# 54 7.6 Conservation of Energy

### Summary

• 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.

# 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 + PE) and energy transferred via work done by nonconservative forces (Wnc). 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 (OE). Then we can state the conservation of energy in equation form as

$\boldsymbol{\textbf{KE}_{\textbf{i}}+\textbf{PE}_{\textbf{i}}+W_{\textbf{nc}}+\textbf{OE}_{\textbf{i}}=\textbf{KE}_{\textbf{f}}+\textbf{PE}_{\textbf{f}}+\textbf{OE}_{\textbf{f}}}.$

All types of energy and work can be included in this very general statement of conservation of energy. Kinetic energy is KE, work done by a conservative force is represented by PE, work done by nonconservative forces is Wnc, and all other energies are included as OE. This equation applies to all previous examples; in those situations 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 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 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.

Table 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.

# 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 Eff of an energy conversion process is defined as

$\boldsymbol{\textbf{Efficiency}(Eff)\:=}\boldsymbol{\frac{\textbf{useful energy or work output}}{\textbf{total energy input}}}\boldsymbol{=}\boldsymbol{\frac{W_{\textbf{out}}}{E_{\textbf{in}}}}.$

Table 2 lists some efficiencies of mechanical devices and human activities.

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 2. Efficiency of the Human Body and Mechanical Devices.

# Section Summary

• The law of conservation of energy states that the total energy is constant in any process. Energy may change in form or be transferred from one system to another, but the total remains the same.
• When all forms of energy are considered, conservation of energy is written in equation form as KEi + PEi + Wnc + OEi = KEf + PEf + OEf, where OE is all other forms of energy besides mechanical energy.
• Commonly encountered forms of energy include electric energy, chemical energy, radiant energy, nuclear energy, and thermal energy.
• Energy is often utilized to do work, but it is not possible to convert all the energy of a system to work.
• The efficiency Eff of a machine or human is defined to be $\boldsymbol{Eff=\frac{W_{\textbf{out}}}{E_{\textbf{in}}}},$ where Wout is useful work output and Ein is the energy consumed.

### Conceptual Questions

1: Describe the energy transfers and transformations for a javelin, starting from the point at which an athlete picks up the javelin and ending when the javelin is stuck into the ground after being thrown.

2: List the energy conversions that occur when riding a bicycle.

### Problems & Exercises

1: Using energy considerations and assuming negligible air resistance, show that a rock thrown from a bridge 20.0 m above water with an initial speed of 15.0 m/s strikes the water with a speed of 24.8 m/s independent of the direction thrown.

## Footnotes

1. 1 Representative values

## Glossary

law of conservation of energy
the general law that total energy is constant in any process; energy may change in form or be transferred from one system to another, but the total remains the same
electrical energy
the energy carried by a flow of charge
chemical energy
the energy in a substance stored in the bonds between atoms and molecules that can be released in a chemical reaction
the energy carried by electromagnetic waves
nuclear energy
energy released by changes within atomic nuclei, such as the fusion of two light nuclei or the fission of a heavy nucleus
thermal energy
the energy within an object due to the random motion of its atoms and molecules that accounts for the object’s temperature
efficiency
a measure of the effectiveness of the input of energy to do work; useful energy or work divided by the total input of energy

### Solutions

Problems & Exercises

2: Equating$\boldsymbol{\Delta\textbf{PE}_{\textbf{g}}}$and$\boldsymbol{\Delta\textbf{KE}},$we obtain$\boldsymbol{v=\sqrt{2gh+v_0^2}=\sqrt{2(9.80\textbf{ m/s}^2)(20.0\textbf{ m})+(15.0\textbf{ m/s})^2}=24.8\textbf{ m/s}}$ 