[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
A single point particle (or, for the angular version of SHM, a single rigid body).
Interactions
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.
Model
Relevant Definitions
Initial Conditions
\begin
[ x_
= x(t=0) = -\frac
{\omega^{2}}]
[ v_
= v(t=0)]\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 |
