Versions Compared

Key

  • This line was added.
  • This line was removed.
  • Formatting was changed.
Comment: Migration of unmigrated content due to installation of a new plugin
unmigratedcomposition-wiki-markupsetup
HTML Table
border1
cellpadding8
cellspacing0
rulescols
framevoid
h2. Description {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=none|indent=10px} | ---- h2. Assumed Knowledge h4. Prior Models * [Point Particle Dynamics] h4. Vocabulary * [system] * [force] * [impulse] * [momentum] * [velocity] ---- h2. Keys to Applicability This model is [generally applicable|generally applicable model] (assuming knowledge of the external forces and system constituents), but is especially useful when: * describing the momentum of systems where external forces are absent (system momentum will be constant). * estimating the force in a process that occurs in a very short time interval as in collisions (impulse will be easier to determine than force). {info}"Very short" is a relative expression. In collisions, we generally use the conservation of the system's momentum to yield information about the _change_ in momentum of the _individual system constituents_. A "very short" collision in that context is one in which the internal forces produce an impulse on the constituent of interest that is so much larger than the impulse from external forces that the external forces can be neglected. Of course, that is a relative criterion as well, and will depend on the desired accuracy of the calculation.{info} ---- h2. Model Specification h4. System Structure *[Constituents|system constituent]:*  System is composed of [Point particles|point particle]. {note}Rigid bodies may be treated as point particles with positions specified by the center of mass positions of the rigid bodies when this model is used.{note} \\ *[Interactions|force]:*  Only [external forces|external force] need be considered, since [internal forces|internal force] do not change the system's momentum. \\ h4. Descriptors *[Object Variables|object variable]:*  None. {note}This model does not require that constituent masses be constant. Thus, mass is a state variable for this model.{note} *[State Variables|state variable]:*  Mass (_m{_}{^}j^) and velocity (_v{_}{^}j^) for each object or momentum (_p{_}{^}j^) for each object inside the system. *[Interaction Variables|interaction variable]:*  External forces (_F{_}{~}ext,k~) or, alternately, impulses may be specified (_J{_}{~}ext,k~).  ---- h2. Model Equations h4. Relationships Among State Variables If not directly given, momenta can be obtained using the definition: {latex}\begin{large}\[ \vec{p}^{\:j} = m^{j}\vec{v}^{\:j}\]\end{large}{latex} The relationship implied by the model is most easily expressed in terms of the *system momentum*, which is the vector sum of the constituent momenta. For a system composed of _N_ point particles: {latex}\begin{large}\[ \vec{p}^{\:\rm sys} = \sum_{j=1}^{N} \vec{p}^{\:j} \]\end{large}{latex} {warning}The number of point particle constituents in the system is not necessarily fixed. A [totally inelastic collision], for example, could be viewed as a process where two separate system constituents exist in the initial state, but only one is present in the final state.{warning} \\ h4. Mathematical Statement of the Model {latex}
Table Row (tr)
valigntop
Table Cell (td)

Introduction to the Model

Description and Assumptions

Excerpt
hiddentrue

System: Any. — Interactions: Any. — Note: Linear momentum evolves separately from angular momentum, so all system constituents are treated as point particles in this model.


This model is generally applicable (assuming knowledge of the external forces and system constituents). The model is especially useful when describing the momentum of systems where external forces are absent (system momentum will be constant) or estimating the force in a process that occurs in a very short time interval such as collisions (impulse will be easier to determine than force).

Learning Objectives

Students will be assumed to understand this model who can:

Relevant Definitions


Latex
\begin{large}\[ \vec{p} = m\vec{v}\]\end{large}

S.I.M. Structure of the Model

Compatible Systems

The system must be effectively composed of point particles, though rigid bodies may be treated as point particles with positions specified by the center of mass positions of the rigid body when this model is used.

Relevant Interactions

Only external forces need be considered, since internal forces do not change the system's momentum.

Laws of Change

Mathematical Representation
Section
Column
Differential Form


Latex
\begin{large}\[ \frac{d\vec{p
}^{\:\rm sys
}}{dt} = 
\:
\sum
_{k=1}^{N_{F}}
 \vec{F}
_
^{\rm ext
,k
}
\]\end{large}
{latex} \\ or, alternately: {latex}\
Column
width40px

Column
Integral Form


Latex
\begin{large}\[ \vec{p
}^{\:\rm sys
}_{f} = \vec{p
}^{\: \rm sys
}_{i} + \sum
_{k=1
 \vec{J}^{
N_{F}} \vec{J
\rm ext}_{
ext,k
fi} = \vec{p}
^{\:\rm sys}
_{i} + \int
\sum
_{t_{
k=1
i}}^{
N
t_{
F
f}} \sum \vec{F}
_
^{\rm ext
,k
}\:dt  \]
\end{large}
{latex} ---- h2. Relevant Examples {contentbylabel:constant_momentum,momentum_force,impulse|showSpace=false|showLabels=true|excerpt=true|operator=OR|maxResults=50} ---- {search-box} \\ \\ | !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/]. | \\
Diagrammatic Representations

Relevant Examples

Toggle Cloak
idconst
Examples Involving Constant Momentum
Cloak
idconst
falsetruetrue50constant_momentum
Toggle Cloak
idimp
Examples Involving Impulse
Cloak
idimp
falsetruetrue50impulse
Toggle Cloak
idoned
Examples Involving 1-D Collisions
Cloak
idoned
falsetruetrue501d_collision
Toggle Cloak
idtwod
Examples Involving 2-D Collisions
Cloak
idtwod
falsetruetrue502d_collision
Toggle Cloak
idelas
Examples Involving Elastic Collisions
Cloak
idelas
falsetruetrue50elastic_collision
Toggle Cloak
idinel
Examples Involving Totally Inelastic Collisions
Cloak
idinel
falsetruetrue50totally_inelastic
Toggle Cloak
idflux
Examples Involving Continuous Momentum Flux
Cloak
idflux
falsetruetrue50momentum_force
Toggle Cloak
idall
All Examples Using this Model
Cloak
idall
falsetruetrueOR50constant_momentum,momentum_force,impulse

Search Box


Table Cell (td)
width235px

Click on the image to learn a handy rule of thumb for pool shots.Image Added

Image Added

Image Added
Photos courtesy:


Wiki Markup
{html}
<script type="text/javascript">
var gaJsHost = (("https:" == document.location.protocol) ? "https://ssl." : "http://www.");
document.write(unescape("%3Cscript src='" + gaJsHost + "google-analytics.com/ga.js' type='text/javascript'%3E%3C/script%3E"));
</script>
<script type="text/javascript">
try {
var pageTracker = _gat._getTracker("UA-11762009-2");
pageTracker._trackPageview();
} catch(err) {}</script>
{html}