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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:
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{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. Description and Assumptions
This model applies to position of a single [point particle], or to the angle of a rigid body, constrained to move in one dimension that experiences a linear restoring force toward its equilibrium. Consequently, its position, velocity, and acceleration will each be a sinusoidal function of time. Simple harmonic motion is sometimes abbreviated SHM, or referred to as "Simple Harmonic Oscillation" (SHO).
h4. Problem Cues
Any object that experiences a _linear_ restoring force or torque so that the equation of motion takes the form
{latex}\begin{large}\[ a = \frac{d^{2}x}{dt^{2}} = - \omega_{0\rm osc}^{2}x \]\end{large}{latex}or
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\begin{large}\[ \alpha = \frac{d^{2}\theta}{dt^{2}} = -\omega_{0\rm osc}^{2}\theta.\] \end{large}{latex}will experience simple harmonic motion with [natural|natural frequency] [angular frequency] ω{~}0~. 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 if the position, the velocity, or the acceleration are sinusoidal in time (in which case they all will be sinusoidal).
h4. 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. Models
h4. Compatible Systems
A single [point particle|point particle] (or, for the angular version of SHM, a single [rigid body]).
h4. 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
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As a consequence of this characteristic equation, the position, velocity, and acceleration (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 equilibrium position and restoring force.
- Define the amplitude, period, natural angular frequency and phase of oscillatory motion.
- Give a formula for the natural 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 angular frequency of a system in the limit of very small displacements from equilibrium.
- Describe the consequences of conservation of mechanical energy for Simple Harmonic Motion (assuming no dissipation).
Relevant Definitions
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\begin{large}\[ A \equiv x_{\rm max} = \sqrt{x_{i}^{2} + \left(\frac{v_{i}}{\omega_{ |
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0\rm osc}}\right)^{2}} \]\end{large} |
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\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{i}}{A}\right) = \sin^{-1}\left(\frac{v_{i}}{\omega_{ |
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0\rm osc} A}\right)\]\end{large} |
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h4. Laws of Change
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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
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\begin{large}\[ x(t) = x_{i}\cos(\omega_{ |
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0\rm osc} (t-t_{i})) + \frac{v_{i}}{\omega_{ |
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00\rm osc} (t-t_{i}))\]\end{large} |
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\begin{large}\[ v(t) = -\omega_{ |
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0\rm osc} x_{i}\sin(\omega_{ |
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0\rm osc} (t-t_{i})) + v_{i}\cos(\omega_{ |
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0\rm osc} (t-t_{i}))\]\end{large} |
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\begin{large}\[ a(t) = -\omega_{ |
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0\rm osc}^{2} x_{i}\cos(\omega_{ |
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0\rm osc} (t-t_{i})) - \omega_{ |
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0\rm osc} v_{i} \sin(\omega_{ |
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0\rm osc} (t-t_{i})) = -\omega_{ |
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0\rm osc}^{2} x(t) \]\end{large} |
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\begin{large}\[ x(t) = A\cos(\omega_{ |
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0\rm osc} t + \phi)\]\end{large} |
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\begin{large}\[ v(t) =-A\omega_{ |
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00\rm osc} t + \phi)\]\end{large} |
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\begin{large}\[ a(t) =-\omega_{ |
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0\rm osc}^{2}A\cos(\omega_{ |
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0\rm osc} t+\phi) =-\omega_{ |
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0\rm osc}^{2} x(t)\]\end{large} |
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Diagrammatic Representations
 Image Added
| Click here for a Mathematica Player application illustrating these representations using phase. |
Relevant Examples
...
Examples involving Pendulums...
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AND50falsetrueexample_problem,SHM,pendulum Pend Spr Examples involving Springs Spr AND50falsetrueexample_problem,SHM,spring Spr RelEx All Examples involving this Model RelEx AND50falsetrueexample_problem,SHM{latex}{center}
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h4. Diagrammatic Representations
* [position versus time graph]
* [velocity versus time graph]
* [acceleration versus time graph]
|[!images^MathematicaPlayer.png!|^HarmonicOscillation.nbp]|[Click here for a Mathematica Player application illustrating these representations.|^HarmonicOscillation.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]|
h1. Relevant Examples
h4. {toggle-cloak:id=Pend} Examples involving Pendulums
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{contentbylabel:example_problem,SHM,pendulum|operator=AND|maxResults=50|showSpace=false|excerpt=true}
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h4. {toggle-cloak:id=Spr} Examples involving Springs
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{contentbylabel:example_problem,SHM,spring|operator=AND|maxResults=50|showSpace=false|excerpt=true}
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h4. {toggle-cloak:id=RelEx} All Examples involving this Model
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{contentbylabel:example_problem,SHM|operator=AND|maxResults=50|showSpace=false|excerpt=true}
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