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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}
 Column
 
Integral Form
 
 Latex

\begin{large}\[ E
Mechanical Energy and Non-Conservative Work

| !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/]. |
\\
h2. Description

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


h4. Prior Models


h4. Vocabulary
*[System.|system]
*[Internal Forces.|internal+force]
*[External Forces.|external+force]
*[Conservative Forces.|conservative+force]
*[Non-conservative forces.|non-conservative+force]

----
h2. Model Specification
h4. Keys to Applicability
Can be applied to any system for which the [work|work] done by the [non conservative forces|non-conservative+force] is known. 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 this particular case the [mechanical energy|mechanical+energy] of the system is constant. 

h4. System Structure

Internal Constituents:  One or more [Point particles|point particle] or [rigid objects|rigid object]. \\

Environment:   [External forces|external force] that do non-conservative work on the system. (??? If I want to be consisten with all the model structure I should leave here only the external forces. My concern is that leaving it in this way may give the impression to the student that they only need to think about the external non conservative forces and forget the possible work done by internal non-conservative forces, what do you thin??)

\\

h4. Descriptors

Object Variables:  Mass or moment of inertia for each object about a given axis of rotation, (_m{_}{^}j^) or (_I{^}j{^}{~}Q{~}_). (????If the objects in the system interact with a spring then the spring constant.)

State Variables:   Kinetic energy for each element of the system and the potential energy of the system. (? Or alternatively, linear speed or angular speed, (_v{_}{^}j^) or (_w{^}j{^}) for each object inside the system and the position of each of the objects in the system).

Interaction Variables:   External and internal non conservative forces, _F{_}{~}NC,ext{~} and _F{_}{~}NC,int{~}) or, alternately, the work done by the external and internal non conservative forces.

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} + 
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}

\sum 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
 
 Toggle Cloak
idgrav
 Examples Involving Gravitational Potential Energy
 
 Cloak
idgrav
 50falsetrueANDgravitational_potential_energy,example_problem
 
 Toggle Cloak
idelas
 Examples Involving Elastic (Spring) Potential Energy
 
 Cloak
idelas
 50falsetrueANDelastic_potential_energy,example_problem
 
 Toggle Cloak
idrot
 Examples Involving Rotational Kinetic Energy
 
 Cloak
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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