| Wiki Markup |
|---|
{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 angleposition] of a [rigid body], which is constrained to move in one dimension thatand experiences a linear [restoring force] towardthat is its equilibrium. Consequently,linearly proportional to its position,displacement velocity,from andan acceleration will each be a sinusoidal function of time. Simple harmonic motion is sometimes abbreviated SHM, or referred to as "Simple Harmonic Oscillation" (SHO).
h5. Problem Cues
A _linear_ restoring force or torque implies that the equation of motion takes the form[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 Any object subject to an equationa consequence of motion with this formcharacteristic willequation, experiencethe simpleposition, harmonicvelocity, motionand with [natural|natural frequency] [angular frequency] ω{~}osc~. The most common systems whose equations of motion take this form are a [mass on a spring] or a [pendulum] (in the small-angle approximation). Any problem requesting or giving position or velocity vs. time for one of these systems will likely require the use of the Simple Harmonic Motion model.
Another cue that Simple Harmonic Motion is occurring is a position, velocity, or acceleration that is a [sinusoidal function] of time (in which case they all will be sinusoidal).
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
Students will be assumed to understand this model who can:
* 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).
h1. Modelsh4. S.I.M. Structure of the Model
h4h5. Compatible Systems
A single [point particle|point particle] (or, for the angular version of SHM, a single [rigid body]).
h4h5. 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}
{panel}
{center}{latex}\begin{large}\[ A = \sqrt{x_{i}^{2} + \left(\frac{v_{i}}{\omega_{0}}\right)^{2}} \]\end{large}{latex}{center}{panel:Amplitude}
{panel}
{column}{column}
{panel:title=Phase|borderStyle=solid|borderWidth=1|bgColor=#FFFFFF}
{panel}{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:Phase}
{panel}
{column}{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}
{panel}
{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:Position}
{panel}
{column}{column}
{panel:title=Velocity|bgColor=#FFFFFF}
{panel}
{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:Velocity}
{panel}
{column}{section}
{section}{column}
\\
{panel:title=Acceleration|bgColor=#FFFFFF}
{panel}
{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:Acceleration}
{panel}
{column}{section}
{panel:Using}
{panel}
{panel:title=Using Phase|borderWidth=1|borderStyle=solid|bgColor=#F0F0F0}
{panel}
{section}{column}
{panel:title=Position|bgColor=#FFFFFF}
{panel}
{center}{latex}\begin{large}\[ x(t) = A\cos(\omega_{0} t + \phi)\]\end{large}{latex}{center}
{panel:Position}
{panel}
{column}{column}
{panel:title=Velocity|bgColor=#FFFFFF}
{panel}
{center}{latex}\begin{large}\[ v(t) =-A\omega_{0}\sin(\omega_{0} t + \phi)\]\end{large}{latex}{center}
{panel:Velocity}
{panel}
{column}{column}
{panel:title=Acceleration|bgColor=#FFFFFF}
{panel}
{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:Acceleration}
{panel}
{column}{section}
{panel:Using Phase}
{panel}
h4
h5. Diagrammatic Representations
* [position versus time graph]
* [velocity versus time graph]
* [acceleration versus time graph]
|[!images^MathematicaPlayer.png!|^SHM-Phase.nbp]|[Click here for a _Mathematica Player_ application \\ illustrating these representations using phase.|^SHM-Phase.nbp]|
|[!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]|
h1h4. Relevant Examples
h4h5. {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}
h4h5. {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}
h4h5. {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} |
Page History
Overview
Content Tools
Activity