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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), its Period which is the time for a single oscillation, its Frequency which is the number of cycles per unit time, and its Phase , which determines the starting point on the sine wave. The period, and its inverse the frequency, are constants determined by the overall system, while the amplitude and phase are determined by the initial conditions (position and Velocity ) of that system.


OVERVIEW

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 .

A general equation describing simple harmonic motion is x(t) = A\sin \left( 2\,\pi \,ft+\delta ight) , where x is the Displacement , A is the Amplitude of oscillation, f is the Frequency , t is the elapsed time, and \delta is the phase of oscillation. If there is no displacement at time t = 0, the phase \delta= 0. A motion with frequency ''f'' has Period T= rac{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 .


MATHEMATICS

It can be shown, by Differentiating , exactly how the acceleration varies with time. Using the Angular Frequency \omega, defined as

: \omega = 2 \pi f\,

the displacement is given by the function

: x(t) = A\sin \left( \omega t +\delta ight)

Differentiating once gives an expression for the velocity at any time.

: v(t) = rac{\mathrm{d}\,x(t)}{\mathrm{d}t} = A\omega \cos \left( \omega t+\delta ight)

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

: a(t) = rac{\mathrm{d}^2\,x(t)}{\mathrm{d} t^2} = - A \omega^2 \sin \left( \omega t+\delta ight)

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

: a(t) = -\omega^2 x(t) \,


EXAMPLES


undergoes simple harmonic motion.]]
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{ rac{k}{M}}.\,

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

: T= rac{1}{f} = 2 \pi \sqrt{ rac{M}{k}}.

The total energy is constant, and given by E = rac{kA^2}{2}, where E is the total energy.

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 frequency \omega 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 \omega.

Mass on a pendulum: In the Small-angle Approximation , the motion of a pendulum is approximated by simple harmonic motion. The period of a mass attached to a string of length \ell with gravitational acceleration g is given by

: T= 2 \pi \sqrt{ rac{\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( heta)=I \alpha

where I is the Moment Of Inertia ; in this case I = m\ell^2. When heta is small, \sin( heta) \approx heta and therefore the expression becomes

:\ell m g heta=I \alpha

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

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


SEE ALSO



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