Chapter 13 Equilibrium

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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

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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
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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
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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.

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7-1 What Is Physics
The great steps of China toward the space
Lauching
Shenzhou five (????) Space ship
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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
Moons orbit
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A report on exploring deep space CE- project
by academician Ou Yang Ziyuan in Yangtze
University
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Academician Ou Yang Ziyuan present Yangtze
University with the all-around picture of the
Moon taking by CE-1
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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.
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Chinas manned spacecraft Shenzhou-7 blasts off
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Chinese taikonauts report they feel "physically
sound"
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Astronauts assemble EVA suit for spacewalk
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Chinese astronaut Zhai Zhigang is traveling in
deep space
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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 !
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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.

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7-2 Newton's Law of Gravitation

is the gravitational constant with a
value of
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7-2 Newton's Law of Gravitation
Nowtons 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).
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7-3 Gravitation and the Principle
of Superposition
3
5
2
i
n
4
a group of n particles

• the Principle of Superposition

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7-4 Gravitation Near Earth's Surface
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7-4 Gravitation Near Earth's Surface
the gravitatonal acceleration
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7-4 Gravitation Near Earth's Surface
(1) Earth is not uniform,
(2) Earth is not a perfect sphere,
(3) Earth rotates.
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7-4 Gravitation Near Earth's Surface
(1) Earth is not uniform
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 Earths
surface.
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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.
equator
Thus, a point at the poles is closer to the dense
core of Earth than is a point on the equator.
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7-4 Gravitation Near Earth's Surface
(3) Earth is rotating.
An object located on Earths surface anywhere
(except at two poles) must rotate in a circle
about the Earths rotation axis and thus have a
centripital acceleration ( requiring a
centripital net force ) directed toward the
center of the ciecle.
equator
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7-4 Gravitation Near Earth's Surface
(3) Earth is rotating.
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7-5 Gravitation Inside Earth
Newtons 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
• 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 particles radial position would not
• exert any net force on the particle.

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7-6 Gravitational Potential Energy
The gravitational potential energy of a
particle-Earth system
(P101)
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7-6 Gravitational Potential Energy
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7-6 Gravitational Potential Energy
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7-6 Gravitational Potential Energy
Proof of (7-17)
Differential displacement
(7-21)
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7-6 Gravitational Potential Energy
Differential displacement
(7-17)
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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
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.
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7-6 Gravitational Potential Energy
From Eq. 4-47
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7-6 Gravitational Potential Energy
potential energy and force

Now lets go the other way derive the force
function from the potential energy function

This is Newtons law of gravitation (7-1) .
( Derivation is the inverse operation of
integration )
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7-6 Gravitational Potential Energy

When the projectile reches infinity, it stops.
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7-6 Gravitational Potential Energy

eastward
For example, rockets are launched eastward at
XiChang to take the advantage of the eastward
speed of 1500km/h due to Earths rotation.
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7-7 Planets and Satellites Kepler's Laws

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).
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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 planets 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.
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7-7 Planets and Satellites Kepler's Laws
Proof of Keplers second law is totally
equivalent to the law of conservation of angular
momentum.
The area of the wedge
The instantaneous rate at which area is been
sweept out is
The magnitude of the angular momen- tum of the
planet about the Sun is
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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 Newtons 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.
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7-7 Planets and Satellites Kepler's Laws
3 THE LAW OF PERIODS

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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.

To find the kinetic energy of the satellite, use
Newtons second law
Compare U and K
(7-32)
(7-33)
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7-8 Satellites Orbits and Energy
The total mechanical energy E of the satellite is

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7-8 Satellites Orbits and Energy