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hiddentrue
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^{
 Wiki Markup
h2. Keys to Applicability

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|non-conservative] 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 especially useful for systems where the non-conservative work is zero, in which case the [mechanical energy] of the system is constant. These can be recognized explicitly by statements like "frictionless surface" "smooth track" or in situations where only forces that may be represented by potential energy are involved.

h2. Description

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{td:align=center|bgcolor=#F2F2F2}*[Model Hierarchy]*
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{pagetree:root=Model Hierarchy|reverse=true}
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|| Page Contents ||
| {toc:style=none|indent=10px} |

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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]

----
h2. Model Specification


h4. System Structure

*[*Constituents*|system constituent]**:*  One or more [point particles|point particle] or [rigid bodies|rigid body], the (internal) interactions between them, whether conservative or not, and any fields applied externally such as a uniform gravitational field.  
{note}The admission of fields avoids requiring the system to contain the source objects of the [conservative|conservative force] interactions that are represented by the fields.  In the example of earth's gravity, this is justified because the earth will have no change of its kinetic energy (its infinite mass implies no change of velocity) or of its potential energy (all potential energy is attributed to objects in the system influenced by the fields).
{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 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. Descriptors

*[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.  

If non-conservative forces are present, each object's vector position (_x{_}{^}j^) must be known *throughout* the time interval of interest (the path must be specified) 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{_}{^}j^ for near-earth [gravity], separation _r{_}{~}jk~ for universal gravity, separation _x{_}{~}jk~ for an elastic interaction, etc.) unless the potential energy (_U{_}{^}jk^) is specified directly.  

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

*[Interaction Variables|interaction variable]:*   Relevant non-conservative forces (_F{_}{^}NC,j{^}{~}k~) or the work done by the non-conservative forces (_W{_}{^}NC,j{^}{~}k~).

----
h2. Model Equations


h4. Relationships Among State Variables

\\
{latex}\begin{large}\[ E = K^{\rm sys} + U^{\rm sys} \]
\[ K^{\rm sys} = \sum_{j=1}^{N}\left(\frac{1}{2}m^{j}(v^{j})^{
2} + \frac{1}{2}
I^{j}(
I\omega^{
j})^{
2}\
right)\]
\[ W^{NC,j}_{k
]\end{large}
 Column
 
Work
 
 Latex
\begin{large}\[W_{fi} = \int_{\rm path} \vec{F}
^{NC,j}_{k}
(\vec{s}) \cdot d\vec{
x}^{j} \]
\[ W^{NC}_{\rm net} 
s} = \
sum
int_{
j=1}^{N} \sum_{k=1
t_{i}}^{
N
t_{
F}^{j
f}} 
W^{NC,j}_{k} 
\vec{F}(t) \cdot \vec{v}(t)\:dt\]\end{large}
{latex}
where _N_ is the number of system constitutents and _N_~F~^j^ is the number of non-conservative forces acting on the _j{_}th system constitutent.

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.
\\

Some common potential energy relationships are:

h5. Near-Earth Gravity

{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.
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}\[ 
U_
\frac{
g
dE}
^
{
j
dt} = 
m^{j}gh^{j}
\sum \
]
left(\
end
vec{
large
F}^{
latex}

h5. Universal Gravity

{latex}\begin{large}\[ U_{g}^{jk} = -G\frac{m^{j}m^{k}}{|\vec{r}_{jk}|}
\rm ext} + \vec{F}^{\rm NC}\right)\cdot \vec{v} \]\end{large}
{latex}

h5. Elastic

{latex}
 Column
 
Integral Form
 
 Latex

\begin{large}\[ 
U
E_{
s}^{jk
f} = 
\frac{1}{2}k^{jk}(|\vec{x}^{j}-\vec{x}^{k}|-x_{0}^{jk})^{2}\]\end{large}{latex}
where _x{_}{^}jk{^}{~}0~ is the natural length of the spring.

h4. Mathematical Statement of the Model


\\
{latex}
\begin
{large} $E_{f} = E_{i} + W^{NC}_{\rm net} $ \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/]. |
\\
E_{i} + \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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