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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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h1. Simple Harmonic Motion
{excerpt:hidden=true}*System:* One [point particle] constrained to move in one dimension. --- *Interactions:* The acceleration must be a [sinusoidal function] of time.{excerpt}
h4. {toggle-cloak:id=desc} Description and Assumptions
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This model applies to a single [point particle] constrained to move in one dimension whose position is a sinusoidal function of time. Simple harmonic motion is sometimes abbreviated SHM.
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h4. {toggle-cloak:id=cues} Problem Cues
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Any object that experiences a _linear_ restoring force or torque so that the equation of motion takes the form
{latex}\begin{large}\[ a = \frac{d^{2}x}{dt^{2}} = - \omega^\omega_{\rm osc}^{2}x \]\end{large}{latex}
or
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or
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\begin{large}\[ \alpha = \frac{d^{2}\theta}{dt^{2}} = -\omega^omega_{\rm osc}^{2}\theta.\] \end{large}{latex}
will experience simple harmonic motion with angular frequency ω. The prototypical example is an object of mass _m_ attached to a spring with force constant _k_, in which case, by [Hooke's Law]:
{latex}\begin{large}\[ a = -\frac{kx}{m} \]\end{large}{latex}
giving simple harmonic motion with angular frequency {latex}$\sqrt{\dfrac{k}{m}}${latex}.
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h4. {toggle-cloak:id=pri} Prior Models
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* [1-D Motion (Constant Velocity)]
* [1-D Motion (Constant Acceleration)]
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h4. {toggle-cloak:id=voc} Vocabulary
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* [restoring force]
* [periodic motion]
* [angular frequency]
* [phase]
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h2. Models
h4. {toggle-cloak:id=sys} {color:red}Compatible Systems{color}
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A single [point particle|point particle] (or, for the angular version of SHM, a single [rigid body]).
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h4. {toggle-cloak:id=int} {color:red}Relevant Interactions{color}
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The system must be subject to a one-dimensional restoring force (or torque) that varies linearly with the displacement (or angular displacement) from an equilibrium position.
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h4. {toggle-cloak:id=def} {color:red} Relevant Definitions{color}
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h5. Initial Conditions
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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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x_{0} = x(t=0) = -\frac{a(t=0)}{\omega^{2}}\qquad\qquad\qquad\]
\[ v_{0} = v(t=0)\]\end{large}{latex}
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h5. Amplitude of Motion
{latex}\begin{large}\[ A A \equiv x_{\rm max} = \sqrt{x_{ |
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00i}}{\omega_{\rm osc}}\right)^{2}} |
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\qquad\qquad\qquad\{latex} \\
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h5. Phase
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\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{ |
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0i}}{A}\right) = \sin^{-1}\left(\frac{v_{ |
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0i}}{\omega_{\rm osc} A}\right)\]\end{large} |
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h4. {toggle-cloak:id=laws} {color:red}Laws of Change{color}
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h5. Position
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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}\[ x(t) = x_{ |
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0i}\cos(\omega_{\rm osc} (t-t_{i})) + \frac{v_{ |
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0i}}{\omega_{\rm osc}}\sin(\omega_{\rm osc} (t-t_{i}))\ |
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qquad\qquad\qquad\{latex}
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or, equivalently
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x A\cos -\omega_{\rm osc} x_{i}\sin(\omega_{\rm osc} (t-t_{i})) + |
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\phiv_{i}\cos(\omega_{\rm osc} (t-t_{i}))\]\end{large} |
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h5. Velocity
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va(t) = -\omega_{\rm osc}^{2} x_{ |
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0sincos(\omega_{\rm osc} (t-t_{i}) |
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+0cossin(\omega_{\rm osc} (t-t_{i})) |
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\qquad\qquad\qquad = -\omega_{\rm osc}^{2} x(t) \]\end{large} |
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or, equivalently:
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v-omega\sincos(\omega_{\rm osc} t + \phi)\]\end{large} |
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h5. Acceleration
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a omega^{2} x_{0}\cos(\omega t) - \omega v_{0} \sin(\omega t) = -\omega^{2} x(t) \qquadomega_{\rm osc}\sin(\omega_{\rm osc} t + \phi)\]\end{large} |
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or, equivalently:
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\begin{large}\[ a(t) = |
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\omega^\omega_{\rm osc}^{2}A\cos(\omega_{\rm osc} t+\phi) = |
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\omega^\omega_{\rm osc}^{2} x(t)\]\end{large} |
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h4. {toggle-cloak:id=diag} {color:red}Diagrammatical Representations{color}
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* Acceleration versus time graph.
* Velocity versus time graph.
* Position versus time graph.
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h2. Relevant Examples
None yet.
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Diagrammatic Representations
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| 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 |