Versions Compared

Old Version 113

changes.mady.by.user Andrew E Pawl

Saved on

compared with

New Version Current

changes.mady.by.user Andrew E Pawl

Saved on

Key

  • This line was added.
  • This line was removed.
  • Formatting was changed.
Comment: Migration of unmigrated content due to installation of a new plugin
{
Wiki Markup
Composition Setup
 HTML Table
border1
cellpadding8
cellspacing0
rulescols
framevoid
}{composition-setup}
{table:rules=cols|cellpadding=8|cellspacing=0|border=1|frame=void}
{tr:valign=top}
{td}

{excerpt:hidden=true}*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.{excerpt}

h4. Introduction to the Model

h5. 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|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.   

h5. Learning Objectives

Students will be assumed to understand this model who can:

* Compute the translational [kinetic energy] of an object.
* Compute the rotational [kinetic energy] of a [rigid body] rotating about an axis.
* Apply the constraint of [rolling without slipping].
* Define the term [non-conservative|non-conservative force].
* Calculate the [work] done by a [force] acting on a moving object.
* State the [Work-Kinetic Energy Theorem].
* Name the [conservative forces|conservative force] commonly encountered in mechanics problems.
* Explain why the zero point of the (near-earth) [gravitational|gravity (near-earth)] [potential energy] is arbitrary.
* Define the variables appearing in the expression for [elastic|Hooke's Law for elastic interactions] [potential energy].
* Calculate the total [mechanical energy] of a [system] containing any number of rotating and translating [rigid bodies|rigid body] near the surface of the earth that interact via springs.
* Construct [intitial-state final-state diagrams|initial-state final-state diagram] to summarize the [mechanical energy] of a [system].
* Describe the conditions under which [mechanical energy] is conserved.

h5. Relevant Definitions

{section}
{column}

h6. Mechanical Energy

{latex}
 Table Row (tr)
valigntop
 Table Cell (td)
 Excerpt
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}
{latex}

{column}{column}
h6. Kinetic Energy
{latex}
 Column
 
Kinetic Energy
 
 Latex
\begin{large}\[ K = \frac{1}{2}mv^{2} + \frac{1}{2}I\omega^{2}\]\end{large}
{latex}

{column}{column}

h6. Work
{latex}
 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}
{latex}

{column}{section}

{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.{note}

h4. 
 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


h5. 
Compatible 
 
 Systems



One 
 
 or 
 
 more 
 [
point 
 
 particles
|point particle] 
 or 
 [
rigid 
 
 bodies
|rigid body]
, 
 
 plus 
 
 any 
 
 conservative 
 
 interactitons 
 
 that 
 
 can 
 
 be 
 
 accounted 
 
 for 
 
 as 
 [
potential 
 
 energies
|potential energy] 
 of 
 
 the 
 
 system. 
 

{info}In 
 Info
In mechanics, 
 
 the 
 
 only 
 
 commonly 
 
 encountered 
 
 conservative 
 
 interactions 
 
 are 
 [
gravity
|gravitation (universal)] 
 and 
 [
springs
|Hooke's Law for elastic interactions].{info}

h5. Relevant Interactions

Any [external force] that performs that perform [work] on the system must be considered, _and also_ any [internal|internal force]  [non-conservative forces|force#nonconservative] that perform work.  Any [internal|internal force] [_conservative_ forces|force#nonconservative] that are present should have their interaction represented by the associated [potential energy] rather than by the [work].

h4. Law of Change

h5. Mathematical Representation

{section}{column}
h6. Differential Form

{latex}
.
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}
{latex}
{column}{column}

h6. Integral Form

{latex}
 Column
 
Integral Form
 
 Latex

\begin{large}\[ E_{f} = E_{i} + \sum W^{\rm ext}_{fi} + \sum W^{\rm NC}_{fi
} = E_{i
}
 + \sum
 \
int_{t_{i}}^{t_{f}} \left(\vec{F}^{\rm ext}(t) + \vec{F}^{\rm NC}(t)\right)\cdot \vec{v}(t)\:dt \
] \end{large}
{latex}
{column}{section}

h5. Diagrammatic Representations

* [
Diagrammatic Representations
 
 
|initial-state final-state diagram].
* [Energy bar graph|Diagrams and Mechanical Energy].


h4. Relevant Examples

h6. {toggle-cloak:id=cons} Examples Involving Constant Mechanical Energy

{cloak:id=cons}
{contentbylabel:
Relevant Examples
 
 Toggle Cloak
idcons
 Examples Involving Constant Mechanical Energy
 
 Cloak
idcons
50falsetrueANDconstant_energy,example_problem
|maxResults=50|showSpace=false|showLabels=true|operator=AND}
{cloak}

h6. {
 Toggle Cloak
:
id
=
noncons
} 
 Examples 
 
 Involving 
 
 Non-Conservative 
 
 Work


{
 Cloak
:
id
=noncons}
{contentbylabel:
noncons
50falsetrueANDnon-conservative_work,example_problem
|maxResults=50|showSpace=false|showLabels=true|operator=AND}
{cloak}

h6. {
 Toggle Cloak
:
id
=
grav
} 
 Examples 
 
 Involving 
 
 Gravitational 
 
 Potential 
 
 Energy


{
 Cloak
:
id
=grav}
{contentbylabel:
grav
50falsetrueANDgravitational_potential_energy,example_problem
|maxResults=50|showSpace=false|showLabels=true|operator=AND}
{cloak}

h6. {
 Toggle Cloak
:
id
=
elas
} 
 Examples 
 
 Involving 
 
 Elastic 
 
 (Spring) 
 
 Potential 
 
 Energy


{
 Cloak
:
id
=elas}
{contentbylabel:
elas
50falsetrueANDelastic_potential_energy,example_problem
|maxResults=50|showSpace=false|showLabels=true|operator=AND}
{cloak}

h6. {
 Toggle Cloak
:
id
=
rot
} 
 Examples 
 
 Involving 
 
 Rotational 
 
 Kinetic 
 
 Energy


{
 Cloak
:
id
=rot}
{contentbylabel:
rot
50falsetrueANDrotational_energy,example_problem
|maxResults=50|showSpace=false|showLabels=true|operator=AND}
{cloak}


h6. {
 Toggle Cloak
:
id
=
all
} 
 All 
 
 Examples 
 
 Using 
 
 this 
 
 Model


{
 Cloak
:
id
=all}
{contentbylabel:
all
50falsetrueANDconstant_energy,example_problem
|maxResults=
50
|showSpace=
false
|showLabels=
true
|operator=
AND
}
{contentbylabel:
non-conservative_work,example_problem
|maxResults=50|showSpace=false|showLabels=true|operator=AND}
{cloak}

\\
\\
{td}
{td:width=150px}
[!coaster.jpg|alt="How does a person's apparent weight change on a roller coaster?  Click the image to investigate."!|Roller Coaster Diet?]
\\
\\
[!bungee.jpg|alt="What are the properties of a bungee cord?  Click the image to investigate."!|Bungee Jump]
\\
\\
!bowarrow.jpg!
\\
Pictures courtesy:

* [Wikimedia Commons|http://commons.wikimedia.org] user [Boris23|http://commons.wikimedia.org/wiki/User:Boris23]
* [Wikimedia Commons|http://commons.wikimedia.org] user [Ellywa|http://nl.wikipedia.org/wiki/Gebruiker:Ellywa]
* [Wikimedia Commons|http://commons.wikimedia.org] user [Evanherk|http://nl.wikipedia.org/wiki/Gebruiker:Evanherk]


{td}
{tr}
{table}
\\



 Table Cell (td)
width150px
Image Added 

 Image Added 

 Image Added 
 Pictures 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}