Simple harmonic motion

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), 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.

Overview

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

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. $F=-kx$ ).

The above differential equation has a sinusoidal solution, pictured below.

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

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

$\omega =2\pi \lambda \,$

the displacement is given by the function

$x(t)=A\cos \left(\omega t+\phi \right).$

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

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

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

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

The results illustrate that when a point on a vibrating string is at the 0,pi,2pi location in the vibration (a moment in time when the string is straight) the point is at maximum velocity and the acceleration is zero. However, when the point is at the location pi/2, the velocity is zero and the acceleration is at maximum.

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

$a(t)=-\omega ^{2}x(t).$

$a(t)=-\left(2\pi \,\lambda \right)^{2}x(t)$

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 its rest position in MKS units.

Examples

An undamped spring-mass system undergoes simple harmonic motion.

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

$\omega =2\pi \lambda ={\sqrt {\frac {k}{M}}}.\,$

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

$T={\frac {1}{\lambda }}=2\pi {\sqrt {\frac {M}{k}}}.$

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

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

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

$\ell mg\sin(\theta )=I\alpha$

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

$\ell mg\theta =I\alpha$

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 with amplitude A with acceleration a:

$k={\frac {Ma}{A}}$

$T_{s}=T_{p}={\frac {1}{\lambda }}=2\pi {\sqrt {\frac {M}{k}}}=2\pi {\sqrt {\frac {M}{\frac {Mg}{A}}}}=2\pi {\sqrt {\frac {A}{g}}}=2\pi {\sqrt {\frac {\ell }{g}}}.$

$E_{tot}={\frac {kA^{2}}{2}}.$

Where:

$k$ is the spring constant.

$M$ is mass.

$a$ is acceleration.

$A$ is amplitude.

$T_{s}$ or $T_{p}$ is the period of the spring or pendulum.

$\lambda$ is the frequency.

$g$ is the acceleration due to gravity on Earth $9.8{\frac {m}{s^{2}}}$.

$\ell$ is the length of the pendulum.

$E_{tot}$ is the total energy.

See also

• Isochronous
• Uniform circular motion
• Complex harmonic motion
• Damping

External links

• Java simulation of spring-mass oscillator
• Explanations and animations of Simple Harmonic Motion