Bilingual Mechanics. Chapter 7 Gravitation. 制作 张昆实 Yangtze University. Chapter 7 Gravitation. 71 What Is Physics? 72 Newton's Law of Gravitation 73 Gravitation and the Principle of Superposition 74 Gravitation Near Earth's Surface
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71 What Is Physics?
72 Newton's Law of Gravitation
73 Gravitation and the Principle of
Superposition
74 Gravitation Near Earth's Surface
75 Gravitation Inside Earth
76 Gravitational Potential Energy
77 Planets and Satellites: Kepler's Laws
78 Satellites: Orbits and Energy
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 lightyears
from its center.
Milky Way galaxy
Andromeda galaxy
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 lightyears across.
★The most distantgalaxies are known to be over
10 billionlight years away !
★What forcebinds together these progressively
larger structures, from star togalaxy to
supercluster ?
★It is the gravitational force that not only holds
you on Earthbut also reaches out across
intergalactic space.
China CE1 project
Exploring the Moon
Orbit around
the Moon:
2007115
Lauching:
20071024
shifting:
2007111
Moon’s orbit
A report on exploring deep space & CE project by
academician Ou Yang Ziyuan in Yangtze University
Academician Ou Yang Ziyuanpresent Yangtze Universitywiththe allaround picture of the Moon taking by CE1
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.
Chinese astronaut Zhai Zhigangis ready
for spacewalk
Congratulations to the successful launching of Shenzhou7 !
The fundamental principles of space flight is Mechanics !
Physics is the cradle of modern science and technology !
Translating this into an equation
(71)
( Nowton’s law of gravitation )
72 Newton's Law of GravitationNowton published the law ofgravitation
In 1687. It may be stated as follows:
Every particlein the universeattractsevery
other particlewitha forcethat isdirectely
proportionalto the product of the masses
of the particles andinversely proportinal
to the square of thedistance between them.
Particle 2 attracts particle 1
with
Particle 1 attracts particle 2
with
and are equal in magnitude
but opposite in direction.
Fig.142
These forces are notchanged even if there are bodies lie between them
(71)
( Nowton’s law of gravitation )
72 Newton's Law of Gravitationis the gravitational constant with a value of
What about an apple and Earth?
Shell theorem:
A uniform spherical shell of matter attracts a particle that is outside the shell as ifall the shell’s mass were concentrated at its center.
72 Newton's Law of GravitationNowton’s law of gravitation applies
strictly to particles; also appliesto
real objects as long as their sizes
are small compared to the distance
between them (Earth and Moon).
Given a group of nparticles, there are gravitational forces between any pair of particles.
1
1
Finding the net force acting on particle 1 from the others
extented body
First, compute the gravitational force that acts on particle 1 due to each of the other particles, in turn.
Then, add these forces vectorialy.
(74)
For particle
(76)
(75)
extented body
73 Gravitation and the Principle of Superposition3
the Principle of Superposition
5
2
i
n
4
a group of nparticles
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
(77)
If the particle is releaced, it will fall
towards the center of Earth with the
gravitatonal acceleration :
Gravitation Near Earth's Surface
(78)
the gravitatonal acceleration
(79)
74Gravitation Near Earth's SurfaceGravitation Near Earth's Surface
(79)
74Gravitation Near Earth's Surfacethe gravitatonal acceleration
We have assumed that Earth is an
inertial frame (negnecting its actual
rotation). This allowed us to assume
the freefall acceleration is the same
as the gravitationalacceleration
However differs from
(79)
Weight differs from
(77)
Because:
74Gravitation Near Earth's Surface(1)Earth is not uniform,
(2)Earth is not a perfect sphere,
(3)Earth rotates.
Oute core
Mantle
Inner core
Thus, varies from region to regionover the surface.
74Gravitation Near Earth's Surface(1)Earth is not uniform
Thedensity of Earth varies
radially:
Inner core 1214 (103 kg/m3)
Outer core 1012 (103 kg/m3)
Mantle 35.5 (103 kg/m3)
and the density of the crust (outer section) of Earth
varies from region to region
over Earth’s surface.
This is one reason the freefall acceleration increasesas one
proceeds, at sea level, from the equator toward either pole.
74Gravitation 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.
equator
Thus, a point at the poles is closer to the dense core of Earth than is a point on the equator.
How Earth’s rotationcauses to differ from ? Put a crate of mass on a scaleat the equator and analyze it.
Freebody diagram
Normal force (outward in direction )
Gravitational force (inward in direction )
Centripital acceleration (inward in direction )
Newton’s secend law for the axis
(710)
74Gravitation Near Earth's Surface(3)Earth is rotating.
An object located on Earth’ssurface anywhere
(except at two poles) must rotate in a circleabout
the Earth’s rotation axis and thus have a centripital
acceleration ( requiring a centripital net force )
directed toward the center of the ciecle.
equator
(711)
=
magnitude of
gravitation force
mass times
centripetal acceleration
mearsure
weight

(712)
Relation between and

=
Newton’s secend law for the axis
gravitation
acceleration
centripetal
acceleration
Freefall
acceleration
(710)
74Gravitation Near Earth's Surface(3)Earth is rotating.
Newton’s shell theorem can also be applied
to a particle located Inside a uniform shell:
A uniform spherical shell of matter exerts nonet gravitationalforce on a particle located insideit.
( particle on Earth’s surface )
(455)
(455)
However, we now choose a referance
configuration with equal to zero as
the seperation distance is large
enough to be approximated as infinite.
gravitational potential energy
At finite
76 Gravitational Potential EnergyThe gravitational potential energy of a particleEarth system
(P101)
(gravitational potential energy)
(717)
For any finite value of , the
value of is negative.
The gravitational potential energy is a
property of the system of the two
particles rather than of either particle along
However, for Earth and a apple,
We often speak of “potential energyof the
apple”, because when a apple moves in
the vicinity of Earth, (apple)
76 Gravitational Potential Energygravitational potential energy
(717)
For a system of three particles, the
gravitational potential energy of
the system isthe sum of the
gravitational potential energies of
all three pairs of particles.
( calculating as if the other
particle were not there )
(718)
76 Gravitational Potential Energypotential energyof a ball at point P, at
radial distance Rfrom Earth’s center.
The work done on the ball by the gravita tional forceas the ball travels from point P to a great (infinite) distance from Earth is
(719)
(720)
76 Gravitational Potential EnergyProof of (717):
Differential displacement
(721)
The work done along each circular arc is zero, because at every point.
Earth
76 Gravitational Potential EnergyPath 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 doneby the gravitational forceon the ball as it moves along ABCDEFG:
the gravitational force is a conservative force, the work done by it on a particle is independent ofthe actual path taken between points A and G.
(722)
Since the work done by a conservative forceis independent ofthe actual path taken.
The change in gravitational potential energy is also independent ofthe actual path taken.
Earth
76 Gravitational Potential EnergyFrom Eq. 447:
We derived the potential energy function
from the force function .
radially
(723)
inward
76 Gravitational Potential Energypotential energy and force
Now let’s go the other way: derive the force functionfrom thepotential energy function
This is Newton’s law of gravitation (71) .
( Derivation is the inverse operation of integration )
Escape Speed:The minimum initial speed that will cause a projectile to move up forever is called the (Earth) escape speed.
Consider a projectile () leaving the surface of a planet with escape speed
Its kinetic energy
Its potential energy
Its kinetic energy
Its potential energy
From the principle of conservation of energy
Escape Speed:
(724)
76 Gravitational Potential EnergyWhen the projectile reches infinity, it stops.
(724)
From Earth:
The escape speed does not depend on the direction in which a projectile is fired from a planet.
However, attaining that speedis easier if the projectile is fired in the directionthe launch siteis moving as the planet rotates about its axis.
76 Gravitational Potential Energyeastward
For example, rockets are launched eastward at XiChang to take the advantage of the eastward speed of 1500km/h due to Earth’s rotation.
The motion of the planets have been a puzzle since the dawn of history.
1 THE LAW OF ORBITS: All planets move in elliptical orbits, with the Sun at one focus.
is the semimajor axis of the orbit
is the eccentricity of the orbit
is the distance from the center of the ellipse to either focus
the eccentricityof Earth’s orbit is only 0.0167
77Planets and Satellites: Kepler's LawsJohannes Kepler(15711630) worked out the empiricallaws that governed these motions based on the data from the observations by Tycho Brahe(15461601).
2 THE LAW OF AREAS: A line that connects a planet to the Sun sweeps outequal 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.
(727)
constant
constant
77 Planets and Satellites: Kepler's LawsProof of Kepler’s second law is totally equivalent to the law of conservation of angular momentum.
The area of the wedge
The instantaneousrate at which area is been sweept out is
The magnitude of the angular momentum of the planet about the Sun is
From
Eq. 1120
(730)
77 Planets and Satellites: Kepler's Laws3 THE LAW OF PERIODS: The square of theperiod of any planet is proportional tothe cube of the semimajor axis of its orbit.
Applying Newton’s second law to the orbiting planet :
The quantity in parentheses is a constant that depends only the mass M of the central body about which the planet orbits.
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）
system (or the satellite) is
(734)
( Circular orbit )
78 Satellites: Orbits and EnergyAs a satelliteorbits Earth on its elliptical path, its speed and the distance from the center of Earthfluctuate with fixed periods. However, the mechanical energy E of the satellite remains constant.
To find the kinetic energy of the satellite, use Newton’s second law
Compare U and K
(732)
(733)
( circular orbit )
Compare E and K
( circular orbit )
(736)
For a satellite in an elliptical orbit of semimajor axis
( elliptical orbit )
(737)
78 Satellites: Orbits and EnergyThe total mechanical energy E of the satellite is