Transcript Chapter 13 Equilibrium

Bilingual Mechanics
Chapter 7
Gravitation
制作 张昆实
Yangtze University
Chapter 7
Gravitation
7-1 What Is Physics?
7-2 Newton's Law of Gravitation
7-3 Gravitation and the Principle of
Superposition
7-4 Gravitation Near Earth's Surface
7-5 Gravitation Inside Earth
7-6 Gravitational Potential Energy
7-7 Planets and Satellites: Kepler's Laws
7-8 Satellites: Orbits and Energy
7-1 What Is Physics
Have you ever
imaged how vast is
the universe?
The sun is one of
millions of stars
that form the Milky
Way Galaxy.
We are near the
edge of the disk of
the galaxy, about
26000 light-years
from its center.
Milky Way galaxy
7-1 What Is Physics
The universe is made
up of many galaxies,
each one containing
millions of stars.
One of the galaxies is
the Andromeda galaxy.
The great galaxy M31
in the Constellation
Andromeda is more than
100000 light-years across.
Andromeda galaxy
7-1 What Is Physics
★ The most distant galaxies are known to be over
10 billion light years away !
★ What force binds together these progressively
larger structures, from star to galaxy to
supercluster ?
★ It is the gravitational force that not only holds
you on Earth but also reaches out across
intergalactic space.
7-1 What Is Physics
The great steps
of China
toward the space
Lauching
Shenzhou five
(神州五号)
Space ship
7-1 What Is Physics
China CE-1 project
Exploring the Moon
Orbit around
the Moon :
2007-11-5
Lauching:
2007-10-24
shifting:
2007-11-1
Moon’s orbit
A report on exploring deep space & CE- project by
academician Ou Yang Ziyuan in Yangtze University
Academician Ou Yang Ziyuan present Yangtze
University with the all-around picture of the Moon
taking by CE-1
Chinese astronauts Jing Haipeng(L), Zhai Zhigang(C) and Liu Boming wave
hands during a press conference in Jiuquan Satellite Launch Center
(JSLC) in Northwest China's Gansu Province, September 24, 2008. The
Shenzhou VII spaceship will blast off Thursday evening from the JSLC to send
the three astronauts into space for China's third manned space mission.
China´s manned spacecraft Shenzhou-7 blasts off
Chinese taikonauts report they feel "physically sound"
Astronauts assemble EVA suit for spacewalk
Chinese astronaut Zhai Zhigang is ready
for spacewalk
Chinese astronaut Zhai Zhigang is traveling
in deep space
Congratulations to the
successful launching
of Shenzhou-7 !
The fundamental
principles of space
flight is Mechanics !
Physics is the cradle
of modern science
and technology !
7-2 Newton's Law of Gravitation
Nowton published the law of gravitation
In 1687. It may be stated as follows:
Every particle in the universe attracts every
other particle with a force that is directely
proportional to the product of the masses
of the particles and inversely proportinal
to the square of the distance between them.
Translating this into an equation
m1m2
F G 2
r
( Nowton’s law of gravitation )
(7-1)
7-2 Newton's Law of Gravitation
m1m2
F  G 2 ( Nowton’s law of gravitation ) (7-1)
r
G is the gravitational constant with a value of
11
G  6.67 10 N  m / kg
11 3
2
 6.67  10 m / kg  s
2
2
(7-2)
 F m2
m1 F
Particle 2 attracts particle 1
with F
r
Particle 1 attracts particle 2
Fig.14-2
with  F
forces are not
F and  F are equal in magnitude These
changed even if there are
but opposite in direction.
bodies lie between them
7-2 Newton's Law of Gravitation
Nowton’s law of gravitation applies
strictly to particles; also applies to
real objects as long as their sizes
are small compared to the distance
between them (Earth and Moon).
What about an apple and Earth?
Shell theorem:
A uniform spherical shell of matter
attracts a particle that is outside the
shell as if all the shell’s mass
were concentrated at its center.
m
msell
sell
7-3 Gravitation and the Principle
of Superposition
Given a group of n particles,
there are gravitational forces
between any pair of particles.
Finding the net force acting on
particle 1 from the others
First, compute the gravitational
force that acts on particle 1 due to
each of the other particles, in turn.
Then, add these forces vectorialy.
2
F1,net  F12  F13  F14  F15 
n
F1,net   F1i
i 2
(7-5)
3
F1,net
1 F1
For particleextented body
5
i
extented
body n
4
a group of n particles
 F1n
(7-4)
F1   dF
the Principle of Superposition
(7-6)
7-4 Gravitation Near Earth's Surface
A particle (m) locates outside Earth
a distance r from Earth’s center. The
magnitude of the gravitational force
from Earth (M) acting on it equals
Mm
F G 2
(7-7)
r
If the particle is releaced, it will fall
towards the center of Earth with the
gravitatonal acceleration a g :
F  mag
(7-8)
m
F
r
M
Gravitation Near
Earth's Surface
the gravitatonal acceleration
GM
ag  2
r
(7-9)
7-4 Gravitation Near Earth's Surface
the gravitatonal acceleration
GM
ag  2
r
(7-9)
m
r
M
7-1
Gravitation Near
Earth's Surface
7-4 Gravitation Near Earth's Surface
We have assumed that Earth is an
inertial frame (negnecting its actual
rotation). This allowed us to assume
the free-fall acceleration is the same
as the gravitational acceleration
g  ag
However
g  9.8m / s
2
m
M
g differs from ag  GM r 2 (7-9)
Weight mg differs from F  GMm r 2 (7-7)
Because: (1) Earth is not uniform,
(2) Earth is not a perfect sphere,
(3) Earth rotates.
7-4 Gravitation Near Earth's Surface
(1) Earth is not uniform
Crust
Oute
core
The density of Earth varies
radially:
Inner core 12-14 (103 kg/m3)
Outer core 10-12 (103 kg/m3)
Mantle
3-5.5 (103 kg/m3)
and the density of the crust
(outer section) of Earth
varies from region to region
over Earth’s surface.
Thus, g varies from region to
region over the surface.
Mantle
Inner
core
7-4 Gravitation Near Earth's Surface
(2) Earth is not a perfect sphere
Earth is approximately an ellipsoid,
flattened at the poles and bulging at
the equattor. Its equatorial radius is
greater than its polar radius by 21km.
M
equator
Thus, a point at the poles is closer
to the dense core of Earth than is a
point on the equator.
This is one reason the free-fall
acceleration g increases as one
proceeds, at sea level, from the
equator toward either pole.
reqt
M
rpol
7-4 Gravitation Near Earth's Surface
(3) Earth is rotating.
An object located on Earth’s surface anywhere
(except at two poles) must rotate in a circle about
the Earth’s rotation axis and thus have a centripital
acceleration ( requiring a centripital net force )
directed toward the center of the ciecle.
M
equator
How Earth’s rotation causes g to differ from a g?
Put a crate of mass m on a scale at the equator
r
and analyze it.
Free-body diagram
N
Normal force N
(outward in r direction )
Gravitational force ma g (inward in  r direction )
Centripital acceleration a (inward in  r direction )
a
Newton’s secend law for the r axis
N  mag  ma  m( R)
2
(7-10)
ma g
7-4 Gravitation Near Earth's Surface
(3) Earth is rotating.
Newton’s secend law for the r axis
FN  mag  ma  m( 2 R)
FN  mg
(7-10)
Reading on the scale
mg  mag  m( 2 R)
mearsure
weight
=
magnitude of
gravitation force
g  ag   2 R
Free-fall
acceleration
=
(7-12)
gravitation
acceleration
(7-11)
-
mass times
centripetal acceleration
Relation between
-
g and a g
centripetal
acceleration
ag  g   2 R  (2 (24  3600))2  6.37 106  0.034m / s 2
7-5 Gravitation Inside Earth
Newton’s shell theorem can also be applied
to a particle located Inside a uniform shell:
A uniform spherical shell of
matter exerts no net gravitational
force on a particle located inside it.
If a particle were to move into Earth, the
gavitational Force would change :
(1) It would tend to increase because the
particle would be moving closer to the
center of Earth.
(2) It would tend to decrease because the
thickening shell of material lying outside
the particle’s radial position would not
exert any net force on the particle.
msell
7-6 Gravitational Potential Energy
The gravitational potential energy of a particle-Earth
system
( particle on Earth’s surface )
U ( y | y 0 )  0
(4-55)
U ( y )  mgy
(4-55)
(P101)
However, we now choose a referance
configuration with U equal to zero as
the seperation distance
is large
enough to be approximated as infinite.
r
r  ,
At finite r ,
gravitational
potential energy
U 0
GMm
U 
r
U 0
r 
r
U 0
7-6 Gravitational Potential Energy
GMm (gravitational
U 
potential energy)
r
r  , U (r )  0
(7-17)
For any finite value of r , the
value of U ( r ) is negative.
The gravitational potential energy U is a
property of the system of the two
particles rather than of either particle along
However, for Earth and a apple, M
m
We often speak of “potential energy of the
apple”, because when a apple moves in
Ek (apple)
the vicinity of Earth, U sys
U 0
r 
r
U 0
7-6 Gravitational Potential Energy
GMm
U 
r
gravitational
potential energy
For a system of three particles, the
gravitational potential energy of
the system is the sum of the
gravitational potential energies of
all three pairs of particles.
U  U12  U13  U 23
(7-17)
m3
r13
m1
r23
r12
( calculating U as if the other
ij
particle were not there )
Gm1m2 Gm1m3 Gm2 m3
U  (


)
r12
r13
r23
(7-18)
m2
7-6 Gravitational Potential Energy
Find the gravitational
Proof of (7-17):
potential energy U of a ball at point P, at
radial distance R from Earth’s center.
The work done on the ball by the gravitational force as the ball travels from point P
to a great (infinite) distance from Earth is
r
dr
1800
Differential
displacement
m
F

W   F (r ) dr
(7-19)
R
F (r )  dr  F (r )dr cos 
W  

(7-20)
GMm
  2 dr
r
GMm
2
r
R
dr  
P
R
M
GMm 
r
R
GMm



R (7-21)
7-6 Gravitational Potential Energy
W  

GMm
2
r
R
dr  
GMm 
r
R
GMm

R
From Eq. 4-47 U  W

(7-21)
r
dr
1800
Differential
displacement
m
F
U   U  W
U  0
P
GMm
U W  
R
M
R
(7-17)
7-6 Gravitational Potential Energy
Path Independence
Moving a ball from A to G along a path :
consisting of three radial lengths and
three circular arcs (cented on Earth).
The work done by the gravitational force
on the ball as it moves along ABCDEFG:
G
F
H
E
WAG  WAB  WBC  WCD  WDE  WEF  WFG
The work done along each circular arc is
zero, because F  ds at every point.
WAG  WAB  WCD  WEF  WAG
the gravitational force is a conservative
force, the work done by it on a particle is
independent of the actual path taken
between points A and G.
D
C
B
A
Earth
7-6 Gravitational Potential Energy
From Eq. 4-47: U
 W
U  U f  U i  W
(4-47)
(7-22)
Since the work W done by a
conservative force is
independent of the actual path
taken.
The change U in gravitational
potential energy is also
independent of the actual path
taken.
G
F
H
E
D
C
B
A
Earth
7-6 Gravitational Potential Energy
potential energy and force
We derived the potential energy function
U (r ) from the force function F ( r ) .

GMm
U (r )  W (r )   F (r )  dr  
r
r
Now let’s go the other way: derive the force
function from the potential energy function
dU
d
GMm radially GMm
F 
  (
) 2
dr
dr
r inward r
(7-23)
This is Newton’s law of gravitation (7-1) .
( Derivation is the inverse operation of integration )
7-6 Gravitational Potential Energy
Escape Speed: The minimum initial speed that will
cause a projectile to move up forever is called the
(Earth) escape speed.
Consider a projectile (m ) leaving the surface of a

planet with escape speed v
2
1
Its kinetic energy
K  2 mv
Its potential energy U   GMm
r
v
R
When the projectile reches infinity, it stops.
Its kinetic energy
K 0
Its potential energy U  0
From the principle of
conservation of energy
Escape Speed:
K  U  12 mv 2  ( GMm
R )0
2GM
v
R
(7-24)
v
M
m
m
R
7-6 Gravitational Potential Energy
Escape Speed
From Earth:
v  11.2km / s
v
The escape speed
does
not depend on the direction
in which a projectile is fired
from a planet.
2GM
v
R
(7-24)
eastward
However, attaining that speed is easier if the
projectile is fired in the direction the launch site is
moving as the planet rotates about its axis .
For example, rockets are launched eastward at
XiChang to take the advantage of the eastward
speed of 1500km/h due to Earth’s rotation.
7-7 Planets and Satellites: Kepler's Laws
The motion of the planets have been a puzzle since the
dawn of history.
Johannes Kepler (1571-1630) worked out the empirical
laws that governed these motions based on the data
from the observations by Tycho Brahe (1546-1601).
1 THE LAW OF ORBITS: All planets move in
elliptical orbits, with the Sun at one focus.
a is the semimajor axis of the orbit
e is the eccentricity of the orbit
ea is the distance from the center
of the ellipse to either focus
the eccentricity
is only 0.0167
e of Earth’s orbit
m
r
M
F ea
a
ea
F
7-7 Planets and Satellites: Kepler's Laws
2 THE LAW OF AREAS: A line that connects a
planet to the Sun sweeps out equal areas in the
plane of the planet’s orbit in equal times; that is,
the rate dA/dt at which it sweeps out area A is
constant.
This second law tell us that
the planet will move most
slowly when it is farthest
from the Sun and most
rapidly when it is nearest to
the Sun.
Sun
M
7-7 Planets and Satellites: Kepler's Laws
Proof of Kepler’s second law is totally equivalent
to the law of conservation of angular momentum.
The area of the wedge
r
A  r  r   r 
1
2
1
2
2
The instantaneous rate at which
area is been sweept out is
dA 1 2 d 1 2
 r
 r 
dt
2
dt
2

Sun 
M
L  rp  r (mv )  r (mr )
dA
L

dt 2m
(7-27)
dA dt
L
A
(7-26)
The magnitude of the angular momentum of the planet about the Sun is
 mr 2
r
constant
constant
p
Sun 
M
r pr
p
m
7-7 Planets and Satellites: Kepler's Laws
3 THE LAW OF PERIODS: The square of the
period of any planet is proportional to the cube of
the semimajor axis of its orbit.
Applying Newton’s second law to
the orbiting planet :
F  ma
From
Eq. 11-20
GMm
2

m
(

r)
2
r
T
2

2

4
2
T 
 GM
m
r

(7-29)
M
2

T
 3 (7-30)
r

The quantity in parentheses is a
constant that depends only the
mass M of the central body
about which the planet orbits.
7-7 Planets and Satellites: Kepler's Laws
3 THE LAW OF PERIODS:
（水星）
（金星）
（地球）
（火星）
（木星）
（土星）
（天王星）
（海王星）
（冥王星）
T 2  4 2 


3
a
 GM 
(7-30)
7-8 Satellites: Orbits and Energy
As a satellite orbits Earth on its elliptical path, its speed
and the distance from the center of Earth fluctuate with
fixed periods. However, the mechanical energy E of the
satellite remains constant.
The potential energy of the
GMm
U 
system (or the satellite) is
r
To find the kinetic energy of the satellite, use Newton’s
second law
Compare U and K
GMm
v 2 (7-32)
F  ma
m
2
U (7-34)
r
r
K 
GMm
K  mv 
2r
1
2
2
2
(7-33)
( Circular orbit )
7-8 Satellites: Orbits and Energy
The total mechanical energy E of the satellite is
GMm GMm
E  K U 

2r
r
GMm
( circular orbit )
E
2r
(7-35)
Compare E and K
GMm
K
2r
For a satellite in an
elliptical orbit of
semimajor axis a
E  K
( circular orbit ) (7-36)
GMm
E
2a
( elliptical orbit ) (7-37)
7-8 Satellites: Orbits and Energy
GMm
U 
r
7-15
GMm
K
2r
7-16
GMm
E
2r