Page History

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.

{table:align=right|cellspacing=0|cellpadding=1|border=1|frame=box|width=40%}
{tr}
{td:align=center|bgcolor=#F2F2F2}*[Model Hierarchy]*
{td}
{tr}
{tr}
{td}
{pagetree:root=Model Hierarchy|reverse=true}
{td}
{tr}
{table}
|| Page Contents ||
| {toc:style=noneoutline|indent=10px} |

----
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} = \int_{\rm path} \vec{F}^{NC} \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_{f} = E_{i} + W^{NC}_{\rm net} $ \end{large}{latex}
\\
----
h2. Diagrammatical Representations

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

----
h2. Relevant Examples

{contentbylabel:constant_energy,non-conservative_work|maxResults=50|showSpace=false|showLabels=true|operator=OR}
\\
\\
\\
| !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/]. |
\\