Definitions

# harmonic motion

harmonic motion, regular vibration in which the acceleration of the vibrating object is directly proportional to the displacement of the object from its equilibrium position but oppositely directed. A single object vibrating in this manner is said to exhibit simple harmonic motion (SHM). More complex harmonic motion can be analyzed as combinations of two or more simple harmonic motions. Examples of objects whose motion approximates SHM are a pendulum swinging in a small arc, a mass bouncing at the end of a stretched spring, and air molecules vibrating back and forth as a sound wave passes. Simple harmonic motion is a periodic motion; that is, it repeats itself at regular intervals. The time required for one complete vibration of the object is the period of the motion. The inverse of the period is the frequency, which is the number of vibrations per unit of time. The maximum displacement of the object from its central position of equilibrium is the amplitude of the motion. At maximum displacement the velocity of the object is zero; the velocity is greatest when the object passes through its equilibrium position. These terms are commonly used to describe any periodic phenomenon, e.g., wave motion and the rotation or revolution of an astronomical body. For any real harmonic motion, various forces act to reduce the amplitude with each vibration, i.e., to damp the motion. If these forces are small compared to the restoring force arising from the original displacement, then the object will vibrate a number of times with successively smaller amplitudes until the motion gradually dies out; this is known as damped harmonic motion. For a certain value of the damping forces, the object returns to its original position in a minimum amount of time and comes to rest at that position; such motion is termed critically damped. If the damping forces are large compared to the restoring force, the object returns slowly to its original position without vibrating at all; the system is said to be overdamped.

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.

Simple harmonic motion is defined by the differential equation $mfrac\left\{d^2 x\right\}\left\{dt^2\right\} = -kx$ , where "k" is a positive constant, "m" is the mass of the body, and "x" is its displacement from the mean position.

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. $F=-kx$ ).

A general equation describing simple harmonic motion is $x\left(t\right) = Acos left\left(2,pi ,ft+phiright\right)$, where x 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 $phi= frac\left\{pi\right\}\left\{2\right\}$. A motion with frequency f has period, $T=frac\left\{1\right\}\left\{f\right\}$.

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.

## 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

$x\left(t\right) = Acos left\left(omega t +phiright\right).$

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

$v\left(t\right) = frac\left\{mathrm\left\{d\right\},x\left(t\right)\right\}\left\{mathrm\left\{d\right\}t\right\} = - Aomega sin left\left(omega t+phiright\right).$

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

$a\left(t\right) = frac\left\{mathrm\left\{d\right\}^2,x\left(t\right)\right\}\left\{mathrm\left\{d\right\} t^2\right\} = - A omega^2 cos left\left(omega t+phiright\right).$

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

$a\left(t\right) = -omega^2 x\left(t\right).$

$a\left(t\right) = -left\left(2pi f right\right)^2 x\left(t\right)$

When and if total energy is constant and kinetic, the formula $E = frac\left\{kA^2\right\}\left\{2\right\}$ 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.

## Examples

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

$omega=2 pi f = sqrt\left\{frac\left\{k\right\}\left\{M\right\}\right\}.,$

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

$T= frac\left\{1\right\}\left\{f\right\} = 2 pi sqrt\left\{frac\left\{M\right\}\left\{k\right\}\right\}.$

The total energy is constant, and given by $E = frac\left\{kA^2\right\}\left\{2\right\},$ 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

$T= 2 pi sqrt\left\{frac\left\{ell\right\}\left\{g\right\}\right\}$

This approximation is accurate only in small angles because of the expression for angular acceleration being proportional to the sine of position:

$ell m g sin\left(theta\right)=I alpha$

where I is the moment of inertia; in this case $I = mell^2$. When $theta$ is small, $sin\left(theta\right) approx theta$ and therefore the expression becomes

$ell m g theta=I alpha$

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

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

## Useful Formulae

Given mass $M$ attached to a spring/pendulum with amplitude $A$ with acceleration $a$:

$k = frac\left\{Ma\right\}\left\{A\right\}$
$f = frac\left\{A\right\}\left\{t\right\} = frac\left\{lambda\right\}\left\{t\right\}$
$T_s = T_p = frac\left\{1\right\}\left\{f\right\} = frac\left\{t\right\}\left\{A\right\} = 2 pi sqrt\left\{ frac\left\{M\right\}\left\{k\right\}\right\} = 2 pi sqrt\left\{ frac\left\{A\right\}\left\{g\right\}\right\} = 2 pi sqrt\left\{ frac\left\{ell\right\}\left\{g\right\}\right\}.$
$E_\left\{tot\right\} = frac\left\{kA^2\right\}\left\{2\right\} = frac\left\{MaA\right\}\left\{2\right\}.$

Where:

$k$ is the spring constant.
$M$ is the mass (usually in kilograms)
$a$ is the acceleration.
$A$ is the amplitude OR $lambda$ is the wavelength.
$f$ is the frequency (usually in hertz).
$t$ is the time in seconds to complete one cycle.
$T_s$ or $T_p$ is the period of the spring or pendulum.
$g$ is the acceleration due to gravity (On Earth at sea level: 9.81 m/s²).
$ell$ is the length of the pendulum.
$E_\left\{tot\right\}$ is the total energy.

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