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Composition Setup |
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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:
Latex |
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}{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 |
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\begin{large}\[ \alpha = \frac{d^{2}\theta}{dt^{2}} = -\omega_{\rm osc}^{2}\theta.\] \end{large}{latex}
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As
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a
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consequence
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of
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this
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characteristic
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equation,
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the
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position,
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velocity,
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and
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acceleration
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(or
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the
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angular
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equivalents)
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will
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each
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be
...
...
...
of
...
time.
...
Simple
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harmonic
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motion
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is
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sometimes
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abbreviated
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SHM,
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or
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referred
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to
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as
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"Simple
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Harmonic
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Oscillation"
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(SHO).
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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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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}
h5. Relevant Definitions
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S.I.M.
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Structure
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of
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the
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Model
Compatible Systems
A single point particle (or,
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for
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the
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angular
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version
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of
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SHM,
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a
...
single
...
...
...
).
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Relevant
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Interactions
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The
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system
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must
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be
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subject
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to
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a
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one-dimensional
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restoring
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force
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(or
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torque)
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that
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varies
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linearly
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with
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the
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displacement
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(or
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angular
...
displacement)
...
from
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a
...
stable
...
equilibrium
...
position.
Laws of Change
Mathematical Representation
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h4. Laws of Change
h5. Mathematical Representation
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Diagrammatic Representations
Click here for a Mathematica Player application illustrating these representations using phase. |
Click here to download the (free) Mathematica Player from Wolfram Research |
Relevant Examples
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{latex}{center}
{panel}
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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}
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