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Introduction to the Model

Description and Assumptions

This model applies to position of a single point particle, or to the angular position of a rigid body, which is constrained to one dimension and experiences a restoring force that is linearly proportional to its displacement from an equilibrium position. This form for the force or torque implies that the equation of motion for the point particle or rigid body will have the form:

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\begin

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[ a = \frac{d^

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

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^

x ]\end

or

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\begin

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[ \alpha = \frac{d^

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

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^

\theta.] \end

As a consequence of this characteristic equation, the position, velocity, and acceleration (or the angular equivalents) will each be sinusoidal functions of time. Simple harmonic motion is sometimes abbreviated SHM, or referred to as "Simple Harmonic Oscillation" (SHO).

Learning Objectives

Students will be assumed to understand this model who can:

Relevant Definitions
Amplitude
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\begin

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[ A \equiv x_

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= \sqrt{x_

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^

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

Phase
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\begin

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

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

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A}\right)]\end

S.I.M. Structure of the Model

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.

Laws of Change

Mathematical Representation
Using Initial Time
Position
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\begin

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[ x(t) = x_

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\cos(\omega_

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(t-t_

)) + \frac{v_{i}}{\omega_{\rm osc}}\sin(\omega_

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(t-t_

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))]\end

Velocity
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\begin

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[ v(t) = -\omega_

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x_

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\sin(\omega_

(t-t_

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)) + v_

\cos(\omega_

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(t-t_

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))]\end


Acceleration
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\begin

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[ a(t) = -\omega_

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^

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x_

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\cos(\omega_

(t-t_

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)) - \omega_

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v_

\sin(\omega_

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(t-t_

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)) = -\omega_

^

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x(t) ]\end



Using Phase
Position
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\begin

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

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t + \phi)]\end

Velocity
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\begin

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[ v(t) =-A\omega_

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\sin(\omega_

t + \phi)]\end

Acceleration
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\begin

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[ a(t) =-\omega_

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^

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A\cos(\omega_

t+\phi) =-\omega_

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^

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x(t)]\end

Diagrammatic Representations

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illustrating these representations using phase.

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Relevant Examples

ExamplesinvolvingPendulums"> Examples involving Pendulums
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