[Model Hierarchy]
Description and Assumptions
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^
x] \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 a system treated as a point particle with position specified by the center of mass).
State Variables
Time (t), position (x) , and velocity (v).
Interactions
Relevant Types
Some time-varying external influence that is confined to one dimension.
Interaction Variables
Acceleration (a(t)).
Model
Laws of Change
Differential Forms:
\begin
[ \frac
= a]\end
\begin
[ \frac
= v]\end
Integral Forms:
\begin
[ v(t) = v(t_
)+\int_{t_{0}}^
a\;dt]\end
\begin
[ x(t) = x(t_
)+\int_{t_{0}}^
v\;dt]\end
Diagrammatical Representations
- Acceleration versus time graph.
- Velocity versus time graph.
- Position versus time graph.
Relevant Examples
None yet.
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