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Physics LibreTexts

6.7: Conservation of Energy

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  • Learning Objectives


    • Explain the law of the conservation of energy.
    • Describe some of the many forms of energy.


    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 \((W_{nc})\) 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 \((OE)\). Then we can state the conservation of energy in equation form as

    \[KE_i + PE_i + W_{nc} + OE_i = KE_f + PE_f + OE_f .\]

    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 \(W_{nc}\) 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.

    Usefulness of the Energy Conservation Principle


    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



    Problem-Solving Strategies for Energy





    \[KE_i + PE_i = KE_f + PE_f.\]



    In most problems, one or more of the terms is zero, simplifying its solution. Do not calculate \(W_c\), the work done by conservative forces; it is already incorporated in the \(PE\) 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 \(h = 0\) at either the initial or final point, so that \(PE_g\) is zero there. Then solve for the unknown in the customary manner.


    Transformation of Energy


    Another example of energy conversion occurs in a solar cell. Sunlight impinging on a solar cell (Figure 7.7.1) 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 \(\PageIndex{1}\): 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 \(10^{68}\)
    Energy released in a supernova \(10^{44}\)
    Fusion of all the hydrogen in Earth’s oceans \(10^{34}\)
    Annual world energy use \(4 \times 10^{20}\)
    Large fusion bomb (9 megaton) \(3.8 \times 10^{16}\)
    1 kg hydrogen (fusion to helium) \(6.4 \times 10^{14}\)
    1 kg uranium (nuclear fission) \(8.0 \times 10^{13}\)
    Hiroshima-size fission bomb (10 kiloton) \(4.2 \times 10^{13}\)
    90,000-ton aircraft carrier at 30 knots \(1.1 \times 10^{10}\)
    1 barrel crude oil \(5.9 \times 10^9\)
    1 ton TNT \(4.2 \times 10^9\)
    1 gallon of gasoline \(1.2 \times 10^8\)
    Daily home electricity use (developed countries) \(7n\times 10^7\)
    Daily adult food intake (recommended) \(1.2 \times 10^7\)
    1000-kg car at 90 km/h \(3.1 \times 10^5\)
    1 g fat (9.3 kcal) \(3.9 \times 10^4\)
    ATP hydrolysis reaction \(3.2 \times 10^4\)
    1 g carbohydrate (4.1 kcal) \(1.7 \times 10^4\)
    1 g protein (4.1 kcal) \(1.7 \times 10^4\)
    Tennis ball at 100 km/h \(22\)
    Mosquito 10–2g at 0.5 m/s) \(1.3 \times 10^{-6}\)
    Single electron in a TV tube beam \(4.0 \times 10^{-15}\)
    Energy to break one DNA strand \(4.0 \times 10^{-19}\)


    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 \(E_{ff}\) of an energy conversion process is defined as

    \[Efficiency \, (E_{ff}) = \dfrac{useful \, energy \, or \, work \, output}{total \, energy \, input} = \dfrac{W_{out}}{E_{in}}.\]


    Activity/device Efficiency (%)
    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

    Efficiency of the Human Body and Mechanical Devices

    PhET Explorations: Masses and Springs


    Figure \(\PageIndex{2}\): Masses and Springs


    • When all forms of energy are considered, conservation of energy is written in equation form as \[KE_i + PE_i + W_{nc} + OE_i = KE_f + PE_f + OE_f ,\] where \(OE\) is all other forms of energy besides mechanical energy.

    The efficiency \(E_{ff}\) of a machine or human is defined to be \(E_{ff} = \frac{W_{out}}{E_{in}},\) where \(W_{out}\) is useful work output and \(E_{in}\) s the energy consumed.


    law of conservation of energy
    electrical energy
    the energy carried by a flow of charge
    chemical energy
    radiant energy
    the energy carried by electromagnetic waves
    nuclear energy
    thermal energy
    the energy within an object due to the random motion of its atoms and molecules that accounts for the object's temperature

    Contributors and Attributions
