NationMaster - Encyclopedia: 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. In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hookes law: where is a positive constant. ... Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In trigonometry, an ideal sine wave is a waveform whose graph is identical to the generalized sine function y = Asin[ω(x − α)] + C, where A is the amplitude, ω is the angular frequency (2π/P where P is the wavelength), α is the phase shift, and C... Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle. ... Periodicity is the quality of occurring at regular intervals (e. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

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 File history File links Simple_harmonic_motion. ...

A general equation describing simple harmonic motion is $y(t) = Asin left( 2,pi ,ft+gammaright)$ , where y 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 $T=frac{1}{f}$ . Wiktionary has related dictionary definitions, such as: period Wiktionary has related dictionary definitions, such as: periodic Period and periodic may refer to: Period (music) Period (rhetoric) Historical period Menstrual cycle, relating to the reproductive system Full stop, also known as a period, that marks the end of a sentence Science...

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. A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...

1 Mathematics
2 Realizations
3 See also
4 External links

Mathematics

It can be shown, by differentiating, exactly how the acceleration varies with time. In mathematics, a derivative is the rate of change of a quantity (e. ...

The displacement is given by the function

$x(t) = Asin left( 2pi ,ft+gammaright)$

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

$frac{d,x(t)}{dt} = dot x = v(t) = Aleft(2pi fright) cos left( 2,pi ,ft+gammaright)$

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

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

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

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

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 it's rest position in MKS units.

Realizations

Simple harmonic motion is exhibited in a variety of simple physical systems and below are some examples: Image File history File linksMetadata Spring-pendulum. ... Image File history File linksMetadata Spring-pendulum. ...

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 $ω$ . The word projection can mean more than one thing. ... The realm of physics consists of two types of circular motion: uniform circular motion and non-uniform circular motion. ... Look up origin in Wiktionary, the free dictionary. ...

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 Small-angle approximation is a useful simplification of the laws of trigonometry which is only approximately true for finite angles, but correct in the limit as the angle approaches zero. ...

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

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). The mathematics of pendulums can be quite complex, but some formula and proofs are given below. ...

External links

Java simulation of spring-mass oscillator

Categories: Classical mechanics | Pendulums

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Encyclopedia > Simple harmonic motion

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Mathematics

Realizations

See also

External links