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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

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slide1

BilingualMechanics

Chapter 7

Gravitation

制作 张昆实

Yangtze University

chapter 7 gravitation

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
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 physics1
7-1 What Is Physics

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 light-years across.

7 1 what is physics2
7-1 What Is Physics

★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.

7 1 what is physics3
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 physics4
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

slide8

A report on exploring deep space & CE- project by

academician Ou Yang Ziyuan in Yangtze University

slide9

Academician Ou Yang Ziyuanpresent Yangtze Universitywiththe all-around picture of the Moon taking by CE-1

slide10

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.

slide16

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

Translating this into an equation

(7-1)

( Nowton’s law of gravitation )

7-2 Newton's Law of Gravitation

Nowton 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.

7 2 newton s law of gravitation1

(7-2)

Particle 2 attracts particle 1

with

Particle 1 attracts particle 2

with

and are equal in magnitude

but opposite in direction.

Fig.14-2

These forces are notchanged even if there are bodies lie between them

(7-1)

( Nowton’s law of gravitation )

7-2 Newton's Law of Gravitation

is the gravitational constant with a value of

7 2 newton s law of gravitation2

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.

7-2 Newton's Law of Gravitation

Nowton’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).

7 3 gravitation and the principle of superposition

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.

(7-4)

For particle-

(7-6)

(7-5)

extented body

7-3 Gravitation and the Principle of Superposition

3

the Principle of Superposition

5

2

i

n

4

a group of nparticles

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

(7-7)

If the particle is releaced, it will fall

towards the center of Earth with the

gravitatonal acceleration :

Gravitation Near Earth's Surface

(7-8)

the gravitatonal acceleration

(7-9)

7-4Gravitation Near Earth's Surface
7 4 gravitation near earth s surface1

7-1

Gravitation Near Earth's Surface

(7-9)

7-4Gravitation Near Earth's Surface

the gravitatonal acceleration

7 4 gravitation near earth s surface2

m

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 gravitationalacceleration

However differs from

(7-9)

Weight differs from

(7-7)

Because:

7-4Gravitation Near Earth's Surface

(1)Earth is not uniform,

(2)Earth is not a perfect sphere,

(3)Earth rotates.

7 4 gravitation near earth s surface3

Crust

Oute core

Mantle

Inner core

Thus, varies from region to regionover the surface.

7-4Gravitation Near Earth's Surface

(1)Earth is not uniform

Thedensity 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.

7 4 gravitation near earth s surface4

This is one reason the free-fall acceleration increasesas one

proceeds, at sea level, from the equator toward either pole.

7-4Gravitation 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.

7 4 gravitation near earth s surface5

How Earth’s rotationcauses to differ from ? Put a crate of mass on a scaleat the equator and analyze it.

Free-body diagram

Normal force (outward in direction )

Gravitational force (inward in direction )

Centripital acceleration (inward in direction )

Newton’s secend law for the axis

(7-10)

7-4Gravitation 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

7 4 gravitation near earth s surface6

Reading on the scale

(7-11)

=

magnitude of

gravitation force

mass times

centripetal acceleration

mearsure

weight

-

(7-12)

Relation between and

-

=

Newton’s secend law for the axis

gravitation

acceleration

centripetal

acceleration

Free-fall

acceleration

(7-10)

7-4Gravitation Near Earth's Surface

(3)Earth is rotating.

7 5 gravitation inside earth
7-5Gravitation 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 nonet gravitationalforce on a particle located insideit.

  • 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 particle’s radial position would not
  • exert any net force on the particle.
7 6 gravitational potential energy

( particle on Earth’s surface )

(4-55)

(4-55)

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

7-6 Gravitational Potential Energy

The gravitational potential energy of a particle-Earth system

(P101)

7 6 gravitational potential energy1

(gravitational potential energy)

(7-17)

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)

7-6 Gravitational Potential Energy
7 6 gravitational potential energy2

gravitational potential energy

(7-17)

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 )

(7-18)

7-6 Gravitational Potential Energy
7 6 gravitational potential energy3

Findthegravitational

potential 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

(7-19)

(7-20)

7-6 Gravitational Potential Energy

Proof of (7-17):

Differential displacement

(7-21)

7 6 gravitational potential energy4

(7-21)

FromEq. 4-47

7-6 Gravitational Potential Energy

Differential displacement

(7-17)

7 6 gravitational potential energy5

The work done along each circular arc is zero, because at every point.

Earth

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 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.

7 6 gravitational potential energy6

(4-47)

(7-22)

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

7-6 Gravitational Potential Energy

From Eq. 4-47:

7 6 gravitational potential energy7

We derived the potential energy function

from the force function .

radially

(7-23)

inward

7-6 Gravitational Potential Energy

potential energy and force

Now let’s go the other way: derive the force functionfrom thepotential energy function

This is Newton’s law of gravitation (7-1) .

( Derivation is the inverse operation of integration )

7 6 gravitational potential energy8

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:

(7-24)

7-6 Gravitational Potential Energy

When the projectile reches infinity, it stops.

7 6 gravitational potential energy9

Escape Speed

(7-24)

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.

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 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.

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

7-7Planets and Satellites: Kepler's Laws

Johannes Kepler(1571-1630) worked out the empiricallaws that governed these motions based on the data from the observations by Tycho Brahe(1546-1601).

7 7 planets and satellites kepler s laws1
7-7 Planets and Satellites: Kepler's Laws

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.

7 7 planets and satellites kepler s laws2

(7-26)

(7-27)

constant

constant

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

The instantaneousrate at which area is been sweept out is

The magnitude of the angular momen-tum of the planet about the Sun is

7 7 planets and satellites kepler s laws3

(7-29)

From

Eq. 11-20

(7-30)

7-7 Planets and Satellites: Kepler's Laws

3 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.

7 7 planets and satellites kepler s laws4

(7-30)

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)

7 8 satellites orbits and energy

The potential energy of the

system (or the satellite) is

(7-34)

( Circular orbit )

7-8 Satellites: Orbits and Energy

As 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

(7-32)

(7-33)

7 8 satellites orbits and energy1

(7-35)

( circular orbit )

Compare E and K

( circular orbit )

(7-36)

For a satellite in an elliptical orbit of semimajor axis

( elliptical orbit )

(7-37)

7-8 Satellites: Orbits and Energy

The total mechanical energy E of the satellite is