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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}

h4. Introduction to the Model

h5. Description and Assumptions

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.   

h5. Learning 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].
* Describe the conditions under which [mechanical energy] is conserved.

h5. Relevant Definitions

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h6. Mechanical Energy

{latex}\begin{large}\[E = K + U\]\end{large}{latex}

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h6. Kinetic Energy
{latex}\begin{large}\[ K = \frac{1}{2}mv^{2} + \frac{1}{2}I\omega^{2}\]\end{large}{latex}

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h6. Work
{latex}\begin{large}\[W_{fi} = \int_{\rm path} \vec{F}(\vec{s}) \cdot d\vec{s} = \int_{t_{i}}^{t_{f}} \vec{F}(t) \cdot \vec{v}(t)\:dt\]\end{large}{latex}

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{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}

h4. S.I.M. Structure of the Model

h5. Compatible Systems

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. 

{info}In mechanics, the only commonly encountered conservative interactions are [gravity|gravitation (universal)] and [springs|Hooke's Law for elastic interactions].{info}

h5. Relevant Interactions

Any [external force] that performs that perform [work] on the system must be considered, _and also_ any [internal|internal force]  [non-conservative forces|force#nonconservative] that perform work.  Any [internal|internal force] [_conservative_ forces|force#nonconservative] that are present should have their interaction represented by the associated [potential energy] rather than by the [work].

h4. Law of Change

h5. Mathematical Representation

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h6. Differential Form

{latex}\begin{large}\[ \frac{dE}{dt} = \vec{F}\cdot \vec{v} \]\end{large}{latex}
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h6. Integral Form

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\begin{large}\[ E_{f} = E_{i} + \sum W^{\rm ext} + \sum W^{\rm NC}_{fi} = E_{i} + \sum \int_{t_{i}}^{t_{f}} \left(\vec{F}^{\rm ext}(t) + \vec{F}^{\rm NC}(t)\right)\cdot \vec{v}(t)\:dt \] \end{large}{latex}
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h5. Diagrammatic Representations

* [Initial-state final-state diagram|initial-state final-state diagram].
* [Energy bar graph|Diagrams and Mechanical Energy].


h4. Relevant Examples

h6. {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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h6. {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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h6. {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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h6. {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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h6. {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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h6. {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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[!coaster.jpg|alt="How does a person's apparent weight change on a roller coaster?  Click the image to investigate."!|Roller Coaster Diet?]
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[!bungee.jpg|alt="What are the properties of a bungee cord?  Click the image to investigate."!|Bungee Jump]
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!bowarrow.jpg!
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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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