hukum gravitasi semesta

THE LAW OF GRAVITY
UNIVERSE
Newton's Law of Universal Gravitation
Every particle in nature attract other particles with a large force is
directly proportional to the product of the two particles and inversely
proportional to the square of the distance between the two masses
m1
r̂12
F12
F
m1m2
r2
G
r12
m2
F21
F G
m1m2
r2
gravitational constant
2
11 N  m
6.672  10
2
kg
mm
F12  G 1 2 2 r̂12
r12
F3  F31  F32
F3  F312  F322  2 F31 F32 cos
F21  F12
massa bumi
How does the gravitational force by the ball-shaped mass?
The force of gravity on the mass m on the surface of the earth
:
F G
M Bm
RB2
Jari-jari bumi
Heavy Objects and Gravitational Force
6.672  10
 6.38  10 m
6
Heavy objects on
the earth's surface
M m
F  G B2
RB
W  mg
g G
11
N  m2
kg 2
 5.98  1024 kg
MB
 9.80 m s 2
2
RB
How heavy objects at a height h from the surface of the earth?
Jarak benda
ke pusat bumi
M Bm
r2
r  RB  h
F G
F G
M Bm
( RB  h ) 2
W   mg 
g  G
MB
( RB  h ) 2
The farther from the earth's surface, the smaller the acceleration of gravity
Kepler's laws
Is it true the earth around the sun?
Why are the planets around the sun?
How to track the orbit of the planets?
1. All the planets circulating in a path elip with the sun as focus.
2. Vector position of each planet to the sun in the same time interval the same swept area.
3. Square of each planet's orbital period is proportional to the cube of the major axis trajectory .
Does Newton's Law of Gravitation in accordance with this statement
a
Suppose the orbit of the planet to the sun is a circle :
M M M P M Pv2
G

2
r
r
M
2
G M  2r T 
r
2r T
KM
 4 2
T  
 GM M
2
 3
 r

c
F1
b
F1
Kepler's Second Law and the Conservation of Angular Momentum
MP
r
v
Moment of force :
τ  r  F  r  F (r )rˆ  0
dL
Always go to orbit centerτ 
L  konstan
0
dt
F
MM
L  r  p  m (r  v )
?
The area exposed to r in time interval dt
dA
dr  vdt
r
MM
h

dr
dA  12 rh
h  dr sin 
dA  12 rdr sin   12 r  dr
 12 r  vdt
dA L
rv  2 
dt m
dA L

= konstan
dt 2m
r
In the same time interval r position sweep the same area
Gravity Field and Gravity Potential
The force experienced
F
g
m
Gravity field :
by the test mass m in the gravitational field g
Medan Gravitasi bumi : g B 
P
Gaya terpusat
F Selalu menuju ke O
dW  F  dr  F ( r )dr
dr
F  F (r )rˆ
Effort only depends on the
initial position danUs end
W   F ( r )dr
r2
r1
Q
r1
U  U f  Ui   r F ( r )dr
r2
r2
1
F
O
r1
RB
F
GM
  2 B rˆ
m
r
m
F
r2
dr
 1 

GM
m
U f  U i  GM B m
B
2
 r  r
r r
 1 1
U f  U i  GM B m     
 rf ri 

r2
1
F
r2
GM B m
rˆ
r2
U (r) 
GM B m
r
1
Energi potensial massa m
pada posisi r
Energy Planet and Satellite Motion
v
Mm
r
GMm mv 2
Law,s Newton II :

2
r
r
E  12 mv 2  G
r
m
M
1
2
E G
mv 2 
GMm
2r
Mm
Mm   GMm
G
2r
2r
r
What is the minimum speed of the object to escape Earth's gravity?
vf  0
M Bm
M Bm
2
1
h  rmak  RB
mv

G


G
2
i
RB
rmak
h
rmak
vi
m
 1
1
vi2  2GM B  
 RB rmak
rmak  
M
vesc



2GM B

RB