8/12/2019 @kul 1 SHM

1/35

GETARAN

AP Physics Chttp://www.jca.umbc.edu/~george/html/courses/phys224/2006spg/lectures/

FENOMENA ALAM,

PEMODELAN & DESKRIPSI FISIS

8/12/2019 @kul 1 SHM

2/35

GETARAN: definisi

Gerak tidak menjalartetapi memiliki pola ke-

berulang-andisekitar titik setimbang

yang ditinjau

Obyek Besaran Fisis

Gerak

benda masif(berukuranbesar/kasatmata) yangmengalamigerak bolak-

balikdisekitar titiksetimbang

posisi (x) (?)

Persamaangerak bolak-balik

kecepatan (v)

percepatan (a)

energi (E)

Pola ke-berulang-an

frekuensi (f) periode (T)

periodik: sinus/cosinus

tidak periodik (tidak dibahas)

8/12/2019 @kul 1 SHM

3/35

Oscillatory Motion is repetitive back and forth motionabout an equilibrium position

Oscillatory Motion isperiodic.

Swinging motion and vibrations are forms ofOscillatory Motion.

Objects that undergo Oscillatory Motion are called

Oscillators.

GETARAN: terminologi (per-istilah-an)

8/12/2019 @kul 1 SHM

4/35

FENOMENA ALAM: getaran teratur & acak

Contoh fenomena alam

dengan sifat gerak berulangyang teratur dan memenuhi

definisi gerak selaras

sederhana

Bila berlangsung pelan, fenomena

gerak ini dapat memenuhi kriteria

selaras sederhana, namun

kenyataannya tidaklah selalu

sederhana

Ilustrasi manakah (kiri / kanan) yang sesuai dengan definisi

gerak bolak-balik (getaran) sesuasi dengan definisi?

8/12/2019 @kul 1 SHM

5/35

EFEK (TERLIHAT) FENOMENA ALAM GETARAN: Gempa Vulkanik

Proses erupsi yang didahuluidengan gempa vulkanik

merupakan hasil dari getaran

hebat dapur magma yang dapat

dirasakan di sekitar area gunung

getaran hebat dapur magma yangdapat dirasakan hingga permukaan

ditangkap dengan seismograp.

Berakibat munculnya rekahan di

permukaan dan/atau di bawah

permukaan

8/12/2019 @kul 1 SHM

6/35

We use natural oscillations to measure time

1.pendulum

2.quartz crystals

Currently, we define 1 second based on

oscillations inside a Cesium atom:

1 second = 9,192,631,770 oscillations

(303) 499-7111: http://tf.nist.gov/

CC:BY-SAtacoekkel(flickr)http:/

/creativecommons.org/licenses/by-s

a/2.0/deed.en

GETARAN ALAMI

http://tf.nist.gov/http://tf.nist.gov/http://tf.nist.gov/http://tf.nist.gov/

8/12/2019 @kul 1 SHM

7/35

Pola Gerak Harmonik

Ilustrasi pola gerak getaran selaras(harmonik) sederhana.

Karena alasan penjabaran

(deksripsi) matematis, dua ilustrasi

bagian atas tidak dibahas.

Hanya gerak dengan pola sinusoidal(sin atau cos) saja yang dibahas

apa perbedaan pola gerak sinus

dan cosinus?

8/12/2019 @kul 1 SHM

8/35

8/12/2019 @kul 1 SHM

9/35

Deskripsi Fisis Umum GHS: Frekuensi (f )Periode (T )

The FREQUENCYof a wave is the inverse of the PERIOD.That means that the frequency is the #cycles per sec.

The commonly used unit is HERTZ(HZ).

Tf

fT

Hzs

ccycfFrequency

scycsTPeriod

11

5.05.0sec5.3

75.1

seconds

cycles

275.1

5.3cycles

seconds

8/12/2019 @kul 1 SHM

10/35

Model Getaran dengan pola GHS: sistem pegas berbeban

An objects maximum

displacement from its

equilibrium position is called the

Amplitude (A) of the motion.

8/12/2019 @kul 1 SHM

11/35

2cos

tx t A

T

We need a position function to

describe the motion above.

Model Getaran GHS: penjabaran fisis pola ke-berulang-an

8/12/2019 @kul 1 SHM

12/35

2cos tx t AT

cos 2x t A ft

cosx t A t

1T

f

2

T

x(t) to symbolize positionas a function of time

A =xmax =xmin

when t = T,

cos(2) = cos(0)

x(t) =A

Model Getaran GHS: besaran simpangan (x)

8/12/2019 @kul 1 SHM

13/35

sinv t A t

cosx t A t d x tv t

dt

In this context we will call

omega Angular Frequency

What is the physical

meaning of the product

(A)?

maxv A

The maximum speed

of an oscillation!

Model Getaran GHS: besaran kecepatan (v) ~ percepatan (a)

8/12/2019 @kul 1 SHM

14/35

When collecting and modeling data of SHM your

mathematical model had a value as shown below:

x(t) Acost

x(t) Acos tC What if your clock didnt start at x=A or x=-A?

This value represents our initial conditions.

We call it the phase angle:

x(t) Acos t

8/12/2019 @kul 1 SHM

15/35

The velocity is 90o

out of phase withthe displacement

The acceleration is

180oout of phasewith thedisplacement

8/12/2019 @kul 1 SHM

16/35

(sin )cos

(cos )sin

axa ax

x

axa ax

x

Review: FUNGSI SINUS

8/12/2019 @kul 1 SHM

17/35

A

TEquil. point

f=1/T

Review: FUNGSI SINUS(tidak semua gerak bolak-balik merupakan fungsi sinus yang mulus)

8/12/2019 @kul 1 SHM

18/35

It is aperiodic motion.

AND

It has arestoringforce thatacts to

restore the oscillator to equilibrium.

The restoring force is given by:

Hookes Law F=-kx

x is the displacement fromequilibrium and kis the force

constant (spring constant).

The period of SHM oscillator does

not depend on the amplitude.

Persamaan GHS: sistem pegas berbeban

8/12/2019 @kul 1 SHM

19/35

Here is what we want to do: DERIVE AN

EXPRESSION THAT DEFINES THE

DISPLACEMENT FROM EQUILIBRIUM OF THESPRING IN TERMS OF TIME.

0)(

2

2

2

xm

k

dt

xd

dt

xdmkx

dt

xd

amakx

maFkxF Netspring

WHAT DOES THIS MEAN? THE SECOND DERIVATIVE OF A FUNCTION

THAT IS ADDED TO A CONSTANT TIMES ITSELF IS EQUAL TO ZERO.

What kind of function will ALWAYS do this?

Persamaan GHS: sistem pegas berbeban: hukum HOOKE

8/12/2019 @kul 1 SHM

20/35

Since all springs exhibit properties

of circle motion we can use theseexpressions to derive the formula

for the period of a spring.

Besaran Fisis: sistem pegas berbeban

8/12/2019 @kul 1 SHM

21/35

apa dimensi dari T?

Besaran Fisis: sistem pegas berbeban: periode ~ frekuensi

PERHATIKAN

- Tatau ftidak bergantung pada nilai g- besaran (nilai) kmenggambarkan keras/lemahnya

lenturan pegas

- periode mengecil untuk pegas yang keras (tidak

mudah lentur) nilai kyang besar

8/12/2019 @kul 1 SHM

22/35

For springs in series:

1/keff= 1/k1+ 1/k2

For springs in

parallel:

keff= k1+ k2

8/12/2019 @kul 1 SHM

23/35

kis the slope of a

F versusx-graph

Besaran Fisis: sistem pegas berbeban: menentukan nilai k

8/12/2019 @kul 1 SHM

24/35

Persamaan GHS:Ayunan Matematis (Pendulum)

What length pendulum will have a period of exactly 1s?

8/12/2019 @kul 1 SHM

25/35

What length pendulum will have a period of exactly 1s?

g

L

T 2 L

g

g T

2

2

L

L9.8m/s2 1s

2

2

0.248m

8/12/2019 @kul 1 SHM

26/35

A physical pendulum is an oscillating body

that rotates according to the location of itscenter of mass rather than a simple

pendulum where all the mass is located at

the end of a light string.

oscillates about a fixed axis that doesnot pass through its center of mass,

and

the object cannot be approximated as

a point mass,

Persamaan GHS: Pendulum Fisis

8/12/2019 @kul 1 SHM

27/35

mgd

IT

TI

mgd

Imgd

ifImgd

LdIdmg

IFr

pendulumphysical

2

2,

0)(

sin,

2,sin

sin

It is im por tant to u nders tand

that d is the lever arm

distance or the distance from

the COM posi t ion to the point of

rotat ion.It is also the same d in

the Parallel Axes theorem.

8/12/2019 @kul 1 SHM

28/35

ExampleA spring is hanging from the ceiling. You know that if you

elongate the spring by 3.0 meters, it will take 330 N of

force to hold it at that position: The spring is then hungand a 5.0-kg mass is attached. The system is allowed toreach equilibrium; then displaced an additional 1.5 metersand released. Calculate the:

kkkxFs )3)((330Spring Constant

Angular frequency 5

1102mk

mk

110 N/m

4.7 rad/s

8/12/2019 @kul 1 SHM

29/35

29

A uniform rod of mass M and length

L is pivoted about one end andoscillates in a vertical plane.

Find the period of oscillation if the

amplitude of the motion is small.

8/12/2019 @kul 1 SHM

30/35

Example

Position of mass at maximum velocity

Maximum acceleration of the mass

Position of mass at maximum acceleration

At the equilibrium position

)5.1()7.4( 22Aa 33.135 m/s/s

At maximum amplitude, 1.5 m

A spring is hanging from the ceiling. You know that if you

elongate the spring by 3.0 meters, it will take 330 N of

force to hold it at that position: The spring is then hung

and a 5.0-kg mass is attached. The system is allowed to

reach equilibrium; then displaced an additional 1.5

meters and released. Calculate the:

8/12/2019 @kul 1 SHM

31/35

b is a constant called

the damping coefficient

Persamaan Gerak Harmonis tidak Sederhana:Ayunan Teredam (damped oscillations)

8/12/2019 @kul 1 SHM

32/35

is called the natural frequency

CC: BY murdoch666 (flickr) http://creativecommons.org/licenses/by/2.0/deed.en

A slowly changing

line that provides a

border to a rapid

oscillation is calledthe envelopeof the

oscillations.

8/12/2019 @kul 1 SHM

33/35

Damped Oscillation (oscillation with friction)

The friction reduces the mechanical energy of the system as time

passes, and the motion is said to be damped.

8/12/2019 @kul 1 SHM

34/35

DampedOscillations

8/12/2019 @kul 1 SHM

35/35

Exercises

Each:M/2, R

k

stretched by x

and then released

cylinders

rolls w/o

slipping

Ra

MRIIfRfrictionfkxfMa

cmcm

2)2/1(,

):(

.)2/3(/

2

2 xx

M

k

RIM

kxa

cm

.2

32

2

k

MT

f

kx