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9.5: Conservation of Energy, the Work-Energy Principle, and the Mechanical Energy Balance

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    When students leave physics mechanics and are introduced to the more general Conservation of Energy principle, they often struggle to understand how it relates to the more restricted Work-Energy Principle and Mechanical Energy Balance. Or as students often more simply put it, “Which energy equation should I use?”

    The following notes (1) illustrate how the more restricted principles are related back to the fundamental Conservation of Energy and (2) give students a list of modeling assumptions that can be recognized and applied to a specific problem to recover the Mechanical Energy Balance.

    Because this course emphasizes constructing problem-specific solutions from fundamental principles, i.e. Conservation of Energy in this case, this material is best introduced after students have read the Section 7.1, 7.2, and 7.3 of Chapter 7 and before they are assigned any work-energy-type homework problems typically found in a basic mechanics course.

    1. Conservation of Energy, the Work-Energy Principle, and the Mechanical Energy Balance — These notes describe (1) how the Work-Energy Principle is developed from the Conservation of Linear Momentum, (2) how the Mechanical Energy Balance can be developed from the Conservation of Energy, (3) how the Work-Energy Principle and the Mechanical Energy Balance are related, and (4) when should students use each. In addition, these notes also introduce spring energy as a type of mechanical energy.
    2. Summary — short summary of the section above.
    3. When can I start my analysis with the Mechanical Energy Balance for a closed system? — These notes provide three different approaches with modeling assumptions for reducing the general Conservation of Energy equation to the more restricted Mechanical Energy Balance. [These notes do make explicit use of the incompressible substance model to relate temperature change to internal energy change.]

    1. Conservation of Energy, the Work-Energy Principle, and the Mechanical Energy Balance

    In your study of engineering and physics, you will run across a number of engineering concepts related to energy. Three of the most common are Conservation of Energy, the Work-Energy Principle, and the Mechanical Engineering Balance. The Conservation of Energy is treated in this course as one of the overarching and fundamental physical laws. The other two concepts are special cases and only apply under limited conditions. The purpose of this note is to review the pedigree of the Work-Energy Principle, to show how the more general Mechanical Energy Balance is developed from Conservation of Energy, and finally to describe the conditions under which the Mechanical Energy Balance is preferred over the Work-Energy Principle.

    Work-Energy Principle for a Particle

    Consider a particle of mass \(m\) and velocity \(\vec{V}_{G}\) moving in a gravitational field of strength \(\vec{g}\) subject to a surface force \(\vec{R}_{\text {surface}}\). Under these conditions, writing Conservation of Linear Momentum for the particle gives the following: \[\frac{d}{dt} \left(m \vec{V}_{G}\right) = \vec{R}_{\text {surface}} + m \vec{g} \label{SM.1.1} \] Forming the dot product of Eq. \((\mathrm{SM.}1.1)\) with the velocity of the particle and rearranging terms gives the rate form of the Work-Energy Principle for a particle: \[\frac{d}{dt} \underbrace{\left(m \frac{\mathrm{V}^{2}}{2}\right)}_{\begin{array}{c} \text{Kinetic} \\ \text{energy} \end{array}} +\frac{d}{dt} \underbrace{(mgz)}_{\begin{array}{c} \text {Gravitational} \\ \text{potential} \\ \text{energy} \end{array}} = \underbrace{\vec{R}_{\text {surface}} \cdot \vec{V}_{G}}_{\begin{array}{c} \text {mechanical} \\ \text{potential} \\ \text{energy} \end{array}} \quad \Rightarrow \quad \frac{d}{dt} \left(E_{K} + E_{GP}\right) = \dot{W}_{\text {mech, in}} \label{SM.1.2} \]

    Recall that mechanical power is defined as \(\dot{W}_{\text {mech, in}} = \vec{R}_{\text {surface}} \cdot \vec{V}_{G}\), the dot product of the surface force with the velocity of its point of application. Because a particle has no extent and only one velocity, the point of application of the surface force and the particle always have the same velocity.

    Integrating Eq. \((\mathrm{SM}.1.1)\) with respect to distance or Eq. \((\mathrm{SM}.1.2)\) with respect to time gives the more familiar relation for change in kinetic energy, change in gravitational potential energy, and mechanical work: \[\begin{array}{c} \displaystyle \Delta \underbrace{\left(m \frac{V_{G}{ }^{2}}{2}\right)}_{=E_{K}} + \Delta \underbrace{\left(mgz_{G}\right)}_{=E_{GP}} = \int\limits_{t_{1}}^{t_{2}} \vec{R}_{\text {surface}} \cdot \underbrace{\vec{V}_{G} \ dt}_{=d \vec{s}} = \underbrace{\int\limits_{1}^{2} \underbrace{\vec{R}_{\text {surface}} \cdot \ d \vec{s}}_{=\delta W_{\text {mech in}}} }_{=W_{\text {mech, in}}} \\ \displaystyle \Downarrow \\ \Delta E_{K} + \Delta E_{GP} = W_{\text {mech, in}} \end{array} \label{SM.1.3} \]

    This is known as the finite-time form of the Work-Energy Principle for a particle. Recall that mechanical work is the time integral of the mechanical power and can be calculated as the dot product of the surface force and the displacement of its point of application. Again, the displacement of the point of application of the surface force is unambiguous for a particle because it is the same as the displacement of the particle.

    Although the Work-Energy Principle uses energy language—energy, work, power—it adds nothing new that could not have been discovered through a careful application of the Conservation of Linear Momentum.

    Conservation of Energy and the Mechanical Energy Balance for a Closed System

    Writing Conservation of Energy for a closed system, we obtain the rate form of Conservation of Energy for a closed system: \[\frac{d}{dt} \left(E_{sys}\right) = \dot{Q}_{\text {net, in}} + \dot{W}_{\text {net, in}} \label{SM.1.4} \] Restricting ourselves to only three types of energy—internal energy \(U\), kinetic energy \(E_{K}\), and gravitational potential energy \(E_{GP}\)—we have the following result: \[\frac{d}{dt} \left(U_{sys} + E_{K, \ sys} + E_{GP, \ sys}\right) = \dot{Q}_{\text {net, in}} + \dot{W}_{\text{net, in}} \nonumber \]

    Although the distinctions are somewhat artificial, we will segregate the energy into two groups: thermal energy and mechanical energy. Internal energy is usually classified as thermal energy because changing internal energy of a system is often associated with a change in temperature. The other two energies are classified as mechanical energy because changing the kinetic energy or gravitational potential energy of a system can be done solely by the application of a surface force and its associated mechanical work. In addition, we will separate the work transfer-rate of energy (power) into two terms: mechanical work where there is an identifiable surface force and non-mechanical work, e.g. electrical work.

    Using these new distinctions between mechanical and thermal phenomena, we can rewrite Eq. \((\mathrm{SM.}1.5)\) and group the mechanical and thermal terms as shown below: \[\begin{array}{c} \dfrac{d}{dt} \left(U_{sys} + E_{K, \ sys} + E_{GP, \ sys}\right) &= \dot{Q}_{\text{net, in}} + \underbrace{\dot{W}_{\text{net, mech, in}} + \dot{W}_{\text{net, nonmech, in}}}_{=\dot{W}_{\text{net, in}}} \\ \underbrace{\frac{d}{dt} \left(E_{K, \ sys} + E_{GP, \ sys}\right)}_{\begin{array}{c} \text{Rate of change} \\ \text{of the} \\ \text{mechanical energy} \\ \text{in the system} \end{array}} &= \underbrace{\dot{W}_{\text{net, mech, in}}}_{\begin{array}{c} \text{Transport rate} \\ \text{of energy by} \\ \text{mechanical work} \end{array}} + \underbrace{ \left[\dot{Q}_{\text{net, in}} + \dot{W}_{\text{net, nonmech, in}} - \dfrac{d}{dt} \left(U_{sys}\right)\right] }_{\begin{array}{c} \text{Net production rate} \\ \text{of mechanical energy} \\ \text{inside the system} \end{array}} \\ &\Downarrow\\ \dfrac{d}{dt} \left(E_{K, \ sys} + E_{GP, \ sys}\right) &= \dot{W}_{\text{net, mech, in}} + \dot{E}_{\text{net, mech, prod}} \end{array} \nonumber \]

    This is called the rate form of the Mechanical Energy Balance for a closed system. It accounts for the storage, transport, and production or destruction of mechanical energy in a closed system. In words,

    the time-rate-of-change of the mechanical energy in the system equals the net transport rate of energy with mechanical work (net mechanical power) into the system plus the net production rate of mechanical energy inside the system.

    In general, the net production rate term can take on both positive and negative values.

    The introduction and presence of a production term does not violate Conservation of Energy because we are only counting one type of energy, and one of the characteristics of energy is that it can be stored in different ways. [Consider a marble rolling up and down the sides of a wooden salad bowl. If there are no losses, there is a continuous interchange between kinetic and gravitational potential energy and if one was only counting kinetic energy it would alternately appear to be produced and then destroyed. The idea of counting only one type of energy is analogous to the idea of counting only a single chemical species (Species Accounting) used earlier in our study of Conservation of Mass.

    Mechanical Energy Balance = Work-Energy Principle?

    We’ve already shown that the Work-Energy Principle for particle is a direct descendant of Conservation of Linear Momentum and the Mechanical Energy Balance for a closed system grew out of the Conservation of Energy. Rewritten below together, we see that they appear to be similar even though one is written for a particle and the other for a more general closed system:

    \[\begin{aligned} &\text{Work-Energy Principle for a particle (Supplementary Materials 1.2):} \\ &\quad\quad\quad\quad \frac{d}{dt} \left(E_{K}+E_{GP}\right) = \dot{W}_{\text {mech, in}} \\ \text{ } \\ &\text{Mechanical Energy Balance for a closed system (Supplementary Materials 1.6):} \\ &\quad\quad\quad\quad \frac{d}{dt} \left(E_{K, \ sys} + E_{GP, \ sys}\right) = \dot{W}_{\text {net, mech, in}} + \dot{E}_{\text {net, mech, prod}} \end{aligned} \nonumber \]

    The only significant difference between the two equations is the net production rate term for mechanical energy.

    If we can find a set of conditions under which mechanical energy is neither produced nor destroyed, the Work-Energy Principle and the Mechanical Energy Balance contain the same information. So the important question is when does this occur? Without creating an inclusive list of conditions, we will state only one set of conditions:

    Mechanical energy will be neither produced nor destroyed within a closed system if (1) the materials in the system are incompressible, (2) there is no internal friction or friction between parts of the closed system, and (3) only mechanical work occurs on the system boundary. (Note that this does not prohibit friction on the boundary of the system.)

    This set of conditions is consistent with the conditions usually invoked when applying the Work-Energy Principle. When these conditions are satisfied, the Mechanical Energy Balance reproduces the results of the Work-Energy Principle with the added advantage that it applies to any closed system. If there is the restriction on internal friction is relaxed, we will show later that mechanical energy can only be destroyed. With this in mind, conditions that do not produce or destroy mechanical energy frequently represent the best or ideal behavior for the system. (This idea will be explored further when we encounter the Second Law of Thermodynamics and the Entropy Accounting Principle.)

    Adding Springs (Elastic Energy) to the Mechanical Energy Balance for a closed system

    Now that we’ve shown the relationship between the Work-Energy Principle and the Mechanical Energy Balance, we wish to include an additional type of energy that can be handled within our Mechanical Energy Balance — elastic or spring energy.

    Typically we will only consider true mechanical springs. Assuming a linear spring with no hysteresis or internal friction, the energy stored in a spring can be calculated from the following equation: \[\begin{array}{ll} & E_{\text {Spring}} &= \dfrac{1}{2} k \left(x - x_{0}\right)^{2} \\ \text { where } \quad\quad & k &= \text{spring constant [Force/Length]} \\ & x_{0} &= \text {unstretched length of the spring} \\ & x &= \text {stretched length of the spring} \end{array} \nonumber \] Note that an unstretched spring stores no mechanical energy, and that a linear, ideal spring stores energy when it compressed or stretched.

    Including this additional form of mechanical energy, we have an expanded rate form of the Mechanical Energy Balance for a closed system: \[\frac{d}{dt} \left(E_{K, \ sys} + E_{GP, \ sys} + E_{Spring, \ sys}\right) = \dot{W}_{\text {net, mech, in}} + \dot{E}_{\text {net, mech, prod}} \nonumber \] This is the most general form we will present.

    Bottom Line — When should and can I use the Mechanical Energy Balance?

    Although the general form can be useful, Eq. \((\mathrm{SM.}1.8)\) is most useful when we can assume that there is no mechanical energy production or destruction. Under these conditions, we have the expanded rate form of the Mechanical Energy Balance for a closed, incompressible system with no internal friction and only mechanical work: \[\frac{d}{dt} \left(E_{K, \ sys} + E_{GP, \ sys} + E_{Spring, \ sys}\right) = \dot{W}_{\text {net, mech, in}} \nonumber \] Integrated with respect to time we recover the finite-time form: \[\Delta E_{K, \ sys} + \Delta E_{GP, \ sys} + \Delta E_{Spring, \ sys} = W_{\text {net, mech, in}} \nonumber \] If your system contains only incompressible objects, there is no internal friction, and only mechanical work occurs on the boundary of the system then you can and should use Eq. \((\mathrm{SM.}1.9)\) or \((\mathrm{SM.}1.10)\) instead of the complete Conservation of Energy. This will also replace the Work-Energy Principle for a particle, unless you find an particular advantage in starting with the Conservation of Linear Momentum and integrating. When applied appropriately and correctly, the Mechanical Energy Balance as presented in Eq. \((\mathrm{SM.}1.9)\) and Eq. \((\mathrm{SM.}1.10)\) can do everything the Work-Energy Principle can and more.


    2. Summary

    Work-Energy Principle for a Particle

    ... Start with the conservation of linear momentum for a particle (Equation \ref{SM.1.1}): \[\frac{d}{dt} \left(m \vec{V}_{G}\right) = \vec{R}_{\text {surface}}+m \vec{g} \nonumber \]

    ... Form the dot product with the velocity of the center of mass \(\vec{V}_{G}\) and define kinetic energy, gravitational potential energy and mechanical power to obtain the rate-form of the work-energy principle for a particle (Equation \ref{SM.1.2}): \[\frac{d}{dt} \underbrace{\left(m \frac{\mathrm{V}^{2}}{2}\right)}_{\begin{array}{c} \text{Kinetic} \\ \text{energy} \end{array}} +\frac{d}{dt} \underbrace{(mgz)}_{\begin{array}{c} \text {Gravitational} \\ \text{potential} \\ \text{energy} \end{array}} = \underbrace{\vec{R}_{\text {surface}} \cdot \vec{V}_{G}}_{\begin{array}{c} \text {mechanical} \\ \text{potential} \\ \text{energy} \end{array}} \quad \Rightarrow \quad \boxed{\frac{d}{dt} \left(E_{K} + E_{GP}\right) = \dot{W}_{\text {mech, in}}} \nonumber \]

    ... Integrate the rate-form of the work-energy principle over a time interval to obtain the finite-time form of the work-energy principle for a particle (Equation \ref{SM.1.3}): \[\begin{array}{c} \displaystyle \Delta \underbrace{\left(m \frac{V_{G}{ }^{2}}{2}\right)}_{=E_{K}} + \Delta \underbrace{\left(mgz_{G}\right)}_{=E_{GP}} = \int\limits_{t_{1}}^{t_{2}} \vec{R}_{\text {surface}} \cdot \underbrace{\vec{V}_{G} \ dt}_{=d \vec{s}} = \underbrace{\int\limits_{1}^{2} \underbrace{\vec{R}_{\text {surface}} \cdot \ d \vec{s}}_{=\delta W_{\text {mech in}}} }_{=W_{\text {mech, in}}} \\ \displaystyle \Downarrow \\ \boxed{\Delta E_{K} + \Delta E_{GP} = W_{\text {mech, in}}} \end{array} \nonumber \]

    Conservation of Energy and Mechanical Energy Balance for a Closed System

    ... Start with the rate form of the conservation of energy for a closed system: \[\frac{d}{dt} \left(E_{sys}\right) = \dot{Q}_{\text {net, in}} + \dot{W}_{\text {net, in}} \nonumber \]

    ... Classify energy into two types — mechanical vs. thermal. Mechanical energy can be accomplished without changing the temperature of the system, while thermal energy typically requires a change in temperature or pressure of the system. Regrouping terms we have the rate-form of the Mechanical Energy Balance (in general mechanical energy, like any one type of energy, is not conserved). (See Equation \ref{SM.1.4})

    \[\begin{aligned} \frac{d}{dt} \left(U_{sys} + E_{K, \ sys} + E_{GP, \ sys}\right) &= \dot{Q}_{\text{net, in}} + \underbrace{\dot{W}_{\text{net, mech, in}} + \dot{W}_{\text{net, nonmech, in}}}_{= \dot{W}_{\text{net, in}}} \\ \frac{d}{dt} \left(E_{K, \ sys} + E_{GP, \ sys}\right) &= \underbrace{\dot{W}_{\text{net, mech, in}}}_{\begin{array}{c} \text{Transport rate} \\ \text{of energy by} \\ \text{mechanical work} \end{array}} + \underbrace{\left[\dot{Q}_{\text{net, in}} + \dot{W}_{\text{net, nonmech, in}} - \frac{d}{dt} \left(U_{sys}\right)\right]}_{\begin{array}{c} \text{Net production rate of mechanical energy} \\ \text{inside the system} \\ ( \text{May be } >, \ <, \text{ or } =0) \end{array}} \\ \frac{d}{dt} \left(E_{K, \ sys} + E_{GP, \ sys}\right) &= \dot{W}_{\text{net, mech, in}} + \dot{E}_{\text{net, mech, prod}} \end{aligned} \nonumber \]

    The value of the mechanical energy production term depends on the process. If the system contains only incompressible substances, the mechanical energy production term is always less than or equal to zero, i.e. mechanical energy is destroyed in real processes. When the mechanical energy production/destruction term is zero the mechanical energy balance and the work-energy principle are identical.

    When does the Mechanical Energy Balance = Work-Energy Principle?

    When mechanical energy is neither created nor destroyed, the Mechanical Energy Balance reproduces the results of the Work-Energy Principle with the added advantage that it applies to any closed system. Although not all inclusive, one useful set of conditions under which this occurs follows:

    Mechanical energy will be neither produced nor destroyed within a system if

    1. the system is closed,
    2. all substances in the system are incompressible,
    3. there is no friction (dissipation) within or between parts of the closed system, and
    4. the only energy transfer on the system boundary is mechanical work. (Note that this does not prohibit friction on the boundary system.)

    If all of these conditions apply to your system, you may and should start your analysis with the Mechanical Energy Balance and set the production term to zero!

    These conditions are consistent with the assumptions that you make in developing and applying the Work-Energy Principle. Conditions that do not produce or destroy mechanical energy frequently represent the best or ideal behavior for the system. In addition, many of the types of problems you solved in physics that involved conservative forces can be solved using the Mechanical Energy Balance.

    A new form of mechanical energy — Springs (Elastic Energy)

    For an ideal linear spring, i.e. no hysteresis or internal friction, the magnitude of the force exerted by the spring, \(|F|\), is proportional to the compression/extension, \(\delta\), of the spring from its unstretched (free) length, i.e. \(|F|=k|\delta|\). An uncompressed or unstretched spring has zero spring (elastic) energy. When a linear, ideal spring is deflected (compressed or stretched) it stores spring energy. Springs may also have kinetic, gravitational potential, and internal energy; however, the amount of spring (elastic) energy only depends on the deflection of the spring. The elastic energy stored in a linear, ideal spring can be calculated as follows: \[\begin{aligned} E_{\text{Spring}} &= \frac{1}{2} k \delta^{2} = \frac{1}{2} k \left|x-x_{o}\right|^{2} \\ \text{where} \quad\quad\quad k &= \text{spring constant } [\mathrm{Force} / \mathrm{Length}] \\ \delta &= \left|x-x_{o}\right| = \text{spring deflection (compression or extension) from its free length} \\ x_{o} &= \text{length of the unstretched spring, sometimes called the "free length"} \\ x &= \text{length of the stretched or compressed spring} \end{aligned} \nonumber \]

    When using conservation of energy (or the mechanical energy balance) to solve a problem with springs, it is usually advantageous to place the springs inside the system. If they remain outside, the spring force (a vector) does work on the system. When placed inside the system, effect of the springs is handled through the changing spring energy in the system.

    Bottom Line — When should I use the Mechanical Energy Balance?

    If your system is (1) closed, contains only (2) incompressible objects, has (3) no internal friction (friction/dissipation within or between parts of the system), and has (4) only mechanical work transfers of energy on the boundary, then mechanical energy is "conserved" and you can, may, and should start your analysis with the Mechanical Energy Balance (rate-form or finite-time form) assuming mechanical production/destruction is identically zero. \[\boxed{\frac{d}{dt} \left(E_{K, \ sys} + E_{GP, \ sys} + E_{Spring, \ sys}\right) = \dot{W}_{\text {net, mech, in}}} \quad \text { or } \quad \boxed{\Delta E_{K, \ sys} + \Delta E_{GP, \ sys} + \Delta E_{Spring, \ sys} = W_{\text {net, mech, in}}} \nonumber \]

    3. Under what conditions is the Mechanical Energy Balance (mechanical energy conserved) valid for a closed system? What assumptions must I make?

    Closed System Mechanical Energy Balance (with mechanical energy conserved)

    \[\begin{array}{ll} \text { Rate Form } & \dfrac{d E_{\text {sys, mech}}}{dt} = \dfrac{d}{dt} \left(E_{\text{Kinetic}} + E_{\text{Gravitational}} + E_{\text{Spring}}\right) = \dot{W}_{\text {mech, net, in}} \\ { } \\ \text { Finite-Time Form } & \Delta E_{\text {sys, mech}} = \Delta E_{\text{Kinetic}} + \Delta E_{\text{Gravitational}}+\Delta E_{\text{Spring}} = W_{\text {mech, net, in}} \end{array} \nonumber \] In all cases where energy is to be counted, we ALWAYS start by applying the full Conservation of Energy Equation, usually in the rate form. Then we travel to the Mechanical Energy Balance (MEB) by one of three approaches:

    APPROACH #1 - (Preferred Approach)

    Starts with the Conservation of Energy Equation and emphasizes mechanics assumptions (no mention of heat transfer, etc.) to move directly to the mechanical energy balance (MEB), where mechanical energy is conserved:

    Assume:

    1. Closed system
    2. Incompressible substance
    3. Only mechanical work/power at boundaries
    4. No internal friction, i.e. within bodies or between surfaces inside the system

    APPROACH #2

    Starts with the Conservation of Energy Equation and emphasizes thermodynamics assumptions without a substance model: \[\frac{d E_{sys}}{dt} = \dot{Q}_{\text{net, in}} + \dot{W}_{\text{net, i }} + \sum_{\text{in}} \dot{m}_{i} \left(h_{i}+\frac{V_{i}^{2}}{2}+g z_{i}\right) - \sum_{\text{out}} \dot{m}_{e} \left(h_{e}+\frac{V_{e}^{2}}{2}+g z_{e}\right) \nonumber \]

    Assume:

    1. Closed system
    2. No heat transfer, i.e. adiabatic system
    3. No change in internal energy, i.e. \(\Delta U=0\)

    APPROACH #3

    Starts with the Conservation of Energy Equation and emphasizes thermodynamics assumptions including the incompressible substance model: \[\frac{d E_{sys}}{dt} = \dot{Q}_{\text{net, in}} + \dot{W}_{\text{net, in}} + \sum_{\text{in}} \dot{m}_{i} \left(h_{i}+\frac{V_{i}^{2}}{2}+g z_{i}\right) - \sum_{\text{out}} \dot{m}_{e} \left(h_{e}+\frac{V_{e}^{2}}{2}+g z_{e}\right) \nonumber \]

    Assume:

    1. Closed system
    2. Incompressible substance
    3. No heat transfer, i.e. adiabatic system
    4. Isothermal process, i.e. no temperature change \((\Delta T=0)\)

    9.5: Conservation of Energy, the Work-Energy Principle, and the Mechanical Energy Balance is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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