You are viewing an old version of this page. View the current version.

Compare with Current View Page History

« Previous Version 97 Next »

Unknown macro: {table}
Unknown macro: {tr}
Unknown macro: {td}

Description and Assumptions

This model applies to position of a single point particle, or to the angle of a rigid body, constrained to move in one dimension that experiences a linear restoring force toward its equilibrium.  Consequently, its position, velocity, and acceleration will each be a sinusoidal function of time. Simple harmonic motion is sometimes abbreviated SHM, or referred to as "Simple Harmonic Oscillation" (SHO).

Problem Cues

Any object that experiences a linear restoring force or torque so that the equation of motion takes the form

Unknown macro: {latex}

\begin

Unknown macro: {large}

[ a = \frac{d^

Unknown macro: {2}

x}{dt^{2}} = - \omega_

Unknown macro: {0}

^

x ]\end

or

Unknown macro: {latex}

\begin

Unknown macro: {large}

[ \alpha = \frac{d^

Unknown macro: {2}

\theta}{dt^{2}} = -\omega_

Unknown macro: {0}

^

\theta] \end

will experience simple harmonic motion with natural angular frequency ω0. The most common systems whose equations of motion take this form are a mass on a spring or a pendulum (in the small-angle approximation). Any problem requesting or giving position or velocity vs. time for one of these systems will likely require the use of the Simple Harmonic Motion model.

Another cue that Simple Harmonic Motion is occurring is if the position, the velocity, or the acceleration are sinusoidal in time (in which case they all will be sinusoidal).

Learning Objectives

Students will be assumed to understand this model who can:

Models

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.

Relevant Definitions

Amplitude
Unknown macro: {center}
Unknown macro: {latex}

\begin

Unknown macro: {large}

[ A = \sqrt{x_

Unknown macro: {i}

^

Unknown macro: {2}

+ \left(\frac{v_{i}}{\omega_{0}}\right)^{2}} ]\end

Phase

Unknown macro: {center}
Unknown macro: {latex}

\begin

Unknown macro: {large}

[ \phi = \cos^{-1}\left(\frac{x_{i}}

Unknown macro: {A}

\right) = \sin^{-1}\left(\frac{v_{i}}{\omega_

Unknown macro: {0}

A}\right)]\end

Laws of Change

Using Initial Time
Position
Unknown macro: {center}
Unknown macro: {latex}

\begin

Unknown macro: {large}

[ x(t) = x_

Unknown macro: {i}

\cos(\omega_

Unknown macro: {0}

(t-t_

)) + \frac{v_{i}}{\omega_{0}}\sin(\omega_

Unknown macro: {0}

(t-t_

Unknown macro: {i}

))]\end

Velocity
Unknown macro: {center}
Unknown macro: {latex}

\begin

Unknown macro: {large}

[ v(t) = -\omega_

Unknown macro: {0}

x_

Unknown macro: {i}

\sin(\omega_

(t-t_

Unknown macro: {i}

)) + v_

\cos(\omega_

Unknown macro: {0}

(t-t_

Unknown macro: {i}

))]\end


Acceleration
Unknown macro: {center}
Unknown macro: {latex}

\begin

Unknown macro: {large}

[ a(t) = -\omega_

Unknown macro: {0}

^

Unknown macro: {2}

x_

Unknown macro: {i}

\cos(\omega_

(t-t_

Unknown macro: {i}

)) - \omega_

Unknown macro: {0}

v_

\sin(\omega_

Unknown macro: {0}

(t-t_

Unknown macro: {i}

)) = -\omega_

^

Unknown macro: {2}

x(t) ]\end

Using Phase
Position
Unknown macro: {center}
Unknown macro: {latex}

\begin

Unknown macro: {large}

[ x(t) = A\cos(\omega_

Unknown macro: {0}

t + \phi)]\end

Velocity
Unknown macro: {center}
Unknown macro: {latex}

\begin

Unknown macro: {large}

[ v(t) =-A\omega_

Unknown macro: {0}

\sin(\omega_

t + \phi)]\end

Acceleration
Unknown macro: {center}
Unknown macro: {latex}

\begin

Unknown macro: {large}

[ a(t) =-\omega_

Unknown macro: {0}

^

Unknown macro: {2}

A\cos(\omega_

t+\phi) =-\omega_

Unknown macro: {0}

^

Unknown macro: {2}

x(t)]\end

Diagrammatic Representations

Unable to render embedded object: File (HarmonicOscillation.nbp) not found.

[Click here for a Mathematica Player application illustrating these representations.]
Click here to download the (free) Mathematica Player from Wolfram Research

Relevant Examples

ExamplesinvolvingPendulums"> Examples involving Pendulums


Unknown macro: {search-box}



  • No labels