Gerak Harmonik

Contoh Gerak Harmonik
Apa itu gerak “periodik” ?????????? “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
xd
m
02
2
=+ kx
dt
xd
m
kx
dt
xd
m −=2
2

Bentuk umum persamaan :
Jika didiferensialkan dua kali di dapat :
dan
( ) ( )δω += tAx t cos
( ) ( ) ( )δωδω +−=+= tAtA
dt
d
tx
dt
d
sincos
( ) ( )δωω +−= tAtx
dt
d
cos2
2
2

Sehingga didapat diferensial kedua dari
adalah :
( ) ( )δω += tAx t cos
( ) ( )δωω +−= tAtx
dt
d
cos2
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 =
( ) ( )0
2 180cos += tAx t ω
( ) tx A
t 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 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
A
A
A

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 Expressions
A 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.

Gerak Harmonik

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Gerak Harmonik