Page History

{composition-setup}{composition-setup}{table:cellspacing=0|cellpadding=8|border=1|frame=void|rules=cols}{tr:valign=top}{td:width=25px|bgcolor=#F2F2F2}
{live-template:Left Column}{td}{td}

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

{cloak:id=desc}
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.
{cloak}, or referred to as "Simple Harmonic Oscillation" (SHO).

h4. {toggle-cloak:id=cues} Problem Cues

{cloak:id=cues}
Any object that experiences a _linear_ restoring force or torque so that the equation of motion takes the form
{latex}\begin{large}\[ a = \frac{d^{2}x}{dt^{2}} = - \omega^{2}x \]\end{large}{latex}
or
{latex}\begin{large}\[ \alpha = \frac{d^{2}\theta}{dt^{2}} = -\omega^{2}\theta\] \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 for elastic interactions]:
{latex}\begin{large}\[ a = -\frac{kx}{m} \]\end{large}{latex}
giving simple harmonic motion with angular frequency
{latex}$\sqrt{\dfrac{k}{m}}${latex}

In the real world, most situations in which there is a stable position with a restoring force can be treated (for small oscillations at least) as if the restoring force is linear. Therefore, even systems with non-ideal springs and the oscillations of a pendulum can be treated as cases of Simple Harmonic Motion as long as the motion is small.

Another cue that Simple Harmonic Motion is occurring is if the poision, the velocity, or the acceleration are sinusoidal in time.  

.
{cloak}

h4. {toggle-cloak:id=pri} Prior Models

{cloak:id=pri}
* [1-D Motion (Constant Velocity)]
* [1-D Motion (Constant Acceleration)]
* [Mechanical Energy and Non-Conservative Work]

{cloak}

h4. {toggle-cloak:id=vocab} Vocabulary

{cloak:id=vocab}
* [restoring force]
* [periodic motion]
* [angular frequency]
* [phase]

{cloak}

h2Learning Objectives

Students will be assumed to understand this model who can:

* Define the terms [equilibrium position] and [restoring force].
* Define the [amplitude], [period], [angular frequency] and [phase] of oscillatory motion.
* 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 [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. {toggle-cloak:id=sys} {color:red}Compatible Systems{color}

{cloak:id=sys}
A single [point particle|point particle] (or, for the angular version of SHM, a single [rigid body]).
{cloak}

h4. {toggle-cloak:id=int} {color:red}Relevant Interactions{color}

{cloak:id=int}
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.
{cloak}

h4. {toggle-cloak:id=def} {color:red}Relevant Definitions{color}

{cloak:id=def}{section}{column}

h5. Frequency

The frequency of oscillation of a simple oscillator is represent by a small Greek letter omega. For a simple mass *m* on a spring of constant *k* the frequency is given by

{latex}\begin{large}\[ \omega = \sqrt{\frac{k}{m}} \]\end{large}{latex}

For a simple pendulum of length *L* making small oscillations under the influence of gravity (with the acceleration due to gravity denoted by *g*) the frequency is

{latex}\begin{large}\[ \omega = \sqrt{\frac{g}{L}} \]\end{large}{latex}

  

h5. Initial Conditions

h6. Acceleration
\\
{latex}\begin{large}\[a_{i} = a(t = t_{i}) \]\end{large}{latex}

h6. Velocity
\\
{latex}\begin{large}\[v_{i} = v(t = t_{i}) \]\end{large}{latex}

h6. Position
\\
{latex}\begin{large}\[x_{i} = x(t = t_{i}) = -\frac{a_{i}}{\omega^{2}} \]\end{large}{latex}


h5. Amplitude of Motion

\\
{latex}\begin{large}\[ A = \sqrt{x_{i}^{2} + \left(\frac{v_{i}}{\omega}\right)^{2}} \]\end{large}{latex}


h5. Phase
\\
{latex}\begin{large}\[ \phi = \cos^{-1}\left(\frac{x_{i}}{A}\right) = \sin^{-1}\left(\frac{v_{i}}{\omega A}\right)\]\end{large}{latex}

h4. {toggle-cloak:id=laws} {color:red}Laws of Change{color}

{cloak:id=laws}{section}

h5. Position

{latex}\begin{large}\[ x(t) = x_{i}\cos(\omega t) + \frac{v_{i}}{\omega}\sin(\omega t)\]\end{large}{latex}
\\
or, equivalently
\\
{latex}\begin{large}\[ x(t) = A\cos(\omega t + \phi)\]\end{large}{latex}

h5. Velocity

{latex}\begin{large}\[ v(t) = -\omega x_{i}\sin(\omega t) + v_{i}\cos(\omega t)\]\end{large}{latex}
\\
or, equivalently:
\\
{latex}\begin{large}\[ v(t) = -A\omega\sin(\omega t + \phi)\]\end{large}{latex}

h5. Acceleration

{latex}\begin{large}\[ a(t) = -\omega^{2} x_{i}\cos(\omega t) - \omega v_{i} \sin(\omega t) = -\omega^{2} x(t) \]\end{large}{latex}
\\
or, equivalently:
\\
{latex}\begin{large}\[ a(t) = -\omega^{2}A\cos(\omega t+\phi) = -\omega^{2} x(t)\]\end{large}{latex}
{cloak}

h4. {toggle-cloak:id=diag} {color:red}Diagrammatical Representations{color}

{cloak:id=diag}
* Acceleration[acceleration versus time graph.]
* Velocity[velocity versus time graph.]
* Position[position versus time graph.]

{cloak}h1. Relevant Examples

h2. {toggle-cloak:id=RelEx}Relevant All Examples involving this Model

{cloak:id=RelEx}
* [Lissajous Figures and the Bowditch Pendulum]
* [Big Ben]
* [Mass Between Two Springs]

{cloak} 



----
{search-box}
\\
\\ {td}{tr}{table}
{live-template:RELATE license}