[Model Hierarchy]
Description and Assumptions
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.
Problem Cues
Any object that experiences a linear restoring force or torque so that the equation of motion takes the form
\begin
[ a = \frac{d^
x}{dt^{2}} = - \omega^
x ]\end
or
\begin
[ \alpha = \frac{d^
\theta}{dt^{2}} = -\omega^
\theta] \end
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]:
\begin
[ a = -\frac
]\end
giving simple harmonic motion with angular frequency
$\sqrt{\dfrac
{m}}$
.
Page Contents |
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Prerequisite Knowledge
Prior Models
Vocabulary and Procedures
System
Constituents
A single point particle (or, for the angular version of SHM, a single rigid body).
State Variables
Time (t), position (x) , velocity (v) and acceleration (a) or their angular equivalents.
Interactions
Relevant Types
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.
Interaction Variables
Force (F) or the angular equivalent.
Model
Relevant Definitions
Initial Conditions
\begin
[ x_
= x(t=0) = -\frac
{\omega^{2}}]
[ v_
= v(t=0)]\end
Amplitude of motion:
\begin
[ A = \sqrt{x_
^
+ \left(\frac{v_{0}}
\right)^{2}}]\end
Phase:
\begin
[ \phi = \cos^{-1}\left(\frac{x_{0}}
\right) = \sin^{-1}\left(\frac{v_{0}}
\right)]\end
Laws of Change
Position:
\begin
[ x(t) = x_
\cos(\omega t) + \frac{v_{0}}
\sin(\omega t)]\end
or, equivalently
\begin
[ x(t) = A\cos(\omega t + \phi) ]\end
Velocity:
\begin
[ v(t) = -\omega x_
\sin(\omega t) + v_
\cos(\omega t)]\end
or, equivalently:
\begin
[ v(t) = -A\omega\sin(\omega t + \phi)]\end
Acceleration:
\begin
[ a(t) = -\omega^
x_
\cos(\omega t) - \omega v_
\sin(\omega t) = -\omega^
x(t) ]\end
or, equivalently:
\begin
[ a(t) = -\omega^
A\cos(\omega t+\phi) = -\omega^
x(t)]\end
Diagrammatical Representations
- Acceleration versus time graph.
- Velocity versus time graph.
- Position versus time graph.
Relevant Examples
None yet.
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RELATE wiki by David E. Pritchard is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License |
