chapter 5 the law of gravity pham hong quang

8/16/2019 Chapter 5 the Law of Gravity Pham Hong Quang

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Fundamental of Physics

PETROVIETNAM UNIVERSITY

FUNDAMENTAL SCIENCE DEPARTMENT

Hanoi, August 2012

Pham Hong QuangE-mail: [email protected]


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Chapter 5 The Law of Gravity

Pham Hong Quang Fundamental Science Department 2

5.1 Newton’s Law of Universal

Gravitation

5.2 Kepler’s Laws

5. Kepler’s !irst Law

5." Kepler’s #e$on% Law

5.5 Kepler’s &hir% Law

5.' &he Gravitational !iel%

5.( Gravitational Potential )nerg*

5.+ )nerg* an% #atellite ,otion


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5.1 Newton’s Law of Unversal Gravtaton


Every particle in the Universe attracts

every other particle with a force of:

1 2

12 122 ˆ

m m F G r

r

×= − × ×

r

G: Gravitational constant G = 6.673·10-11 N·!"#$!

1% !: asses of particles 1 an& !

': &istance separatin$ these particles

: (nit vector in r &irection12

r̂


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5.1 Newton’s Law of Universal Gravitation


•

)n 17*+ ,enryaven&ish eas(re& G.• he two sall spheresare /e& at the en&s of ali$ht horiontal ro&.

• wo lar$e asses wereplace& near the sallones.• he an$le of rotation

was eas(re& 2y the&eection of a li$ht2ea reecte& fro airror attache& to thevertical s(spension.


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5.1 Newton’s Law of Universal Gravitation


!in%ing g from

G

4hat happen if we ta#e in acco(nt the rotation of

the Earth 5

5 l


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5.2 Kepler’s Laws


eplers 8irst 9aw

ll planets ove in elliptical or2its with the ;(n

at one foc(s.

eplers ;econ& 9aw

he ra&i(s vector &rawn fro the ;(n to a

planet sweeps o(t e


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5.2 Kepler’s Laws


Notes -out

)llipses•F 1 an& F ! are each a foc(s

of the ellipse.

hey are locate& a

&istance c fro the

center.

he s( of r1 an& r!

reains constant.

• he lon$est &istance

thro($h the center is the


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5.2 Kepler’s Laws


he shortest &istancethro($h the center is theminor a/is.

b is the semi-minoraxis.

he e$$entri$it* of theellipse is &e/ne& as e = c "a.

8or a circle% e = 0 he ran$e of val(es ofthe eccentricity forellipses is 0 > e > 1.

he hi$her the val(eof e% the lon$er an&

Notes -out )llipses0

ont.


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5.3 Kepler’s First Law


circ(lar or2it is a special case of the $eneral

elliptical or2its.

ltho($h we &o not prove it here% )t is a &irectres(lt of the inverse s


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5.3 Kepler’s First Law


he ;(n is at one foc(s.)t is not at the center of the ellipse.Nothin$ is locate& at the other foc(s.

-phelion is the point farthest away fro the;(n. he &istance for aphelion is a B c.

8or an or2it aro(n& the Earth% this point iscalle& the apo$ee.

Perihelion is the point nearest the ;(n. he &istance for perihelion is a C c.

8or an or2it aro(n& the Earth% this point iscalle& the peri$ee.

Notes -out )llipses0 Planet

rits

54K l ’S dL


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5.4 Kepler’s Second Law


his law is a conse


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5.4 Kepler’s Second Law, cont.


55K l ’Th dL


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5.5 Kepler’s Thrd Law


his law can 2e pre&icte& fro the inverse s


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5.5 Kepler’s Third Law,cont.


his can 2e eten&e& to an elliptical or2it.'eplace r with a.'ee2er a is the sei-aor ais.

K s is in&epen&ent of the ass of the planet% an&

so is vali& for any planet.)f an o2ect is or2itin$ another o2ect% the val(e

of K will &epen& on the o2ect 2ein$ or2ite&.8or eaple% for the Doon aro(n& the Earth% K ;(n

is replace& with K earth.

2

2 3 3

Sun

4S

T a K a

GM

π = =

÷


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5.5 Kepler’s Third Law,cont.


Usin$ the &istance 2etween the Earth an& the

;(n% an& the perio& of the Earths or2it% eplers

hir& 9aw can 2e (se& to /n& the ass of the

;(n.

;iilarly% the ass of any o2ect 2ein$ or2ite&

can 2e fo(n& if yo( #now inforation a2o(t

o2ects or2itin$ it.

)/ample0 ,ass of the#un

2 3

Sun 2

4 r M

GT

π =


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5.6 The Gravitational Field


gravitational 3el%

eists at every point inspace.

he $ravitational /el& is &e/ne& as

“The gravitational eld is the gravitationalforce experienced by a test particle placed atthat point divided by the mass of the test

particle”. he presence of the test particle is not necessaryfor the /el& to eist..

g

m≡ F

gr


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5.6 The Gravitational Field


he $ravitational /el&

vectors point in the&irection of theacceleration a particlewo(l& eperience ifplace& in that /el&.

he a$nit(&e is that ofthe free fall accelerationat that location.


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5.7 Gravitational Potential Energy


he $ravitational force is conservative.

The change in gravitational potential energyof a system associated with a given

displacement of a member of the system is

dened as the negative of the internal work

done by the gravitational force on that

member during the displacement.( )∆ = − = −∫ f

i

r

f i

r

U U U F r dr


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s a particle oves fro

to % its $ravitationalpotential ener$y chan$es 2y∆U.hoose the ero for the$ravitational potentialener$y where the force isero.

his eans Ui = 0 where

r i = F∞

his is vali& only for r

RE an& not vali& for r >

( ) E GM m

U r r

= −


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Graph of the

$ravitational potential

ener$y U vers(s r for

an o2ect a2ove the

Earths s(rface.

he potential ener$y

$oes to ero as r

approaches in/nity.


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8or any two particles% the $ravitational potentialener$y f(nction 2ecoes

he $ravitational potential ener$y 2etween any twoparticles varies as 1"r.

'ee2er the force varies as 1"r !.

he potential ener$y is ne$ative 2eca(se the force is

attractive an& we chose the potential ener$y to 2eero at in/nite separation.n eternal a$ent (st &o positive wor# to increasethe separation 2etween two o2ects.

Gravitational Potential )nerg*0

General

1 2Gm m

U r

= −


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he total $ravitational potentialener$y of the syste is the s(over all pairs of particles.Each pair of particles contri2(tes

a ter of U.ss(in$ three particles:

he a2sol(te val(e of Utotal

represents the wor# nee&e& toseparate the particles 2y an

in/nite &istance.

#*stems with &hree or

,ore Parti$les

total 12 13 23

1 3 2 31 2

12 13 23

U U U U

m m m mm mG

r r r

= + +

= − + +

÷


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5.8 Energy and Satellite Motion


ss(e an o2ect of ass m ovin$ with a

spee& v in the vicinity of a assive o2ect ofass M.

M AA mlso ass(e M is at rest in an inertial referencefrae.

he total ener$y is the s( of the systes#inetic an& potential ener$ies.

otal ener$y E = K BU

)n a 2o(n& syste% E is necessarily less than 0.

21

2

MmE mv G

r = −


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he total echanical ener$y

is ne$ative in the case of a

circ(lar or2it.

he #inetic ener$y is positive

an& is e


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)s$ape #pee% from

)arthn o2ect of ass m is

proecte& (pwar& fro the

Earths s(rface with an

initial spee&% v i.

Use ener$y consi&erations

to /n& the ini( val(eof the initial spee& nee&e&

to allow the o2ect to ove

in/nitely far away fro the


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eca(se the total ener$y of the syste is constant

his epression can 2e (se& to calc(latethe ai( altit(&e h because wenow that

9ettin$

an& ta#in$ % weo2tain:

∞→maxr

esci vv =


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!ham "on# $uan# 2+

Thank you

chapter 5 the law of gravity pham hong quang

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