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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^

x ]\end

or

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\begin

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[ \alpha = \frac{d^

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\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 for elastic interactions:

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\begin

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[ a = -\frac

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]\end

giving simple harmonic motion with angular frequency

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$\sqrt{\dfrac

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{m}}$

In the real world, most situations in which there is a stable position with a restoring force can be treated (for small oscillations at least) as if the restoring force is linear. Therefore, even systems with non-ideal springs and the oscillations of a pendulum can be treated as cases of Simple Harmonic Motion as long as the motion is small.

Another cue that Simple Harmonic Motion is occurring is if the poision, 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

Angular Frequency

The angular frequency of oscillation of a simple oscillator is represent by a small Greek letter omega. For a simple mass m on a spring of constant k the frequency is given by

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\begin

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[ \omega = \sqrt{\frac

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{m}} ]\end

For a simple pendulum of length L making small oscillations under the influence of gravity (with the acceleration due to gravity denoted by g) the frequency is

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\begin

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[ \omega = \sqrt{\frac

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{L}} ]\end

Initial Conditions
Acceleration


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\begin

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[a_

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= a(t = t_

) ]\end

Velocity


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\begin

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[v_

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= v(t = t_

) ]\end

Position


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\begin

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[x_

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= x(t = t_

) = -\frac{a_{i}}{\omega^{2}} ]\end

Amplitude of Motion


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\begin

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[ A = \sqrt{x_

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^

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

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\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}}

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\right)]\end

Laws of Change

Position
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\begin

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[ x(t) = x_

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\cos(\omega t) + \frac{v_{i}}

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\sin(\omega t)]\end


or, equivalently

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\begin

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[ x(t) = A\cos(\omega t + \phi)]\end

Velocity
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\begin

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[ v(t) = -\omega x_

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\sin(\omega t) + v_

\cos(\omega t)]\end


or, equivalently:

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\begin

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[ v(t) = -A\omega\sin(\omega t + \phi)]\end

Acceleration
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\begin

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[ a(t) = -\omega^

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x_

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\cos(\omega t) - \omega v_

\sin(\omega t) = -\omega^

x(t) ]\end


or, equivalently:

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\begin

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[ a(t) = -\omega^

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A\cos(\omega t+\phi) = -\omega^

x(t)]\end

Diagrammatical Representations

Relevant Examples

All Examples involving this Model


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