulum is a simple harmonic oscillator. A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in . Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period.
We begin by defining the displacement to be the arc length . We see from  that the net force on the bob is tangent to the arc and equals . (The weight  has components  along the string and  tangent to the arc.) Tension in the string exactly cancels the component  parallel to the string. This leaves a net restoring force back toward the equilibrium position at .
Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about ), ( and  differ by about 1% or less at smaller angles). Thus, for angles less than about , the restoring force  is
The displacement  is directly proportional to . When  is expressed in radians, the arc length in a circle is related to its radius ( in this instance) by:
so that
For small angles, then, the expression for the restoring force is:
This expression is of the form:
where the force constant is given by  and the displacement is given by . For angles less than about , the restoring force is directly proportional to the displacement, and the simple pendulum is a simple harmonic oscillator.
Using this equation, we can find the period of a pendulum for amplitudes less than about . For the simple pendulum:
Thus,
for the period of a simple pendulum. This result is interesting because of its simplicity. The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass. As with simple harmonic oscillators, the period  for a pendulum is nearly independent of amplitude, especially if  is less than about . Even simple pendulum clocks can be finely adjusted and accurate.
Note the dependence of  on . If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity. Consider the following example.

### Test Prep for AP Courses

### Section Summary

1. A mass  suspended by a wire of length  is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about
    The period of a simple pendulum is
    where

### Conceptual Questions

### Problems & Exercises

As usual, the acceleration due to gravity in these problems is taken to be , unless otherwise specified.


# Oscillatory Motion and Waves

## Energy and the Simple Harmonic Oscillator

### Learning Objectives

By the end of this section, you will be able to:
1. Determine the maximum speed of an oscillating system.
To study the energy of a simple harmonic oscillator, we first consider all the forms of energy it can have We know from Hooke’s Law: Stress and Strain Revisited that the energy stored in the deformation of a simple harmonic oscillator is a form of potential energy given by:
Because a simple harmonic oscillator has no dissipative forces, the other important form of energy is kinetic energy . Conservation of energy for these two forms is:
or
This statement of conservation of energy is valid for all simple harmonic oscillators, including ones where the gravitational force plays a role
Namely, for a simple pendulum we replace the velocity with , the spring constant with , and the displacement term with . Thus
In the case of undamped simple harmonic motion, the energy oscillates back and forth between kinetic and potential, going completely from one to the other as the system oscillates. So for the simple example of an object on a frictionless surface attached to a spring, as shown again in , the motion starts with all of the energy stored in the spring. As the object starts to move, the elastic potential energy is converted to kinetic energy, becoming entirely kinetic energy at the equilibrium position. It is then converted back into elastic potential energy by the spring, the velocity becomes zero when the kinetic energy is completely converted, and so on. This concept provides extra insight here and in later applications of simple harmonic motion, such as alternating current circuits.

The conservation of energy principle can be used to derive an expression for velocity . If we start our simple harmonic motion with zero velocity and maximum displacement (), then the total energy is
This total energy is constant and is shifted back and forth between kinetic energy and potential energy, at most times being shared by each. The conservation of energy for this system in equation form is thus:
Solving this equation for  yields:
Manipulating this expression algebraically gives:
and so
where
From this expression, we see that the velocity is a maximum () at , as stated earlier in . Notice that the maximum velocity depends on three factors. Maximum velocity is directly proportional to amplitude. As you might guess, the greater the maximum displacement the greater the maximum velocity. Maximum velocity is also greater for stiffer systems, because they exert greater force for the same displacement. This observation is seen in the expression for  it is proportional to the square root of the force constant . Finally, the maximum velocity is smaller for objects that have larger masses, because the maximum velocity is inversely proportional to the square root of . For a given force, objects that have large masses accelerate more slowly.
A similar calculation for the simple pendulum produces a similar result, namely:

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### Section Summary

1. Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:

2. Maximum velocity depends on three factors: it is directly proportional to amplitude, it is greater for stiffer systems, and it is smaller for objects that have larger masses:

### Conceptual Questions

### Problems & Exercises


# Oscillatory Motion and Waves

## Uniform Circular Motion and Simple Harmonic Motion

### Learning Objectives

By the end of this section, you will be able to:
1. Compare simple harmonic motion with uniform circular motion.

There is an easy way to produce simple harmonic motion by using uniform circular motion.  shows one way of using this method. A ball is attached to a uniformly rotating vertical turntable, and its shadow is projected on the floor as shown. The shadow undergoes simple harmonic motion. Hooke’s law usually describes uniform circular motions ( constant) rather than systems that have large visible displacements. So observing the projection of uniform circular motion, as in , is often easier than observing a precise large-scale simple harmonic oscillator. If studied in sufficient depth, simple harmonic motion produced in this manner can give considerable insight into many aspects of oscillations and waves and is very useful mathematically. In our brief treatment, we shall indicate some of the major features of this relationship and how they might be useful.
 shows the basic relationship between uniform circular motion and simple harmonic motion. The point P travels around the circle at constant angular velocity . The point P is analogous to an object on the merry-go-round. The projection of the position of P onto a fixed axis undergoes simple harmonic motion and is analogous to the shadow of the object. At the time shown in the figure, the projection has position  and moves to the left with velocity . The velocity of the point P around the circle equals .The projection of  on the -axis is the velocity  of the simple harmonic motion along the -axis.
To see that the projection undergoes simple harmonic motion, note that its position  is given by
where ,  is the constant angular velocity, and  is the radius of the circular path. Thus,
The angular velocity  is in radians per unit time; in this case  radians is the time for one revolution . That is, . Substituting this expression for , we see that the position  is given by:
This expression is the same one we had for the position of a simple harmonic oscillator in Simple Harmonic Motion: A Special Periodic Motion. If we make a graph of position versus time as in , we see again the wavelike character (typical of simple harmonic motion) of the projection of uniform circular motion onto the -axis.
Now let us use  to do some further analysis of uniform circular motion as it relates to simple harmonic motion. The triangle formed by the velocities in the figure and the triangle formed by the displacements ( and ) are similar right triangles. Taking ratios of similar sides, we see that
We can solve this equation for the speed  or
This expression for the speed of a simple harmonic oscillator is exactly the same as the equation obtained from conservation of energy considerations in Energy and the Simple Harmonic Oscillator.You can begin to see that it is possible to get all of the characteristics of simple harmonic motion from an analysis of the projection of uniform circular motion.
Finally, let us consider the period  of the motion of the projection. This period is the time it takes the point P to complete one revolution. That time is the circumference of the circle  divided by the velocity around the circle, . Thus, the period  is
We know from conservation of energy considerations that
Solving this equation for  gives
Substituting this expression into the equation for  yields
Thus, the period of the motion is the same as for a simple harmonic oscillator. We have determined the period for any simple harmonic oscillator using the relationship between uniform circular motion and simple harmonic motion.
Some modules occasionally refer to the connection between uniform circular motion and simple harmonic motion. Moreover, if you carry your study of physics and its applications to greater depths, you will find this relationship useful. It can, for example, help to analyze how waves add when they are superimposed.

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### Section Summary

A projection of uniform circular motion undergoes simple harmonic oscillation.

### Problems & Exercises


# Oscillatory Motion and Waves

## Damped Harmonic Motion

### Learning Objectives

By the end of this section, you will be able to:
1. Compare and discuss underdamped and overdamped oscillating systems.
2. Explain critically damped system.
A guitar string stops oscillating a few seconds after being plucked. To keep a child happy on a swing, you must keep pushing. Although we can often make friction and other non-conservative forces negligibly small, completely undamped motion is rare. In fact, we may even want to damp oscillations, such as with car shock absorbers.
For a system that has a small amount of damping, the period and frequency are nearly the same as for simple harmonic motion, but the amplitude gradually decreases as shown in . This occurs because the non-conservative damping force removes energy from the system, usually in the form of thermal energy. In general, energy removal by non-conservative forces is described as
where  is work done by a non-conservative force (here the damping force). For a damped harmonic oscillator,  is negative because it removes mechanical energy (KE + PE) from the system.
If you gradually increase the amount of damping in a system, the period and frequency begin to be affected, because damping opposes and hence slows the back and forth motion. (The net force is smaller in both directions.) If there is very large damping, the system does not even oscillate—it slowly moves toward equilibrium.  shows the displacement of a harmonic oscillator for different amounts of damping. When we want to damp out oscillations, such as in the suspension of a car, we may want the system to return to equilibrium as quickly as possible Critical damping is defined as the condition in which the damping of an oscillator results in it returning as quickly as possible to its equilibrium position The critically damped system may overshoot the equilibrium position, but if it does, it will do so only once. Critical damping is represented by Curve A in . With less-than critical damping, the system will return to equilibrium faster but will overshoot and cross over one or more times. Such a system is underdamped; its displacement is represented by the curve in . Curve B in  represents an overdamped system. As with critical damping, it too may overshoot the equilibrium position, but will reach equilibrium over a longer period of time.
Critical damping is often desired, because such a system returns to equilibrium rapidly and remains at equilibrium as well. In addition, a constant force applied to a critically damped system moves the system to a new equilibrium position in the shortest time possible without overshooting or oscillating about the new position. For example, when you stand on bathroom scales that have a needle gauge, the needle moves to its equilibrium position without oscillating. It would be quite inconvenient if the needle oscillated about the new equilibrium position for a long time before settling. Damping forces can vary greatly in character. Friction, for example, is sometimes independent of velocity (as assumed in most places in this text). But many damping forces depend on velocity—sometimes in complex ways, sometimes simply being proportional to velocity.

### Test Prep for AP Courses

### Section Summary

1. Damped harmonic oscillators have non-conservative forces that dissipate their energy.
2. Critical damping returns the system to equilibrium as fast as possible without overshooting.
3. An underdamped system will oscillate through the equilibrium position.
4. An overdamped system moves more slowly toward equilibrium than one that is critically damped.

### Conceptual Questions

### Problems & Exercises


# Oscillatory Motion and Waves

## Forced Oscillations and Resonance

### Learning Objectives

By the end of this section, you will be able to:
1. Observe resonance of a paddle ball on a string.
2. Observe amplitude of a damped harmonic oscillator.
Sit in front of a piano sometime and sing a loud brief note at it with the dampers off its strings. It will sing the same note back at you—the strings, having the same frequencies as your voice, are resonating in response to the forces from the sound waves that you sent to them. Your voice and a piano’s strings is a good example of the fact that objects—in this case, piano strings—can be forced to oscillate but oscillate best at their natural frequency. In this section, we shall briefly explore applying a periodic driving force acting on a simple harmonic oscillator. The driving force puts energy into the system at a certain frequency, not necessarily the same as the natural frequency of the system. The natural frequency is the frequency at which a system would oscillate if there were no driving and no damping force.
Most of us have played with toys involving an object supported on an elastic band, something like the paddle ball suspended from a finger in . Imagine the finger in the figure is your finger. At first you hold your finger steady, and the ball bounces up and down with a small amount of damping. If you move your finger up and down slowly, the ball will follow along without bouncing much on its own. As you increase the frequency at which you move your finger up and down, the ball will respond by oscillating with increasing amplitude. When you drive the ball at its natural frequency, the ball’s oscillations increase in amplitude with each oscillation for as long as you drive it. The phenomenon of driving a system with a frequency equal to its natural frequency is called resonance. A system being driven at its natural frequency is said to resonate. As the driving frequency gets progressively higher than the resonant or natural frequency, the amplitude of the oscillations becomes smaller, until the oscillations nearly disappear and your finger simply moves up and down with little effect on the ball.
 shows a graph of the amplitude of a damped harmonic oscillator as a function of the frequency of the periodic force driving it. There are three curves on the graph, each representing a different amount of damping. All three curves peak at the point where the frequency of the driving force equals the natural frequency of the harmonic oscillator. The highest peak, or greatest response, is for the least amount of damping, because less energy is removed by the damping force.
It is interesting that the widths of the resonance curves shown in  depend on damping: the less the damping, the narrower the resonance. The message is that if you want a driven oscillator to resonate at a very specific frequency, you need as little damping as possible. Little damping is the case for piano strings and many other musical instruments. Conversely, if you want small-amplitude oscillations, such as in a car’s suspension system, then you want heavy damping. Heavy damping reduces the amplitude, but the tradeoff is that the system responds at more frequencies.
These features of driven harmonic oscillators apply to a huge variety of systems. When you tune a radio, for example, you are adjusting its resonant frequency so that it only oscillates to the desired station’s broadcast (driving) frequency. The more selective the radio is in discriminating between stations, the smaller its damping. Magnetic resonance imaging (MRI) is a widely used medical diagnostic tool in which atomic nuclei (mostly hydrogen nuclei) are made to resonate by incoming radio waves (on the order of 100 MHz). A child on a swing is driven by a parent at the swing’s natural frequency to achieve maximum amplitude. In all of these cases, the efficiency of energy transfer from the driving force into the oscillator is best at resonance. Speed bumps and gravel roads prove that even a car’s suspension system is not immune to resonance. In spite of finely engineered shock absorbers, which ordinarily convert mechanical energy to thermal energy almost as fast as it comes in, speed bumps still cause a large-amplitude oscillation. On gravel roads that are corrugated, you may have noticed that if you travel at the “wrong” speed, the bumps are very noticeable whereas at other speeds you may hardly feel the bumps at all.  shows a photograph of a famous example (the Tacoma Narrows Bridge) of the destructive effects of a driven harmonic oscillation. The Millennium Bridge in London was closed for a short period of time for the same reason while inspections were carried out.
In our bodies, the chest cavity is a clear example of a system at resonance. The diaphragm and chest wall drive the oscillations of the chest cavity which result in the lungs inflating and deflating. The system is critically damped and the muscular diaphragm oscillates at the resonant value for the system, making it highly efficient.

### Test Prep for AP Courses

### Section Summary

1. A system’s natural frequency is the frequency at which the system will oscillate if not affected by driving or damping forces.
2. A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate.
3. The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies.

### Conceptual Questions

### Problems & Exercises


# Oscillatory Motion and Waves

## Waves

### Learning Objectives

By the end of this section, you will be able to:
1. State the characteristics of a wave.
2. Calculate the velocity of wave propagation.
What do we mean when we say something is a wave? The most intuitive and easiest wave to imagine is the familiar water wave. More precisely, a wave is a disturbance that propagates, or moves from the place it was created. For water waves, the disturbance is in the surface of the water, perhaps created by a rock thrown into a pond or by a swimmer splashing the surface repeatedly. For sound waves, the disturbance is a change in air pressure, perhaps created by the oscillating cone inside a speaker. For earthquakes, there are several types of disturbances, including disturbance of Earth’s surface and pressure disturbances under the surface. Even radio waves are most easily understood using an analogy with water waves. Visualizing water waves is useful because there is more to it than just a mental image. Water waves exhibit characteristics common to all waves, such as amplitude, period, frequency and energy. All wave characteristics can be described by a small set of underlying principles.
A wave is a disturbance that propagates, or moves from the place it was created. The simplest waves repeat themselves for several cycles and are associated with simple harmonic motion. Let us start by considering the simplified water wave in . The wave is an up and down disturbance of the water surface. It causes a sea gull to move up and down in simple harmonic motion as the wave crests and troughs (peaks and valleys) pass under the bird. The time for one complete up and down motion is the wave’s period . The wave’s frequency is , as usual. The wave itself moves to the right in the figure. This movement of the wave is actually the disturbance moving to the right, not the water itself (or the bird would move to the right). We define wave velocity  to be the speed at which the disturbance moves. Wave velocity is sometimes also called the propagation velocity or propagation speed,  because the disturbance propagates from one location to another.
The water wave in the figure also has a length associated with it, called its wavelength

  , the distance between adjacent identical parts of a wave. ( is the distance parallel to the direction of propagation.) The speed of propagation  is the distance the wave travels in a given time, which is one wavelength in the time of one period. In equation form, that is
or
This fundamental relationship holds for all types of waves. For water waves,  is the speed of a surface wave; for sound,  is the speed of sound; and for visible light,  is the speed of light, for example.

### Transverse and Longitudinal Waves

A simple wave consists of a periodic disturbance that propagates from one place to another. The wave in  propagates in the horizontal direction while the surface is disturbed in the vertical direction. Such a wave is called a transverse wave or shear wave; in such a wave, the disturbance is perpendicular to the direction of propagation. In contrast, in a longitudinal wave or compressional wave, the disturbance is parallel to the direction of propagation.  shows an example of a longitudinal wave. The size of the disturbance is its amplitude X and is completely independent of the speed of propagation .
Waves may be transverse, longitudinal, or a combination of the two. (Water waves are actually a combination of transverse and longitudinal. The simplified water wave illustrated in
shows no longitudinal motion of the bird.) The waves on the strings of musical instruments are transverse—so are electromagnetic waves, such as visible light.
Sound waves in air and water are longitudinal. Their disturbances are periodic variations in pressure that are transmitted in fluids. Fluids do not have appreciable shear strength, and thus the sound waves in them must be longitudinal or compressional. Sound in solids can be both longitudinal and transverse.
Earthquake waves under Earth’s surface also have both longitudinal and transverse components (called compressional or P-waves and shear or S-waves, respectively). These components have important individual characteristics—they propagate at different speeds, for example. Earthquakes also have surface waves that are similar to surface waves on water.

### Test Prep for AP Courses

### Section Summary

1. A wave is a disturbance that moves from the point of creation with a wave velocity .
2. A wave has a wavelength , which is the distance between adjacent identical parts of the wave.
3. Wave velocity and wavelength are related to the wave’s frequency and period by

           or

4. A transverse wave has a disturbance perpendicular to its direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.

### Conceptual Questions

### Problems & Exercises