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{excerpt:hidden=true}*System:* Any system that does not undergo significant changes in [internal energy]. --- *Interactions:* Any interactions that can be parameterized as mechanical work.  Notable exceptions include heat transfer or radiation.{excerpt}

h1. Mechanical Energy and Non-Conservative Work

h4. {toggle-cloak:id=desc} Description and Assumptions

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If we ignore non-mechanical processes like heat transfer, radiative losses, etc., then we arrive at a model involving only [mechanical energy] which changes due to the application (or extraction) of the [work|work] done by [non-conservative forces|force#nonconservative] The non-conservative forces can be external forces exerted on the system or internal forces resulting from the interactions between the elements inside the system. 
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h4. {toggle-cloak:id=cues} Problem Cues

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The model is especially useful for systems where the non-conservative work is zero, in which case the [mechanical energy] of the system is constant.  The most important cue for mechanical energy conservation is the dominance of gravity or spring forces (both [conservative forces|force#nonconservative]) in a problem.  Since friction is a common source of non-conservative work, another important cue for problems in which mechancial energy is conserved is an explicit statement such as "frictionless surface" or "smooth track".  
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h4. {toggle-cloak:id=pri} Prior Models

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* [Point Particle Dynamics]

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h4. {toggle-cloak:id=vocab} Vocabulary

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* [system]
* [force]
* [work]
* [kinetic energy]
* [rotational kinetic energy]
* [gravitational potential energy|gravitation (universal)]
* [elastic potential energyLearning Objectives

Students will be assumed to understand this model who can:

* Compute the translational [kinetic energy] of an object.
* Compute the rotational [kinetic energy] of a [rigid body] rotating about an axis.
* Apply the constraint of [rolling without slipping].
* Define the term [non-conservative|non-conservative force].
* Calculate the [work] done by a [force] acting on a moving object.
* State the [Work-Kinetic Energy Theorem].
* Name the [conservative forces|conservative force] commonly encountered in mechanics problems.
* Explain why the zero point of the (near-earth) [gravitational|gravity (near-earth)] [potential energy] is arbitrary.
* Define the variables appearing in the expression for [elastic|Hooke's Law for elastic interactions] [potential energy].
* Calculate the total [mechanical energy] of a [system] containing any number of rotating and translating [rigid bodies|rigid body] near the surface of the earth that interact via springs.
* Construct [intitial-state final-state diagrams|initial-state final-state diagram] to summarize the [mechanical energy] of a [system].

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h2* Describe the conditions under which [mechanical energy] is conserved.


h1. Model

h4. {toggle-cloak:id=sys} {color:red} Compatible Systems {color}

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One or more [point particles|point particle] or [rigid bodies|rigid body], plus any conservative interactitons that can be accounted for as [potential energies|potential energy] of the system. 
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{info}In introductory mechanics, the only commonly encountered conservative interactions are [gravity|gravitation (universal)] and springs.{info}


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h4. {toggle-cloak:id=int} {color:red} Relevant Interactions {color}

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All [non-conservative forces|force#nonconservative] that perform [work] on the system must be considered, _including_ [internal forces|internal force] that perform such work. [Conservative forces|force#nonconservative] that are present should have their interaction represented by the associated [potential energy] rather than by the [work].
{note}Occasionally it is easier to consider the work of conservative forces directly, omitting their potential energy.
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h4. {toggle-cloak:id=def} {color:red} Relevant Definitions {color}

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{latex}\begin{large}\begin{alignat*}{1} & E = \sum_{\rm system} K + \sum_{\rm system} U \\
 & K = \frac{1}{2}mv^{2} + \frac{1}{2}I\omega^{2}\\
 &W = \int_{\rm path} \vec{F} \cdot d\vec{s}
\end{alignat*}\end{large}{latex}

{note}The system potential energy is the sum of all the potential energies produced by interactions between system constituents.  Even when there are two system constituents involved (for example in a double star) each *interaction* produces only one potential energy.{note}
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h4. {toggle-cloak:id=law} {color:red} Law of Change {color}

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\begin{large}\[ E_{f} = E_{i} + \sum_{\rm non-cons} W \] \end{large}{latex}
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h4. {toggle-cloak:id=diag} {color:red} Diagrammatic Representations {color}

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* [Initial-state final-state diagram|initial-state final-state diagram].
* [Energy bar graph|energy bar graph].

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h2h1. Relevant Examples

h4. {toggle-cloak:id=cons} Examples Involving Constant Mechanical Energy

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{contentbylabel:constant_energy,example_problem|maxResults=50|showSpace=false|showLabels=true|operator=AND}
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h4. {toggle-cloak:id=noncons} Examples Involving Non-Conservative Work

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{contentbylabel:non-conservative_work,example_problem|maxResults=50|showSpace=false|showLabels=true|operator=AND}
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h4. {toggle-cloak:id=grav} Examples Involving Gravitational Potential Energy

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{contentbylabel:gravitational_potential_energy,example_problem|maxResults=50|showSpace=false|showLabels=true|operator=AND}
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h4. {toggle-cloak:id=elas} Examples Involving Elastic (Spring) Potential Energy

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{contentbylabel:elastic_potential_energy,example_problem|maxResults=50|showSpace=false|showLabels=true|operator=AND}
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h4. {toggle-cloak:id=rot} Examples Involving Rotational Kinetic Energy

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{contentbylabel:rotational_energy,example_problem|maxResults=50|showSpace=false|showLabels=true|operator=AND}
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h4. {toggle-cloak:id=all} All Examples Using this Model

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{contentbylabel:constant_energy,example_problem|maxResults=50|showSpace=false|showLabels=true|operator=AND}
{contentbylabel:non-conservative_work,example_problem|maxResults=50|showSpace=false|showLabels=true|operator=AND}
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Pictures courtesy:

* [Wikimedia Commons|http://commons.wikimedia.org] user [Boris23|http://commons.wikimedia.org/wiki/User:Boris23]
* [Wikimedia Commons|http://commons.wikimedia.org] user [Ellywa|http://nl.wikipedia.org/wiki/Gebruiker:Ellywa]
* [Wikimedia Commons|http://commons.wikimedia.org] user [Evanherk|http://nl.wikipedia.org/wiki/Gebruiker:Evanherk]


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