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Simple Harmonic Motion




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 y(t) = A\sin \left( 2\,\pi \,ft+\phi ight) , where y is the displacement, A is the amplitude of oscillation, f is the frequency, t is the elapsed time, and \phi is the phase of oscillation. If there is no displacement at time t = 0, the phase \phi = 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 .


REALIZATIONS


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= 2 \pi \sqrt{ rac{m}{k}}.

Uniform Circular Motion: Simple harmonic motion can in some cases be considered to be the one-dimensional class="copylinks">Projection Of [[uniform Circular Motion . If an object moves with angular speed \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.

Pendulum: In the Small Angle Approximation , the motion of a pendulum is simple harmonic motion.


SEE ALSO