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Wiki Markup
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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.
Introduction to the Model
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 done by non-conservative forces 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. 
Learning Objectives
Students will be assumed to understand this model who can:
Relevant Definitions
 Section
 Column
 
Mechanical Energy
 
 Latex
\begin{large}\[E = K + U\]\end{large}
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Kinetic Energy
 
 Latex
\begin{large}\[
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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

{cloak:id=desc}
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

{cloak:id=cues}
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

{cloak:id=pri}

* [Point Particle Dynamics]

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

{cloak:id=vocab}

* [system]
* [force]
* [work]
* [kinetic energy]
* [rotational kinetic energy]
* [gravitational potential energy|Gravitation]
* [elastic potential energy|Hooke's Law]
* [mechanical energy]

{cloak}

h2. Model

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

{cloak:id=sys}

One or more [point particles|point particle] or [rigid bodies|rigid body], plus any interactitons that can be accounted for as [potential energies|potential energy] of the system.
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!system^energy.png|width=200!

{cloak}

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

{cloak:id=int}

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.
{note}
{cloak}

h4. {toggle-cloak:id=def} {color:red} Relevant Definitions {color}

{cloak:id=def}
\\
{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}\]\end{large}
 Column
 
Work
 
 Latex
\
\
 &W
begin{large}\[W_{fi} = \int_{\rm path} \vec{F}(\vec{s}) \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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{cloak}

h4. {toggle-cloak:id=law} {color:red} Law of Change {color}

{cloak:id=law}
\\
{latex}
 = \int_{t_{i}}^{t_{f}} \vec{F}(t) \cdot \vec{v}(t)\:dt\]\end{large}
 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.
S.I.M. Structure of the Model
Compatible Systems
One or more point particles or rigid bodies, plus any conservative interactitons that can be accounted for as potential energies of the system. 
 Info
In mechanics, the only commonly encountered conservative interactions are gravity and springs.
Relevant Interactions
Any external force that performs that perform work on the system must be considered, and also any internal non-conservative forces that perform work. Any internal conservative forces that are present should have their interaction represented by the associated potential energy rather than by the work.
Law of Change
Mathematical Representation
 Section
 Column
 
Differential Form
 
 Latex
\begin{large}\[ \frac{dE}{dt} = \sum \left(\vec{F}^{\rm ext} + \vec{F}^{\rm NC}\right)\cdot \vec{v} \]\end{large}
 Column
 
Integral Form
 
 Latex

\begin{large}\[ E_{f} = E_{i} + \sum
_
 W^{\rm 
non-cons
ext}_{fi} 
W
+ \
]
sum 
\end{large}{latex}
\\
{cloak}

h4. {toggle-cloak:id=diag} {color:red} Diagrammatic Representations {color}

{cloak:id=diag}

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

{cloak}

h2. Relevant Examples

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

{cloak:id=cons}
{contentbylabel:
W^{\rm NC}_{fi} \] \end{large}
Diagrammatic Representations
Relevant Examples
 
 Toggle Cloak
idcons
 Examples Involving Constant Mechanical Energy
 
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idcons
50falsetrueANDconstant_energy,example_problem
|maxResults=50|showSpace=false|showLabels=true|operator=AND}
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h4. {
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noncons
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 Examples 
 
 Involving 
 
 Non-Conservative 
 
 Work


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


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


{
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=elas}
{contentbylabel:
elas
50falsetrueANDelastic_potential_energy,example_problem
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h4. {
 Toggle Cloak
:
id
=
rot
} 
 Examples 
 
 Involving 
 
 Rotational 
 
 Kinetic 
 
 Energy


{
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=rot}
{contentbylabel:
rot
50falsetrueANDrotational_energy,example_problem
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h4. {
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all
} 
 All 
 
 Examples 
 
 Using 
 
 this 
 
 Model


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50falsetrueANDconstant_energy,example_problem
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{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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