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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}
 Column
 
Kinetic Energy
 
 Latex
\begin{large}\[ K = \frac{1}{2}mv^{2} + \frac{1}{2}I\omega^{2}\]\end{large}
 Column
 
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}
 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}
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Integral Form
 
 Latex

\begin{large}\[ E
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h2. Keys to Applicability

Can be applied to any system for which the change in [mechanical energy] can be attributed to [work|work] done by [non conservative forces|non-conservative] (as opposed to processes like heat transfer, radiative losses, etc.). 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. It is specially useful for systems where the non-conservative work is zero, in which case the [mechanical energy] of the system is constant. 


h2. Description

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{td:align=center|bgcolor=#F2F2F2}*[Model Hierarchy]*
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h2. Assumed Knowledge


h4. Prior Models
* [Point Particle Dynamics]

h4. Vocabulary
* [system]
* [internal force]
* [external force]
* [conservative force]
* [non-conservative]
* [kinetic energy]
* [gravitational potential energy]
* [elastic potential energy]
* [mechanical energy]

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h2. Model Specification

h4. System Structure

*[Constituents|system constituent]:*  One or more [point particles|point particle] or [rigid bodies|rigid body]. Technically, the system must be defined in such a way as to contain all objects that participate in any non-negligible [conservative|conservative force] interactions that are present.  
{note}For example, for systems subject to earth's gravity, the earth should technically be included in the system, though it is usually sufficient to treat it as a rigid body that is at rest and has infinite mass.  If this approximation is made, the earth will have zero kinetic energy (it will not change its velocity, since it has infinite mass).{note}  \\

*[Interactions|interaction]:*   All forces that do [non-conservative] [work] on the system must be considered, _including_ [internal forces|internal force] that perform such work.  [Conservative forces|conservative force] that are present should have their interaction represented by a [potential energy] rather than by [work].  

\\

h4. Descriptors

*[Object Variables|object variable]*:  None.
{note}Object masses and moment of inertia can technically change in this model, so they are state variables.{note}
*[State Variables|state variable]*:   Mass (_m_^j^) and possibly moment of inertia (_I_{^}j{^}) for each object plus  linear (_v_^j^) and possibly rotational (ω^j^) speeds for each object, or alternatively, the kinetic energy (_K_^j^) may be specified directly.  When a conservative interaction is present, some sort of position or separation is required for each object (_h_^j^ for near-earth gravity, _r_~jk~ for universal gravity, _x_~jk~ for an elastic interaction, etc.) unless the potential eenrgy (_U_^jk^) is specified directly.

*[Interaction Variables|interaction variable]*:   Relevant non-conservative forces (_F_^NC^~k~) plus information about the objects' vector displacements (_s_^j^) or the work done by the non-conservative forces (_W_^NC^~k~).  

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h2. Model Equations

h4. Laws of Interaction
Here should go the possible type of forces, friction, tension, etcs
{latex}\begin{large}$ $\end{large}{latex}


h4. Laws of Change

{latex}
\begin
{large} $E
_{f} = E_{i} + \sum 
W_{i,f}^{NC} $ \end{large}{latex}\\

where _W{^}NC{^}{~}i,f{~}_ is the [work] done by the all the non-conservative forces on the system between the initial state defined by _E{~}i{~}_ and the final state defined by _E{~}f{~}_ and is given by

{latex}\begin{large}$ W_{i,f}^{NC} = \int_{i}^{f} \sum \vec{F}^{NC} . d\vec{r}  $ \end{large}{latex}

| !copyright and waiver^copyrightnotice.png! | RELATE wiki by David E. Pritchard is licensed under a [Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License|http://creativecommons.org/licenses/by-nc-sa/3.0/us/]. |
\\

W^{\rm ext}_{fi} + \sum W^{\rm NC}_{fi} \] \end{large}
Diagrammatic Representations
Relevant Examples
 
 Toggle Cloak
idcons
 Examples Involving Constant Mechanical Energy
 
 Cloak
idcons
 50falsetrueANDconstant_energy,example_problem
 
 Toggle Cloak
idnoncons
 Examples Involving Non-Conservative Work
 
 Cloak
idnoncons
 50falsetrueANDnon-conservative_work,example_problem
 
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idgrav
 Examples Involving Gravitational Potential Energy
 
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idgrav
 50falsetrueANDgravitational_potential_energy,example_problem
 
 Toggle Cloak
idelas
 Examples Involving Elastic (Spring) Potential Energy
 
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idelas
 50falsetrueANDelastic_potential_energy,example_problem
 
 Toggle Cloak
idrot
 Examples Involving Rotational Kinetic Energy
 
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idrot
 50falsetrueANDrotational_energy,example_problem
 
 Toggle Cloak
idall
 All Examples Using this Model
 
 Cloak
idall
 50falsetrueANDconstant_energy,example_problem 50falsetrueANDnon-conservative_work,example_problem


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