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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 GroupThe Milky Way and the Andromeda galaxy Cluster: Hydra and Centaurus
5-2 Newton's law of gravitation
221
r
mmG F
Newton (23) —1665
22-11 /kgmN106.67=G
Cavendish’s experiment
Density Profile
14-3 Gravitation and the principle of superposition
FdF
FFFFF=F
1
n
2=i1i1n1413121
例 5-1 A discrete system (5 particles)
例 5-2 A particle and a uniform rod
L
M
dr
dmdm
r
GmdF ,
21
1
NdLd
MGmddLL
MGm
rL
MGmr
dr
L
MGmdr
L
M
r
GmdFF
dLd
dL
d
dL
d
101
11
21
21
11
100.3)(
]11
[]1
[
14-4 Gravitation near earth’s surface
22 r
GM,ama,F
r
Mm GF gg
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
例 5-3 A pulsar
%031.0)2
( 2
gg
g
a
R
Ta
ga
211 /102.9 smag
例 5-4 The difference in ag (a) In the vicinity of
Earth
r m 6 77 106.dr
r
GM2da ,
r
GMa
3E
g2E
g
263
Eg /1037.4 )70.1(
r
GM2da smm
(b) A black hole – The tidal force
hh RrmR 229,1095.2 4
2
3E
g
/5.14
)70.1(r
GM2da
sm
m
5-5 Gravitation inside EarthA uniform shell of matter exerts no net gravitational force on a particle inside it.例 5-5
Express delivery
KrrmG
r
rGm
r
MGmF
rVM
)3
4(
3
4
3
4
2
3
2
3
5-6 Gravitational potential energy
dxxF
dxxFxdxF
xdxFWU
r
GMmU
R
)(
))(180)(cos()(
)(
R
GMmx
GMmdx
x
GMmU
dxx
GMmxdxF
RR
][
)(
2
2
Potential energy and force
2)(
r
GMm
r
GMm
dr
d
dr
dUF
Escape velocity
R
GMv
R
GMmmvUK
2
0)(2
1 2
Some Escape Speeds
例 5-6 Asteroids
skmvR
GMmmv
R
GMmmv
UKUK
f
if
iiff
/16102
1
2
1 22
5-7 Planets and Satellites: Kepler’s Laws
1. The law of orbits.
elliptical orbits eccentricity semimajor(semiminor) axis
2. The law of areas.
22
2
2
1
2
1
dt
2
1
rdt
dr
dA
drdA
constant2dt
))((
))((2
m
LdA
mrrmr
mvrrpL
3. The law of periods.
32
2
222
)4
(
)2
())((
rGM
T
rT
mrmr
GMm
例 5-7 Finding the mass:
)4
(2
32
GT
rM
例 5-8 Comet Halley
97.0 ,
103.52
2 ,)4
(
12
3/12
2
eRaea
mRaR
aRRGMT
a
p
pa
pa
例 5-8 A binary system and a black hole
312
2
221
32
21
21
12
1221
4
)( ,
))((
rGTmm
m
mm
rmr
rmr
mGmF
holeblack a - 9
47.3109.6
2)6( ,
2
2
30
3
22
321
s
s
S
Mm
Mkg
G
Tv
mM
m
v
rT
312
2
221
32
21
21
12
1221
4
)( ,
))((
rGTmm
m
mm
rmr
rmr
mGmF
14-8 Satellites: Orbits and Energy
The potential energy r
GMmU
r
vm
r
GMm 2
2
The kinetic energy
r
GmmmvK
22
1 2
The total energy
orbit)(circular 2
2
Kr
GMmE
r
GMm
r
GMmUKE
The elliptic orbits:
orbit) l(elliptica 2
orbit)(circular 2
a
GmmE
Kr
GmmE
5-9 Einstein and Gravitation• General/special theory of relativity
• Principle of equivalence: gravitation and acceleration are
equivalent.
Gravitation is due to a curvature of spacetime that is caused by the masses.
Curvature of Space
敬請期待物理 VI– 流力
