chapter 5 the law of gravity pham hong quang
TRANSCRIPT
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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
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5.1 Newton’s Law of Universal
Gravitation
5.2 Kepler’s Laws
5. Kepler’s !irst Law
5." Kepler’s #e$on% Law
5.5 Kepler’s &hir% Law
5.' &he Gravitational !iel%
5.( Gravitational Potential )nerg*
5.+ )nerg* an% #atellite ,otion
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5.1 Newton’s Law of Unversal Gravtaton
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Every particle in the Universe attracts
every other particle with a force of:
1 2
12 122 ˆ
m m F G r
r
×= − × ×
r
G: Gravitational constant G = 6.673·10-11 N·!"#$!
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
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•
)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
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!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
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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
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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
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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
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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
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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
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his law is a conse
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5.4 Kepler’s Second Law, cont.
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55K l ’Th dL
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5.5 Kepler’s Thrd Law
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his law can 2e pre&icte& fro the inverse s
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5.5 Kepler’s Third Law,cont.
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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.
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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
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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
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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
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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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5.7 Gravitational Potential Energy
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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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5.7 Gravitational Potential Energy
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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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5.7 Gravitational Potential Energy
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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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5.7 Gravitational Potential Energy
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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$les
total 12 13 23
1 3 2 31 2
12 13 23
U U U U
m m m mm mG
r r r
= + +
= − + +
÷
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5.8 Energy and Satellite Motion
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ss(e an o2ect of ass m ovin$ 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$y E = K BU
)n a 2o(n& syste% E is necessarily less than 0.
21
2
MmE mv G
r = −
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5.8 Energy and Satellite Motion
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he total echanical ener$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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5.8 Energy and Satellite Motion
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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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5.8 Energy and Satellite Motion
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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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5.8 Energy and Satellite Motion
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!ham "on# $uan# 2+
Thank you