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 characterised 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.
Encyclopedia
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 characterised 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 general equation describing simple harmonic motion is , where y 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 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 attached to a spring of spring constant exhibits simple harmonic motion in space with .
Alternately, if the other factors are known and the period is to be found, this equation can be used:
.
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 around a circle of radius centered at the origin of the
x-y plane, then its motion along the
x and the
y coordinates is simple harmonic with amplitude and angular speed .
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 with gravitation acceleration is given by:
See also
