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5 Gravitation

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5 Gravitation. The Milky Way galaxy and the black hole. www.ifa.hawaii.edu/~barnes/ast110_06/bhaq.html. 5 - 1 The world and the gravitational force. The Local Supercluster The Local Group The Milky Way and the Andromeda galaxy Cluster: Hydra and Centaurus. - PowerPoint PPT Presentation
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5 Gravitation
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Page 2: 5 Gravitation

The Milky Way galaxy and the black hole

Page 3: 5 Gravitation

www.ifa.hawaii.edu/~barnes/ast110_06/bhaq.html

Page 4: 5 Gravitation

5-1 The world and the gravitational force

•The Local Supercluster•The Local GroupThe Milky Way and the Andromeda galaxy Cluster: Hydra and Centaurus

Page 5: 5 Gravitation

5-2 Newton's law of gravitation

221

rmmG F

Newton (23) —1665

22-11 /kgmN106.67=G

Page 6: 5 Gravitation

Cavendish’s experiment

Page 7: 5 Gravitation

Density Profile

Page 8: 5 Gravitation

14-3 Gravitation and the principle of superposition

FdF

FFFFF=F

1

n

2=i1i1n1413121

Page 9: 5 Gravitation

例 5-1 A discrete system (5 particles)

Page 10: 5 Gravitation

例 5-2 A particle and a uniform rod

LM

drdmdm

rGmdF ,2

1 1

NdLd

MGmddLL

MGmrL

MGmrdr

LMGmdr

LM

rGmdFF

dLd

dL

d

dL

d

101

11

21

21

11

100.3)(

]11[]1[

Page 11: 5 Gravitation

14-4 Gravitation near earth’s surface

22 rGM,ama,F

rMm GF gg

Page 12: 5 Gravitation

Three reasons that 1. Earth is not

uniform.2. Earth is not a

sphere. 3. Earth is rotating.

gag

22

2

/034.0)2(

F

smRT

Rga

mamgma

maNma

g

g

g

Page 13: 5 Gravitation

例 5-3 A pulsar %031.0)2( 2

gg

g

aR

Taga

211 /102.9 smag

例 5-4 The difference in ag (a) In the vicinity of

Earth r m 6 77 106. dr

rGM2da ,

rGMa 3

Eg2

Eg

263

Eg /1037.4 )70.1(

rGM2da smm

Page 14: 5 Gravitation

(b) A black hole – The tidal force

hh RrmR 229,1095.2 4

2

3E

g

/5.14

)70.1(r

GM2da

sm

m

Page 15: 5 Gravitation

5-5 Gravitation inside EarthA uniform shell of matter exerts no net gravitational force on a particle inside it.例 5-5

Express delivery

Page 16: 5 Gravitation

KrrmGr

rGmr

MGmF

rVM

)3

4(

34

34

2

3

2

3

Page 17: 5 Gravitation

5-6 Gravitational potential energy

dxxFdxxFxdxF

xdxFWU

rGMmU

R

)( ))(180)(cos()(

)(

Page 18: 5 Gravitation

RGMm

xGMmdx

xGMmU

dxx

GMmxdxF

RR

][

)(

2

2

Page 19: 5 Gravitation

Potential energy and force

2)(r

GMmr

GMmdrd

drdUF

Escape velocity

RGMv

RGMmmvUK

2

0)(21 2

Page 20: 5 Gravitation

Some Escape Speeds

Page 22: 5 Gravitation

5-7 Planets and Satellites: Kepler’s Laws1. The law of orbits.

elliptical orbits eccentricity semimajor(semiminor) axis

Page 23: 5 Gravitation

2. The law of areas.

22

2

21

21

dt

21

rdtdrdA

drdA

Page 24: 5 Gravitation

constant2dt

))((

))((2

mLdA

mrrmr

mvrrpL

Page 25: 5 Gravitation

3. The law of periods.

32

2

222

)4(

)2())((

rGM

T

rT

mrmr

GMm

Page 26: 5 Gravitation

例 5-7 Finding the mass:

)4( 2

32

GTrM

例 5-8 Comet Halley

97.0 ,

103.52

2 ,)4

(

12

3/12

2

eRaea

mRaR

aRRGMTa

p

pa

pa

Page 27: 5 Gravitation
Page 28: 5 Gravitation

例 5-8 A binary system and a black hole

312

2

221

32

21

21

12

1221

4)(

,

))((

rGTmm

mmmrmr

rmr

mGmF

Page 29: 5 Gravitation

holeblack a - 947.3109.6

2)6( ,2

2

30

3

22

321

s

s

S

MmMkg

GTv

mMm

vrT

312

2

221

32

21

21

12

1221

4)(

,

))((

rGTmm

mmmrmr

rmr

mGmF

Page 30: 5 Gravitation

14-8 Satellites: Orbits and Energy

The potential energy r

GMmU

rvm

rGMm 2

2

The kinetic energy

rGmmmvK

221 2

Page 31: 5 Gravitation

The total energy

orbit)(circular 2

2

Kr

GMmE

rGMm

rGMmUKE

Page 32: 5 Gravitation

The elliptic orbits:

orbit) l(elliptica 2

orbit)(circular 2

aGmmE

Kr

GmmE

Page 33: 5 Gravitation

5-9 Einstein and Gravitation• General/special theory of relativity

• Principle of equivalence: gravitation and acceleration are

equivalent.

Page 34: 5 Gravitation

Gravitation is due to a curvature of spacetime that is caused by the masses.

Curvature of Space

Page 35: 5 Gravitation
Page 36: 5 Gravitation

敬請期待物理 VI– 流力


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