Simple harmonic motion

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Simple harmonic motion.
Simple harmonic motion.

Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. The motion is periodic, as it repeats itself at standard intervals in a specific manner - described as being sinusoidal, with constant amplitude. It is characterized by its amplitude which is always positive and depends on how motion starts initially, its period which is the time for a single oscillation, and its phase which depends on displacement as well as velocity of the moving object.

One definition of simple harmonic motion is "motion in which the acceleration of the oscillator is proportional to, and opposite in direction to the displacement from its equilibrium position", or a \propto -x.

Image:Simple harmonic motion.png

A general equation describing simple harmonic motion is x(t) = A\sin \left( 2\,\pi \,ft+\gamma\right), where x is the displacement, A is the amplitude of oscillation, f is the frequency, t is the elapsed time, and γ is the phase of oscillation. If there is no displacement at time t = 0, the phase γ = 0. A motion with frequency f has period T=\frac{1}{f}.

Simple harmonic motion can serve as a mathematical model of a variety of motions and provides the basis of the characterisation of more complicated motions through the techniques of Fourier analysis.

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It can be shown, by differentiating, exactly how the acceleration varies with time.

The displacement is given by the function

x(t) = A\sin \left( 2\pi \,ft+\gamma\right)

We then differentiate once to get an expression for the velocity at any time.

\frac{d\,x(t)}{dt} = \dot x = v(t) = A\left(2\pi f\right) \cos \left( 2\,\pi \,ft+\gamma\right)

And once again to get the acceleration at a given time.

\frac{d^2\,x(t)}{d t^2} = \ddot x = a(t) = - A \left( 2\pi f \right)^2 \sin \left( 2\,\pi \,ft+\gamma\right)

These results can of course be simplified, giving us an expression for acceleration in terms of displacement.

a = \ddot x = -\left( 2\pi \,f \right)^2 x

When and if total energy is constant and kinetic the formula E = \frac{kA^2}{2} applies for simple harmonic motion, where E is considered the total energy while all energy is in its kinetic form. A representing the mean displacement of the spring from its rest position in MKS units.

Simple harmonic motion is exhibited in a variety of simple physical systems and below are some examples:

Mass on a spring: A mass M attached to a spring of spring constant k exhibits simple harmonic motion in space with

\omega=2 \pi f = \sqrt{\frac{k}{M}}.\,

With ω representing angular frequency.

Alternately, if the other factors are known and the period is to be found, this equation can be used:

T= 2 \pi \sqrt{\frac{M}{k}}.

Uniform circular motion: Simple harmonic motion can in some cases be considered to be the one-dimensional projection of uniform circular motion. If an object moves with angular speed ω around a circle of radius R centered at the origin of the x-y plane, then its motion along the x and the y coordinates is simple harmonic with amplitude R and angular speed ω.

Mass on a pendulum: In the small-angle approximation, the motion of a pendulum is shown to approximate simple harmonic motion. The period of a mass attached to a string of length \ell with gravitation acceleration g is given by

T= 2 \pi \sqrt{\frac{\ell}{g}}.

This approximation is accurate only in small angles because of the expression for angular acceleration being proportional to the sine of position.

\ell m g sin(\theta)=I \alpha

With θ being small, sin(\theta) \approx \theta and therefore the expression becomes

\ell m g \theta=I \alpha

which makes angular acceleration directly proportional to θ, satisfying the definition of Simple Harmonic Motion

For an exact solution not relying on a small-angle approximation, see pendulum (mathematics).

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