Contoh Gerak Harmonik

Apa itu gerak “per iodik” ?????????? “harmonik” ????????? “Getaran” ??????????

Gerak periodik, adalah gerak berulang pada

waktu yang tetap.

Getaran, adalah gerak bolak-balik pada

jalan yang sama.

Gerak harmonik, adalah gerak dengan

persamaan berupa fungsi

sinus.

Hooke's Law

One of the properties of elasticity is that it takes about twice as much force to stretch a spring twice as far. That linear dependence of displacement upon stretching force is called Hooke's law.

Contoh gerak harmonis

Gaya yang bekerja : - Gaya balik : F = - kx - Gaya Newton : F = ma

Dalam kondisi setimbang :

F = - kx =

atau

atau

Persamaan ini dipenuhi oleh fungsi “sinusoidal”.

2

2

dt

xdm

02

2

=+ kxdt

xdm

kxdt

xdm −=

2

2

Bentuk umum persamaan :

Jika didiferensialkan dua kali di dapat :

dan

( ) ( )δω += tAx t cos

( ) ( ) ( )δωδω +−=+= tAtAdt

dtx

dt

dsincos

( ) ( )δωω +−= tAtxdt

dcos2

2

2

Sehingga didapat diferensial kedua dari

adalah :

( ) ( )δω += tAx t cos

( ) ( )δωω +−= tAtxdt

dcos2

2

2

Arti fisis dari tetapan dapat dilihat dalam persamaan :

Jadi, fungsi kembali pada nilai semula setelah selang waktu ( = T )

( ){ }δωπω ++= /2cos tAx

( )δπω ++= 2cos tA

( )δω += tAcos

ωπ /2

ω

Besaran disebut fasa dari gerak harmonik.

Tetapan disebut tetapan fasa.

( )δω +t

δ( ) tAx t ωcos1 =

( ) ( )02 180cos += tAx t ω

( ) tx At ωcos23 =

( ) tAx t ω2cos4 =

( )tx1

( )tx1

( )tx1

( )tx2

( )tx3

( )tx4

Simple Harmonic Motion

When a mass is acted upon by an elastic force which tends to bring it back to its equilibrium configuration, and when that force is proportional to the distance from equilibrium (e.g., doubles when the distance from equilibrium doubles, a Hooke's Law force), then the object will undergo simple harmonic motion when released.

A mass on a spring is the standard example of such periodic motion. If the displacement of the mass is plotted as a function of time, it will trace out a pure sine wave. It turns out that the motion of the medium in a traveling wave is also simple harmonic motion as the wave passes a given point in the medium.

Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's Law. The motion is sinusoidal in time and demonstrates a single resonant frequency.

Simple Harmonic Motion Equations

The motion equation for simple harmonic motion contains a complete description of the motion, and other parameters of the motion can be calculated from it.

The velocity and acceleration are given by

Simple Pendulum

The motion of a simple pendulum is like simple harmonic motion in that the equation for the angular displacement is

which is the same form as the motion of a mass on a spring:

The anglular frequency of the motion is then given by

compared to for a mass on a spring.

The frequency of the pendulum in Hz is given by

and the period of motion is then

Period of Simple Pendulum

A point mass hanging on a massless string is an idealized example of a simple pendulum.

When displaced from its equilibrium point, the restoring force which brings it back to the center is given by:

For small angles θ, we can use

the approximation

in which case Newton's 2nd law takes the form

Even in this approximate case, the solution of the equation uses calculus and differential equations. The differential equation is

and for small angles θ the solution is:

Pendulum Geometry

Pendulum Equation

The equation of motion for the simple pendulum for sufficiently small amplitude has the form

which when put in angular form becomes

This differential equation is like that for the simple harmonic oscillator and has the solution:

Deskripsi Gerak Harmonik Dengan Menggunakan Vektor :

Tugas !!!!!!!A

AAA

Gerak Harmonik Teredam & Terpaksa

Tugas !!!!!!!!!!

Analisa dalam bentuk matematik dengan caramu sendiri

Traveling Wave Relationship

A single frequency traveling wave will take the form of a sine wave. A snapshot of the wave in space at an instant of time can be used to show the relationship of the wave properties frequency, wavelength and propagation velocity.

Traveling Wave Relationship

The motion relationship "distance = velocity x time" is the key to the basic wave relationship. With the wavelength as distance, this relationship becomes =vT. Then using f=1/T gives the standard wave relationship

This is a general wave relationship which applies to sound and light waves, other electromagnetic waves, and waves in mechanical media.

String Wave Solutions

A solution to the wave equation for an ideal string can take the form of a traveling wave

For a string of length L which is fixed at both ends, the solution can take the form of standing waves:

For different initial conditions on such a string, the standing wave solution can be expressed to an arbitrary degree of precision by a Fourier series

Traveling Wave Solution for String

A useful solution to the wave equation for an ideal string is

It can be shown to be a solution to the one-dimensional wave equation by direct substitution:

Setting the final two expressions equal to each other and factoring out the common terms gives

These two expressions are equal for all values of x and t and therefore represent a valid solution if the wave velocity is

Wave velocity for a stretched string

String Traveling Wave Velocity

For a point of constant height moving to the right:

For a point of constant height moving to the left:

From the traveling wave solution, the phase velocity for a string wave is given by:

Traveling Wave Parameters

A traveling wave solution to the wave equation may be written in several different ways with different choices of related parameters. These include the basic periodic motion parameters amplitude, period and frequency.

Equivalent forms of wave solution:

Wave parameters:

*Amplitude A*Period T = 1/f

*Frequency f = 1/T*Propagation speed v

*Angular frequency ω = 2πf*Wave relationship v = fλ

Plane Wave ExpressionsA traveling wave which is confined to one plane in space and varies sinusoidally in both space and time can be expressed as combinations of

It is sometimes convenient to use the complex form

which may be shown to be a combination of the above forms by the use of the Euler identity

In the case of classical waves, either the real or the imaginary part is chosen since the wave must be real, but for application to quantum mechanical wavefunctions such as that for a free particle, the complex form may be retained.