College Physics for AP® Courses

# 7.6Conservation of Energy

College Physics for AP® Courses7.6 Conservation of Energy

### Learning Objectives

By the end of this section, you will be able to:

• Explain the law of the conservation of energy.
• Describe some of the many forms of energy.
• Define efficiency of an energy conversion process as the fraction left as useful energy or work, rather than being transformed, for example, into thermal energy.

The information presented in this section supports the following AP® learning objectives and science practices:

• 4.C.1.2 The student is able to predict changes in the total energy of a system due to changes in position and speed of objects or frictional interactions within the system. (S.P. 6.4)
• 4.C.2.1 The student is able to make predictions about the changes in the mechanical energy of a system when a component of an external force acts parallel or antiparallel to the direction of the displacement of the center of mass. (S.P. 6.4)
• 4.C.2.2 The student is able to apply the concepts of conservation of energy and the work-energy theorem to determine qualitatively and/or quantitatively that work done on a two-object system in linear motion will change the kinetic energy of the center of mass of the system, the potential energy of the systems, and/or the internal energy of the system. (S.P. 1.4, 2.2, 7.2)
• 5.A.2.1 The student is able to define open and closed systems for everyday situations and apply conservation concepts for energy, charge, and linear momentum to those situations. (S.P. 6.4, 7.2)
• 5.B.5.4 The student is able to make claims about the interaction between a system and its environment in which the environment exerts a force on the system, thus doing work on the system and changing the energy of the system (kinetic energy plus potential energy). (S.P. 6.4, 7.2)
• 5.B.5.5 The student is able to predict and calculate the energy transfer to (i.e., the work done on) an object or system from information about a force exerted on the object or system through a distance. (S.P. 2.2, 6.4)

### Law of Conservation of Energy

Energy, as we have noted, is conserved, making it one of the most important physical quantities in nature. The law of conservation of energy can be stated as follows:

Total energy is constant in any process. It may change in form or be transferred from one system to another, but the total remains the same.

We have explored some forms of energy and some ways it can be transferred from one system to another. This exploration led to the definition of two major types of energy—mechanical energy $KE+PEKE+PE size 12{ left ("KE"+"PE" right )} {}$ and energy transferred via work done by nonconservative forces $(Wnc)(Wnc) size 12{ $$W rSub { size 8{"nc"} }$$ } {}$. But energy takes many other forms, manifesting itself in many different ways, and we need to be able to deal with all of these before we can write an equation for the above general statement of the conservation of energy.

### Other Forms of Energy than Mechanical Energy

At this point, we deal with all other forms of energy by lumping them into a single group called other energy ($OEOE size 12{"OE"} {}$). Then we can state the conservation of energy in equation form as

$KEi+PEi+Wnc+OEi=KEf+PEf+OEf.KEi+PEi+Wnc+OEi=KEf+PEf+OEf. size 12{"KE" rSub { size 8{i} } +"PE" rSub { size 8{i} } +W rSub { size 8{"nc"} } +"OE" rSub { size 8{i} } ="KE" rSub { size 8{f} } +"PE" rSub { size 8{f} } +"OE" rSub { size 8{f} } } {}$
7.65

All types of energy and work can be included in this very general statement of conservation of energy. Kinetic energy is $KEKE size 12{"KE"} {}$, work done by a conservative force is represented by $PEPE size 12{"PE"} {}$, work done by nonconservative forces is $WncWnc size 12{W rSub { size 8{"nc"} } } {}$, and all other energies are included as $OEOE size 12{"OE"} {}$. This equation applies to all previous examples; in those situations $OEOE size 12{"OE"} {}$ was constant, and so it subtracted out and was not directly considered.

### Making Connections: Usefulness of the Energy Conservation Principle

The fact that energy is conserved and has many forms makes it very important. You will find that energy is discussed in many contexts, because it is involved in all processes. It will also become apparent that many situations are best understood in terms of energy and that problems are often most easily conceptualized and solved by considering energy.

When does $OEOE size 12{"OE"} {}$ play a role? One example occurs when a person eats. Food is oxidized with the release of carbon dioxide, water, and energy. Some of this chemical energy is converted to kinetic energy when the person moves, to potential energy when the person changes altitude, and to thermal energy (another form of $OEOE size 12{"OE"} {}$).

### Some of the Many Forms of Energy

What are some other forms of energy? You can probably name a number of forms of energy not yet discussed. Many of these will be covered in later chapters, but let us detail a few here. Electrical energy is a common form that is converted to many other forms and does work in a wide range of practical situations. Fuels, such as gasoline and food, carry chemical energy that can be transferred to a system through oxidation. Chemical fuel can also produce electrical energy, such as in batteries. Batteries can in turn produce light, which is a very pure form of energy. Most energy sources on Earth are in fact stored energy from the energy we receive from the Sun. We sometimes refer to this as radiant energy, or electromagnetic radiation, which includes visible light, infrared, and ultraviolet radiation. Nuclear energy comes from processes that convert measurable amounts of mass into energy. Nuclear energy is transformed into the energy of sunlight, into electrical energy in power plants, and into the energy of the heat transfer and blast in weapons. Atoms and molecules inside all objects are in random motion. This internal mechanical energy from the random motions is called thermal energy, because it is related to the temperature of the object. These and all other forms of energy can be converted into one another and can do work.

### Real World Connections: Open or Closed System?

Consider whether the following systems are open or closed: a car, a spring-operated dart gun, and the system shown in Figure 7.15(a).

A car is not a closed system. You add energy in the form of more gas in the tank (or charging the batteries), and energy is lost due to air resistance and friction.

A spring-operated dart gun is not a closed system. You have to initially compress the spring. Once that has been done, however, the dart gun and dart can be treated as a closed system. All of the energy remains in the system consisting of these two objects.

Figure 7.15(a) is an example of a closed system, once it has been started. All of the energy in the system remains there; none is brought in from outside or leaves.

Table 7.1 gives the amount of energy stored, used, or released from various objects and in various phenomena. The range of energies and the variety of types and situations is impressive.

### Problem-Solving Strategies for Energy

You will find the following problem-solving strategies useful whenever you deal with energy. The strategies help in organizing and reinforcing energy concepts. In fact, they are used in the examples presented in this chapter. The familiar general problem-solving strategies presented earlier—involving identifying physical principles, knowns, and unknowns, checking units, and so on—continue to be relevant here.

Step 1. Determine the system of interest and identify what information is given and what quantity is to be calculated. A sketch will help.

Step 2. Examine all the forces involved and determine whether you know or are given the potential energy from the work done by the forces. Then use step 3 or step 4.

Step 3. If you know the potential energies for the forces that enter into the problem, then forces are all conservative, and you can apply conservation of mechanical energy simply in terms of potential and kinetic energy. The equation expressing conservation of energy is

$KEi+PEi=KEf+PEf.KEi+PEi=KEf+PEf. size 12{"KE" rSub { size 8{i} } +"PE" rSub { size 8{i} } ="KE" rSub { size 8{f} } +"PE" rSub { size 8{f} } } {}$
7.66

Step 4. If you know the potential energy for only some of the forces, possibly because some of them are nonconservative and do not have a potential energy, or if there are other energies that are not easily treated in terms of force and work, then the conservation of energy law in its most general form must be used.

$KEi+PEi+Wnc+OEi=KEf+PEf+OEf.KEi+PEi+Wnc+OEi=KEf+PEf+OEf. size 12{"KE" rSub { size 8{i} } +"PE" rSub { size 8{i} } +W rSub { size 8{"nc"} } +"OE" rSub { size 8{i} } ="KE" rSub { size 8{f} } +"PE" rSub { size 8{f} } +"OE" rSub { size 8{f} } } {}$
7.67

In most problems, one or more of the terms is zero, simplifying its solution. Do not calculate $WcWc size 12{W rSub { size 8{c} } } {}$, the work done by conservative forces; it is already incorporated in the $PEPE size 12{"PE"} {}$ terms.

Step 5. You have already identified the types of work and energy involved (in step 2). Before solving for the unknown, eliminate terms wherever possible to simplify the algebra. For example, choose $h=0h=0 size 12{h=0} {}$ at either the initial or final point, so that $PEgPEg size 12{"PE" rSub { size 8{g} } } {}$ is zero there. Then solve for the unknown in the customary manner.

Step 6. Check the answer to see if it is reasonable. Once you have solved a problem, reexamine the forms of work and energy to see if you have set up the conservation of energy equation correctly. For example, work done against friction should be negative, potential energy at the bottom of a hill should be less than that at the top, and so on. Also check to see that the numerical value obtained is reasonable. For example, the final speed of a skateboarder who coasts down a 3-m-high ramp could reasonably be 20 km/h, but not 80 km/h.

### Transformation of Energy

The transformation of energy from one form into others is happening all the time. The chemical energy in food is converted into thermal energy through metabolism; light energy is converted into chemical energy through photosynthesis. In a larger example, the chemical energy contained in coal is converted into thermal energy as it burns to turn water into steam in a boiler. This thermal energy in the steam in turn is converted to mechanical energy as it spins a turbine, which is connected to a generator to produce electrical energy. (In all of these examples, not all of the initial energy is converted into the forms mentioned. This important point is discussed later in this section.)

Another example of energy conversion occurs in a solar cell. Sunlight impinging on a solar cell (see Figure 7.20) produces electricity, which in turn can be used to run an electric motor. Energy is converted from the primary source of solar energy into electrical energy and then into mechanical energy.

Figure 7.20 Solar energy is converted into electrical energy by solar cells, which is used to run a motor in this solar-power aircraft. (credit: NASA)
Object/phenomenon Energy in joules
Big Bang $10 68 10 68 size 12{"10" rSup { size 8{"68"} } } {}$
Energy released in a supernova $10 44 10 44 size 12{"10" rSup { size 8{"44"} } } {}$
Fusion of all the hydrogen in Earth’s oceans $10 34 10 34 size 12{"10" rSup { size 8{"34"} } } {}$
Annual world energy use $4 × 10 20 4 × 10 20 size 12{4 times "10" rSup { size 8{"20"} } } {}$
Large fusion bomb (9 megaton) $3 . 8 × 10 16 3 . 8 × 10 16 size 12{3 "." 8 times "10" rSup { size 8{"16"} } } {}$
1 kg hydrogen (fusion to helium) $6 . 4 × 10 14 6 . 4 × 10 14 size 12{6 "." 4 times "10" rSup { size 8{"14"} } } {}$
1 kg uranium (nuclear fission) $8 . 0 × 10 13 8 . 0 × 10 13 size 12{8 "." 0 times "10" rSup { size 8{"13"} } } {}$
Hiroshima-size fission bomb (10 kiloton) $4 . 2 × 10 13 4 . 2 × 10 13 size 12{4 "." 2 times "10" rSup { size 8{"13"} } } {}$
90,000-ton aircraft carrier at 30 knots $1 . 1 × 10 10 1 . 1 × 10 10 size 12{1 "." 1 times "10" rSup { size 8{"10"} } } {}$
1 barrel crude oil $5 . 9 × 10 9 5 . 9 × 10 9 size 12{5 "." 9 times "10" rSup { size 8{9} } } {}$
1 ton TNT $4 . 2 × 10 9 4 . 2 × 10 9 size 12{4 "." 2 times "10" rSup { size 8{9} } } {}$
1 gallon of gasoline $1 . 2 × 10 8 1 . 2 × 10 8 size 12{1 "." 2 times "10" rSup { size 8{8} } } {}$
Daily home electricity use (developed countries) $7 × 10 7 7 × 10 7 size 12{7 times "10" rSup { size 8{7} } } {}$
Daily adult food intake (recommended) $1 . 2 × 10 7 1 . 2 × 10 7 size 12{1 "." 2 times "10" rSup { size 8{7} } } {}$
1000-kg car at 90 km/h $3 . 1 × 10 5 3 . 1 × 10 5 size 12{3 "." 1 times "10" rSup { size 8{5} } } {}$
1 g fat (9.3 kcal) $3 . 9 × 10 4 3 . 9 × 10 4 size 12{3 "." 9 times "10" rSup { size 8{4} } } {}$
ATP hydrolysis reaction $3 . 2 × 10 4 3 . 2 × 10 4 size 12{3 "." 2 times "10" rSup { size 8{4} } } {}$
1 g carbohydrate (4.1 kcal) $1 . 7 × 10 4 1 . 7 × 10 4 size 12{1 "." 7 times "10" rSup { size 8{4} } } {}$
1 g protein (4.1 kcal) $1 . 7 × 10 4 1 . 7 × 10 4 size 12{1 "." 7 times "10" rSup { size 8{4} } } {}$
Tennis ball at 100 km/h $22 22$
Mosquito $(10–2g at 0.5 m/s)(10–2g at 0.5 m/s)$ $1 . 3 × 10 − 6 1 . 3 × 10 − 6 size 12{1 "." 3 times "10" rSup { size 8{-6} } } {}$
Single electron in a TV tube beam $4 . 0 × 10 − 15 4 . 0 × 10 − 15 size 12{4 "." 0 times "10" rSup { size 8{-"15"} } } {}$
Energy to break one DNA strand $10 − 19 10 − 19 size 12{"10" rSup { size 8{-"19"} } } {}$
Table 7.1 Energy of Various Objects and Phenomena

### Efficiency

Even though energy is conserved in an energy conversion process, the output of useful energy or work will be less than the energy input. The efficiency $EffEff size 12{ ital "Eff"} {}$ of an energy conversion process is defined as

$Efficiency ( Eff ) = useful energy or work output total energy input = W out E in . Efficiency ( Eff ) = useful energy or work output total energy input = W out E in . size 12{"Efficiency " $$ital "Eff"$$ = { {"useful energy or work output"} over {"total energy input"} } = { {W rSub { size 8{"out"} } } over {E rSub { size 8{"in"} } } } "." } {}$
7.68

Table 7.2 lists some efficiencies of mechanical devices and human activities. In a coal-fired power plant, for example, about 40% of the chemical energy in the coal becomes useful electrical energy. The other 60% transforms into other (perhaps less useful) energy forms, such as thermal energy, which is then released to the environment through combustion gases and cooling towers.

Activity/device Efficiency (%)1
Cycling and climbing 20
Swimming, surface 2
Swimming, submerged 4
Shoveling 3
Weightlifting 9
Steam engine 17
Gasoline engine 30
Diesel engine 35
Nuclear power plant 35
Coal power plant 42
Electric motor 98
Compact fluorescent light 20
Gas heater (residential) 90
Solar cell 10
Table 7.2 Efficiency of the Human Body and Mechanical Devices
Masses and Springs

A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energies for each spring.

Figure 7.21

### Footnotes

• 1Representative values
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