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{
Wiki Markup
Composition Setup
 Excerpt
hiddentrue
System: One point particle constrained to move in one dimension. — Interactions: The particle must experience a force (or torque) that attempts to restore it to equilibrium and is directly proportional to its displacement from that equilibrium.
Introduction to the Model
Description and Assumptions
This model applies to position of a single point particle, or to the angular position of a rigid body, which is constrained to one dimension and experiences a restoring force that is linearly proportional to its displacement from an equilibrium position. This form for the force or torque implies that the equation of motion for the point particle or rigid body will have the form:
 Latex
}{composition-setup}

{excerpt:hidden=true}{*}System:* One [point particle] constrained to move in one dimension. --- *Interactions:* The particle must experience a force (or torque) that attempts to [restore|restoring force] it to equilibrium and is directly proportional to its displacement from that equilibrium.{excerpt}

h4. Introduction to the Model

h5. Description and Assumptions

This [model] applies to [position] of a single [point particle], or to the [angular position] of a [rigid body], which is constrained to one dimension and experiences a [restoring force] that is linearly proportional to its displacement from an [equilibrium position].  This form for the force or torque implies that the equation of motion for the [point particle] or [rigid body] will have the form:
{latex}\begin{large}\[ a = \frac{d^{2}x}{dt^{2}} = - \omega_{\rm osc}^{2}x \]\end{large}{latex}or
{latex}
or
 Latex
\begin{large}\[ \alpha = \frac{d^{2}\theta}{dt^{2}} = -\omega_{\rm osc}^{2}\theta.\] \end{large}{latex}

As 
...
 a 
...
 consequence 
...
 of 
...
 this 
...
 characteristic 
...
 equation, 
...
 the 
...
 position, 
...
 velocity, 
...
 and 
...
 acceleration 
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 (or 
...
 the 
...
 angular 
...
 equivalents) 
...
 will 
...
 each 
...
 be 
...
sinusoidal 
...
 functions
...
 of 
...
 time. 
...
 Simple 
...
 harmonic 
...
 motion 
...
 is 
...
 sometimes 
...
 abbreviated 
...
 SHM, 
...
 or 
...
 referred 
...
 to 
...
 as 
...
 "Simple 
...
 Harmonic 
...
 Oscillation" 
...
 (SHO).
...
Learning 
...
 Objectives
...
Students 
...
 will 
...
 be 
...
 assumed 
...
 to 
...
 understand 
...
 this 
...
 model 
...
 who 
...
 can:
...
  • Define 
...
  •  the 
...
  •  terms 
...
...
...
  •  and 
...
...
...
  • .
...
  • Define 
...
  •  the 
...
...
  • , 
...
...
  • , 
...
...
...
...
  •  and 
...
...
  •  of 
...
  •  oscillatory 
...
  •  motion.
...
  • Give 
...
  •  a 
...
  •  formula 
...
  •  for 
...
  •  the 
...
...
...
...
  •  of 
...
  •  the 
...
  •  oscillation 
...
  •  of 
...
  •  a 
...
...
  •  or 
...
...
...
...
...
  • .
...
  • Write 
...
  •  mathematical 
...
  •  expressions 
...
  •  for 
...
  •  the 
...
...
  • , 
...
...
  •  and 
...
...
  •  of 
...
  •  Simple 
...
  •  Harmonic 
...
  •  Motion 
...
  •  as 
...
  •  functions 
...
  •  of 
...
  •  time 
...
  •  for 
...
  •  the 
...
  •  special 
...
  •  cases 
...
  •  that 
...
  •  the 
...
  •  initial 
...
  •  velocity 
...
  •  is 
...
  •  zero 
...
  •  or 
...
  •  the 
...
  •  initial 
...
  •  position 
...
  •  is 
...
  •  equilibrium.
...
  • Graphically 
...
  •  represent 
...
  •  the 
...
  •  position, 
...
  •  velocity 
...
  •  and 
...
  •  acceleration 
...
  •  of 
...
  •  Simple 
...
  •  Harmonic 
...
  •  Motion.
...
  • Use 
...
  •  the 
...
  •  laws 
...
  •  of 
...
...
  •  to 
...
  •  determine 
...
  •  the 
...
...
...
...
  •  of 
...
  •  a 
...
...
  •  in 
...
  •  the 
...
  •  limit 
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  •  of 
...
  •  very 
...
  •  small 
...
  •  displacements 
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  •  from 
...
  •  equilibrium.
...
  • Describe 
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  •  the 
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  •  consequences 
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  •  of 
...
...
  •  of 
...
...
...
  •  for 
...
  •  Simple 
...
  •  Harmonic 
...
  •  Motion 
...
  •  (assuming 
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  •  no 
...
  •  dissipation).
...
Relevant 
...
 Definitions
 Section


{section}{column}

{panel:title=Amplitude|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF|borderColor=#FFFFFF|titleBGColor=#FFFFFF}
{center}{latex}
 Column
 Panel
borderColor#FFFFFF
bgColor#FFFFFF
borderWidth1
titleBGColor#FFFFFF
titleAmplitude
borderStylesolid
 Center
 Latex
\begin{large}\[ A \equiv x_{\rm max} = \sqrt{x_{i}^{2} + \left(\frac{v_{i}}{\omega_{\rm osc}}\right)^{2}} \]\end{large}
{latex}{center}{panel}

{column}{column}
{panel:title=Phase|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF|borderColor=#FFFFFF}
{center}
{latex}
 Column
 Panel
borderColor#FFFFFF
bgColor#FFFFFF
borderWidth1
titlePhase
borderStylesolid
 Center
 Latex
\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{i}}{A}\right) = \sin^{-1}\left(\frac{v_{i}}{\omega_{\rm osc} A}\right)\]\end{large}
{latex}
{center}{panel}

{column}{section}


h4. 
S.I.M. 
...
 Structure 
...
 of 
...
 the 
...
Model
Compatible Systems
A single point particle (or, 
...
 for 
...
 the 
...
 angular 
...
 version 
...
 of 
...
 SHM, 
...
 a 
...
 single 
...
rigid 
...
 body
...
).
...
Relevant 
...
 Interactions
...
The 
...
 system 
...
 must 
...
 be 
...
 subject 
...
 to 
...
 a 
...
 one-dimensional 
...
 restoring 
...
 force 
...
 (or 
...
 torque) 
...
 that 
...
 varies 
...
 linearly 
...
 with 
...
 the 
...
 displacement 
...
 (or 
...
 angular 
...
 displacement) 
...
 from 
...
 a 
...
 stable 
...
 equilibrium 
...
 position.
Laws of Change
Mathematical Representation
 Panel
borderWidth1
titleUsing Initial Time
borderStylesolid
 Section



h4. Laws of Change

h5. Mathematical Representation

{panel:title=Using Initial Time|borderWidth=1|borderStyle=solid}
{panel}
{section}{column}
{panel:title=Position|bgColor=#FFFFFF}
{center}{latex}
 Column
 Panel
bgColor#FFFFFF
titlePosition
 Center
 Latex
\begin{large}\[ x(t) = x_{i}\cos(\omega_{\rm osc} (t-t_{i})) + \frac{v_{i}}{\omega_{\rm osc}}\sin(\omega_{\rm osc} (t-t_{i}))\]\end{large}
{latex}{center}
{panel}

{column}{column}

{panel:title=Velocity|bgColor=#FFFFFF}
{center}{latex}
 Column
 Panel
bgColor#FFFFFF
titleVelocity
 Center
 Latex
\begin{large}\[ v(t) = -\omega_{\rm osc} x_{i}\sin(\omega_{\rm osc} (t-t_{i})) + v_{i}\cos(\omega_{\rm osc} (t-t_{i}))\]\end{large}
{latex}{center}
{panel}


{
 Section
column}{section}
{section}{column}
\\
{panel:title=Acceleration|bgColor=#FFFFFF}
{center}{latex}
 Column

 Panel
bgColor#FFFFFF
titleAcceleration
 Center
 Latex
\begin{large}\[ a(t) = -\omega_{\rm osc}^{2} x_{i}\cos(\omega_{\rm osc} (t-t_{i})) - \omega_{\rm osc} v_{i} \sin(\omega_{\rm osc} (t-t_{i})) = -\omega_{\rm osc}^{2} x(t) \]\end{large}
{latex}{center}
{panel}
{column}{section}
\\
\\

{panel:title=Using Phase|borderWidth=1|borderStyle=solid|bgColor=#F0F0F0}
{panel}
{section}{column}
{panel:title=Position|bgColor=#FFFFFF}
{center}{latex}


 Panel
bgColor#F0F0F0
borderWidth1
titleUsing Phase
borderStylesolid
 Section
 Column
 Panel
bgColor#FFFFFF
titlePosition
 Center
 Latex
\begin{large}\[ x(t) = A\cos(\omega_{\rm osc} t + \phi)\]\end{large}
{latex}{center}
{panel}

{column}{column}

{panel:title=Velocity|bgColor=#FFFFFF}
{center}{latex}
 Column
 Panel
bgColor#FFFFFF
titleVelocity
 Center
 Latex
\begin{large}\[ v(t) =-A\omega_{\rm osc}\sin(\omega_{\rm osc} t + \phi)\]\end{large}
{latex}{center}
{panel}

{column}{column}

{panel:title=Acceleration|bgColor=#FFFFFF}
{center}{latex}
 Column
 Panel
bgColor#FFFFFF
titleAcceleration
 Center
 Latex
\begin{large}\[ a(t) =-\omega_{\rm osc}^{2}A\cos(\omega_{\rm osc} t+\phi) =-\omega_{\rm osc}^{2} x(t)\]\end{large}
Diagrammatic Representations
Image Added
Click here for a Mathematica Player application illustrating these representations using phase.
Image Added
Click here to download the (free) Mathematica Player from Wolfram Research
Relevant Examples
...
 Toggle Cloak
idPend
 Examples involving Pendulums
...
 Cloak
idPend
{latex}{center}
{panel}

{column}{section}

h5. Diagrammatic Representations

* [position versus time graph]
* [velocity versus time graph]
* [acceleration versus time graph]


|[!images^MathematicaPlayer.png!|^SHM-Phase.nbp]|[Click here|^SHM-Phase.nbp] for a _Mathematica Player_ application illustrating these representations using _phase_.|

|[!images^download_now.gif!|http://www.wolfram.com/products/player/download.cgi]|[Click here|http://www.woldfram.com/products/player/download.cgi] to download the (free) _Mathematica Player_ from [Wolfram Research|http://www.wolfram.com]|


h4. Relevant Examples

h6. {toggle-cloak:id=Pend} Examples involving Pendulums

{cloak:id=Pend}
{contentbylabel:AND50falsetrueexample_problem,SHM,pendulum|operator=AND|maxResults=50|showSpace=false|excerpt=true}
{cloak:Pend}
h6. {toggle-cloak:id=
Spr
} 
 Examples 
 
 involving 
 
 Springs


{cloak:id=Spr}
{contentbylabel: Spr AND50falsetrueexample_problem,SHM,spring|operator=AND|maxResults=50|showSpace=false|excerpt=true}
{cloak:Spr}
h6. {toggle-cloak:id=
RelEx
} 
 All 
 
 Examples 
 
 involving 
 
 this 
 
 Model


{cloak:id=RelEx}
{contentbylabel: RelEx AND50falsetrueexample_problem,SHM|operator=AND|maxResults=50|showSpace=false|excerpt=true}
{cloak}