Repetitive back-and-forth movement through a central, or equilibrium, position in which the maximum displacement on one side is equal to the maximum displacement on the other. Each complete vibration takes the same time, the period; the reciprocal of the period is the frequency of vibration. The force that causes the motion is always directed toward the equilibrium position and is directly proportional to the distance from it. A pendulum displays simple harmonic motion; other examples include the electrons in a wire carrying alternating current and the vibrating particles of a medium carrying sound waves.
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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.
In words, simple harmonic motion is "motion where the force acting on a body and thereby acceleration of the body is proportional to, and opposite in direction to the displacement from its equilibrium position" (i.e. ).
A general equation describing simple harmonic motion is , where x is the displacement, A is the amplitude of oscillation, f is the frequency, t is the elapsed time, and is the phase of oscillation. If there is no displacement at time t = 0, the phase . A motion with frequency f has period, .
Simple harmonic motion can serve as a mathematical model of a variety of motions and provides the basis of the characterization of more complicated motions through the techniques of Fourier analysis.
the displacement is given by the function
Differentiating once gives an expression for the velocity at any time.
And once again to get the acceleration at a given time.
These results can of course be simplified, giving us an expression for acceleration in terms of displacement.
When and if total energy is constant and kinetic, the formula 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.
Alternately, if the other factors are known and the period is to be found, this equation can be used:
The total energy is constant, and given by where E is the total energy.
This approximation is accurate only in small angles because of the expression for angular acceleration being proportional to the sine of position:
where I is the moment of inertia; in this case . When is small, and therefore the expression becomes
which makes angular acceleration directly proportional to , satisfying the definition of Simple Harmonic Motion.
For a solution not relying on a small-angle approximation, see pendulum (mathematics).
Given mass attached to a spring/pendulum with amplitude with acceleration :
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