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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 = \
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h2. Description and Assumptions

If we ignore 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 just 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. 

h2. Problem 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.  Since friction is a common source of non-conservative work, problems in which mechancial energy is conserved can often be recognized by explicit statements like "frictionless surface" "smooth track" or in situations where only gravity and/or springs ([conservative forces|force#nonconservative] that can be represented by [potential energy]) are involved.

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

h4. Prior Models

* [Point Particle Dynamics]

h4. Vocabulary

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

----
h2. System

h4. Constituents

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.

h4. State Variables

Mass (_m{_}) and possibly moment of inertia (_I{_}) for each object plus  linear (_v{_}) and possibly rotational (ω) speeds for each object, or alternatively, the kinetic energy (_K{_}) may be specified directly.  

If non-conservative forces are present, each object's path of travel (_s{_}) must be known *throughout* the time interval of interest unless the work done by each force is specified directly.  

When a conservative interaction is present, some sort of specific position or separation is required for each object (height _h{_} for near-earth [gravity], separation _r{_} for universal gravity, departure from equilibrium _x{_} for an elastic interaction, etc.) unless the relevant potential energy (_U{_}) is specified directly.  

Alternately, in place of separate kinetic and potential energies, the mechanical energy of the system (_E_) can be specified directly.

----

h2. Interactions

h4. Relevant Types

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}

h4. Interaction Variables

Relevant non-conservative forces (_F{_}{^}NC^) or the work done by the non-conservative forces (_W{_}{^}NC{^}).

----
h2. Model

h4. Relevant Definitions

\\
{latex}\begin{large}\begin{alignat*}{1} & E = K_{\rm sys} + U_{\rm sys} \\
 & K_{\rm sys} = \sum_{\rm constituents} \left(\
frac{1}{2}mv^{2} + \frac{1}{2}I\omega^{2}\
right)\\
 &W^{NC
]\end{large}
 Column
 
Work
 
 Latex
\begin{large}\[W_{fi} = \int_{\rm path} \vec{F}
^
(\vec{
NC
s}) \cdot d\vec{s} = \
\
& W^{NC}_{\rm net} = \sum_{\rm NC forces} W^{NC} \end{alignat*}\end{large}{latex}

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.
\\

h4. Law of Change
\\
{latex}
\begin
{large} $E
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}\[ E_{f} = E_{i} + \sum W^{
NC
\rm ext}_{fi} + \sum W^{\rm 
net
NC}_{fi} 
$
\] \end{large}
{latex}
\\
----
h2. Diagrammatical Representations

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

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h2. Relevant Examples

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| !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/]. |
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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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