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Simple Harmonic Motion

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, 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

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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

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 a 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.

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 an equilibrium position.

Relevant Definitions

Amplitude
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\begin

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[ A = \sqrt{x_

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^

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+ \left(\frac{v_{i}}{\omega_{0}}\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

Laws of Change

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_{0}}\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 x_

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\sin(\omega_

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(t-t_

)) + v_

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\cos(\omega_

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(t-t_

))]\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

Diagrammatical Representations

Relevant Examples

All Examples involving this Model


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