Page History

{composition-setup}{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}\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 (or the angular equivalents) will each be [sinusoidal functions|sinusoidal function] of time. Simple harmonic motion is sometimes abbreviated SHM, or referred to as "Simple Harmonic Oscillation" (SHO).

h5. Learning Objectives

{toggle-cloak:id=obj}Students will be assumed to understand this model who can:

{cloak:id=obj}

* Define the terms [equilibrium position] and [restoring force].
* Define the [amplitude], [period], [natural|natural frequency] [angular frequency] and [phase] of oscillatory motion.
* Give a formula for the [natural|natural frequency] [angular frequency] of the oscillation of a [pendulum] or [mass on a spring].
* Write mathematical expressions for the [position], [velocity] and [acceleration] 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 [dynamics] to determine the [natural|natural frequency] [angular frequency] of a [system] in the limit of very small displacements from equilibrium.
* Describe the consequences of [conservation|conserved] of [mechanical energy] for Simple Harmonic Motion (assuming no dissipation).

{cloak:obj}


h4. S.I.M. Structure of the Model


h5. Compatible Systems

A single [point particle|point particle] (or, for the angular version of SHM, a single [rigid body]).

h5. 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.


h4. Relevant Definitions

{section}{column}

{panel:title=Amplitude|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF|borderColor=#FFFFFF|titleBGColor=#FFFFFF}
{center}{latex}\begin{large}\[ A = \sqrt{x_{i}^{2} + \left(\frac{v_{i}}{\omega_{0}}\right)^{2}} \]\end{large}{latex}{center}{panel}

{column}{column}
{panel:title=Phase|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF|borderColor=#FFFFFF}
{center}
{latex}\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{i}}{A}\right) = \sin^{-1}\left(\frac{v_{i}}{\omega_{0} A}\right)\]\end{large}{latex}
{center}{panel}

{column}{section}


h4. S.I.M. Structure of the Model


h5. Compatible Systems

A single [point particle|point particle] (or, for the angular version of SHM, a single [rigid body]).

h5. 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.


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}\begin{large}\[ x(t) = x_{i}\cos(\omega_{0} (t-t_{i})) + \frac{v_{i}}{\omega_{0}}\sin(\omega_{0} (t-t_{i}))\]\end{large}{latex}{center}
{panel}

{column}{column}

{panel:title=Velocity|bgColor=#FFFFFF}
{center}{latex}\begin{large}\[ v(t) = -\omega_{0} x_{i}\sin(\omega_{0} (t-t_{i})) + v_{i}\cos(\omega_{0} (t-t_{i}))\]\end{large}{latex}{center}
{panel}


{column}{section}
{section}{column}
\\
{panel:title=Acceleration|bgColor=#FFFFFF}
{center}{latex}\begin{large}\[ a(t) = -\omega_{0}^{2} x_{i}\cos(\omega_{0} (t-t_{i})) - \omega_{0} v_{i} \sin(\omega_{0} (t-t_{i})) = -\omega_{0}^{2} x(t) \]\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}\begin{large}\[ x(t) = A\cos(\omega_{0} t + \phi)\]\end{large}{latex}{center}
{panel}

{column}{column}

{panel:title=Velocity|bgColor=#FFFFFF}
{center}{latex}\begin{large}\[ v(t) =-A\omega_{0}\sin(\omega_{0} t + \phi)\]\end{large}{latex}{center}
{panel}

{column}{column}

{panel:title=Acceleration|bgColor=#FFFFFF}
{center}{latex}\begin{large}\[ a(t) =-\omega_{0}^{2}A\cos(\omega_{0} t+\phi) =-\omega_{0}^{2} x(t)\]\end{large}{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:example_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:example_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:example_problem,SHM|operator=AND|maxResults=50|showSpace=false|excerpt=true}
{cloak}