制作张昆实 Yangtze University

制作张昆实制作张昆实

Yangtze UniversityYangtze University

制作张昆实制作张昆实

Yangtze UniversityYangtze University

BilingualBilingual MechanicsMechanics

BilingualBilingual MechanicsMechanics Chapter 7

GravitationGravitation

Chapter 7

GravitationGravitation

Chapter 7 Chapter 7 GravitationGravitation Chapter 7 Chapter 7 GravitationGravitation

7-1 7-1 What Is Physics?

7-2 7-2 Newton's Law of Gravitation Newton's Law of Gravitation

7-37-3 Gravitation and the Principle of Gravitation and the Principle of

SuperpositionSuperposition

7-4 7-4 Gravitation Near Earth's Surface Gravitation Near Earth's Surface

7-57-5 Gravitation Inside Earth

7-67-6 Gravitational Potential Energy

7-77-7 Planets and Satellites: Kepler's Laws

7-87-8 Satellites: Orbits and Energy

7-1 7-1 What Is Physics?

7-2 7-2 Newton's Law of Gravitation Newton's Law of Gravitation

7-37-3 Gravitation and the Principle of Gravitation and the Principle of

SuperpositionSuperposition

7-4 7-4 Gravitation Near Earth's Surface Gravitation Near Earth's Surface

7-57-5 Gravitation Inside Earth

7-67-6 Gravitational Potential Energy

7-77-7 Planets and Satellites: Kepler's Laws

7-87-8 Satellites: Orbits and Energy

Have you ever imaged how vastvast isthe universeuniverse?

The sunsun is one of millions of starsstarsthat form the Milky the Milky Way GalaxyWay Galaxy.

We are near the edge of the disk of the galaxythe galaxy, about 26000 light-years from its center.

Have you ever imaged how vastvast isthe universeuniverse?

The sunsun is one of millions of starsstarsthat form the Milky the Milky Way GalaxyWay Galaxy.

We are near the edge of the disk of the galaxythe galaxy, about 26000 light-years from its center. Milky Way galaxyMilky Way galaxy

7-1 What Is Physics 7-1 What Is Physics

The universe is made

up of many galaxiesgalaxies,

each one containing containing

millions of stars.millions of stars.

One of the galaxies is

the Andromeda galaxy.

The great galaxy M31 The great galaxy M31

in in the the Constellation

Andromeda is more than

100000 light-years across.

The universe is made

up of many galaxiesgalaxies,

each one containing containing

millions of stars.millions of stars.

One of the galaxies is

the Andromeda galaxy.

The great galaxy M31 The great galaxy M31

in in the the Constellation

Andromeda is more than

100000 light-years across.

Andromeda galaxy


★ The most distantdistant galaxiesgalaxies are known to be over

10 billion10 billion light yearslight years away !

★ What forceWhat force binds togetherbinds together these progressively

larger structures, from star star toto galaxygalaxy to

superclustersupercluster ?

★ It is the gravitational forcethe gravitational force that not onlynot only holds

you on Earthon Earth but alsobut also reaches out across across

intergalactic spaceintergalactic space.

★ The most distantdistant galaxiesgalaxies are known to be over

10 billion10 billion light yearslight years away !

★ What forceWhat force binds togetherbinds together these progressively

larger structures, from star star toto galaxygalaxy to

superclustersupercluster ?

★ It is the gravitational forcethe gravitational force that not onlynot only holds

you on Earthon Earth but alsobut also reaches out across across

intergalactic spaceintergalactic space.


The great steps The great steps

of China of China

toward the spacetoward the space

The great steps The great steps

of China of China

toward the spacetoward the space

Shenzhou fiveShenzhou five

(( 神州五号神州五号 ))

Space shipSpace ship

Shenzhou fiveShenzhou five

(( 神州五号神州五号 ))

Space shipSpace ship

LauchingLauching


China CE-1 project China CE-1 project Exploring the MoonExploring the Moon

Lauching: Lauching: 2007-10-242007-10-24

Orbit aroundOrbit aroundthe Moonthe Moon : : 2007-11-52007-11-5

shifting: shifting: 2007-11-12007-11-1

Moon’s orbit Moon’s orbit


A report on exploring deep space & CE- project byacademician Ou Yang Ziyuan in Yangtze UniversityA report on exploring deep space & CE- project byacademician 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

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.

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 offChina´s manned spacecraft Shenzhou-7 blasts off

Chinese taikonauts report they feel "physically sound"Chinese taikonauts report they feel "physically sound"

Astronauts assemble EVA suit for spacewalkAstronauts assemble EVA suit for spacewalk

Chinese astronaut Zhai Zhigang is ready for spacewalk

Chinese astronaut Zhai Zhigang is ready for spacewalk

Chinese astronaut Zhai Zhigang is traveling in deep spaceChinese astronaut Zhai Zhigang is traveling in deep space

The fundamental principles of space flight is Mechanics !

The fundamental principles of space flight is Mechanics !

Physics is the cradle of modern science and technology !

Physics is the cradle of modern science and technology !

Congratulations to the successful launching of Shenzhou-7 !

Congratulations to the successful launching of Shenzhou-7 !

NowtonNowton published the law ofthe law of gravitationgravitationIn 1687. In 1687. It may be stated as follows:It may be stated as follows:

Every particleEvery particle in the universein the universe attractsattracts every every other particleother particle withwith a forcea force that isthat is directely directely proportionalproportional toto the product of the the product of the massesmasses of the particles of the particles andand inversely proportinalinversely proportinal toto the the square of thesquare of the distancedistance between them between them.

NowtonNowton published the law ofthe law of gravitationgravitationIn 1687. In 1687. It may be stated as follows:It may be stated as follows:

Every particleEvery particle in the universein the universe attractsattracts every every other particleother particle withwith a forcea force that isthat is directely directely proportionalproportional toto the product of the the product of the massesmasses of the particles of the particles andand inversely proportinalinversely proportinal toto the the square of thesquare of the distancedistance between them between them.

7-27-2 Newton's Law of Gravitation Newton's Law of Gravitation7-27-2 Newton's Law of Gravitation Newton's Law of Gravitation

1 22

m mF G

r ( Nowton’s law of gravitation )( Nowton’s law of gravitation ) (7-1)

Translating this into an equationTranslating this into an equation

is the gravitational constant with a value of is the gravitational constant with a value of

7-27-2 Newton's Law of Gravitation Newton's Law of Gravitation 7-27-2 Newton's Law of Gravitation Newton's Law of Gravitation

1 22

m mF G

r ( Nowton’s law of gravitation )( Nowton’s law of gravitation ) (7-1)

G

(7-2)

11 2 26.67 10 /G N m kg 11 3 26.67 10 /m kg s

F

F

r1m 2m

Fig.14-2

Particle 2 attracts particle 1

with


with

and are equal in magnitude

but opposite in direction.


with


with

and are equal in magnitude

but opposite in direction.

F

F

F

F

These forces are not changed even if there are bodies lie between them

These forces are not changed even if there are bodies lie between them

Nowton’s law of gravitation Nowton’s law of gravitation appliesappliesstrictlystrictly to to particlesparticles; ; also appliesalso applies to to real objectsreal objects as long as their sizes as long as their sizes are small compared to the distance are small compared to the distance between them between them (Earth and Moon).(Earth and Moon).

Nowton’s law of gravitation Nowton’s law of gravitation appliesappliesstrictlystrictly to to particlesparticles; ; also appliesalso applies to to real objectsreal objects as long as their sizes as long as their sizes are small compared to the distance are small compared to the distance between them between them (Earth and Moon).(Earth and Moon).

Shell theorem:Shell theorem:Shell theorem:Shell theorem:

7-27-2 Newton's Law of Gravitation Newton's Law of Gravitation 7-27-2 Newton's Law of Gravitation Newton's Law of Gravitation

What about an apple and Earth?What about an apple and Earth?

sellmsellm

A uniform spherical shellspherical shell of matter attracts a particleparticle that is outsideoutside the shell as ifas if all the shell’s mass weall the shell’s mass were concentrated at its centerre concentrated at its center.

11,netF

First, compute the gravitational forcgravitational forcee that acts on particle 1 due to each of the other particles, in turnin turn.

7-37-3 Gravitation and the Principle Gravitation and the Principle of Superposition of Superposition 7-37-3 Gravitation and the Principle Gravitation and the Principle of Superposition of Superposition

the Principle of Superpositionthe Principle of Superposition

Given a group of nn particlesparticles, there are gravitational forcesgravitational forces between any pairany pair of particles.

2

5

4n

3

iFinding the net forceFinding the net force acting on particle 11 from the othersthe others

Then, add these forces vectorialyvectorialy.

1, 12 13 14 15 1net nF F F F F F

(7-4)

1F dF

(7-6)1, 1

2

n

net ii

F F

(7-5)extented body

For particle-

a group of nn particlesparticles

1

extented body

1F

the gravitatonal accelerationthe gravitatonal acceleration

If the particle is releaced, it will fallfall towards the center of Earththe center of Earth with the the gravitatonal accelerationgravitatonal acceleration :

If the particle is releaced, it will fallfall towards the center of Earththe center of Earth with the the gravitatonal accelerationgravitatonal acceleration :ga

A particle (m) locates outside EarthA particle (m) locates outside Eartha distance a distance r r from Earth’s center. The from Earth’s center. The magnitude of magnitude of the gravitational forcethe gravitational force fr from Earth (M) acting onom Earth (M) acting on it equals

A particle (m) locates outside EarthA particle (m) locates outside Eartha distance a distance r r from Earth’s center. The from Earth’s center. The magnitude of magnitude of the gravitational forcethe gravitational force fr from Earth (M) acting onom Earth (M) acting on it equals

7-47-4 Gravitation Near Earth's SurfaceGravitation Near Earth's Surface 7-47-4 Gravitation Near Earth's SurfaceGravitation Near Earth's Surface

2

MmF G

r (7-7)

m

r

M

gF ma (7-8)

F

Gravitation NeaGravitation Near Earth's Surfacr Earth's Surfacee

2g

GMa

r (7-9)

the gravitatonal accelerationthe gravitatonal accelerationthe gravitatonal accelerationthe gravitatonal acceleration


m

r

M

Gravitation NeaGravitation Near Earth's Surfacr Earth's Surfacee

2g

GMa

r (7-9)

7-1

BecauseBecause:

We have assumed that Earth is an inertial frame (negnecting its actual rotation). This allowed us to assumethe free-fall acceleration is the same as the gravitational acceleration

We have assumed that Earth is an inertial frame (negnecting its actual rotation). This allowed us to assumethe free-fall acceleration is the same as the gravitational acceleration

(1) Earth is Earth is not uniform, uniform,(1) Earth is Earth is not uniform, uniform,


M

m

29.8 /g m sgg aHowever However differs from fromHowever However differs from from 2

ga GM r (7-9)g

(2) Earth is not a perfect sphere,Earth is not a perfect sphere,(2) Earth is not a perfect sphere,Earth is not a perfect sphere,(3) Earth rotates.Earth rotates.(3) Earth rotates.Earth rotates.

Weight Weight differs fromdiffers from Weight Weight differs fromdiffers from 2F GMm r (7-7)mg


(1) Earth is not uniformEarth is not uniform(1) Earth is not uniformEarth is not uniformThe densitydensity of Earth varies varies radiallyradially:: Inner coreInner core 12-14 (103 kg/m3) Outer coreOuter core 10-12 (103 kg/m3) MantleMantle 3-5.5 (103 kg/m3)

The densitydensity of Earth varies varies radiallyradially:: Inner coreInner core 12-14 (103 kg/m3) Outer coreOuter core 10-12 (103 kg/m3) MantleMantle 3-5.5 (103 kg/m3)

CrustCrust

Inner Inner corecore

Oute Oute corecore

MantleMantleand the density of the crustthe crust (outer sectionouter section) of Earthvariesvaries from regionregion to region region over Earth’s surface.

and the density of the crustthe crust (outer sectionouter section) of Earthvariesvaries from regionregion to region region over Earth’s surface.

Thus, variesvaries from regionregion to reregiongion over the surfaceover the surface.Thus, variesvaries from regionregion to reregiongion over the surfaceover the surface.

g


M

(2) Earth is not a perfect sphereEarth is not a perfect sphere(2) Earth is not a perfect sphereEarth is not a perfect sphere

equator

EarthEarth is approximately an ellipsoidellipsoid, flflattened at the polesattened at the poles and bulging at thbulging at the equattore equattor. Its equatorial radiusequatorial radius is greater than its polar radiuspolar radius by 21km21km.

EarthEarth is approximately an ellipsoidellipsoid, flflattened at the polesattened at the poles and bulging at thbulging at the equattore equattor. Its equatorial radiusequatorial radius is greater than its polar radiuspolar radius by 21km21km.

Thus, a point at the polesa point at the poles is closercloser to the dense coredense core of Earth thanthan is a a point on the equatorpoint on the equator.

Thus, a point at the polesa point at the poles is closercloser to the dense coredense core of Earth thanthan is a a point on the equatorpoint on the equator.

Mpolr

eqtr This is one reasonone reason the free-fall acceleration increasesincreases as oneas oneproceedsproceeds, at sea level, from the eqfrom the equator toward either poleuator toward either pole.

This is one reasonone reason the free-fall acceleration increasesincreases as oneas oneproceedsproceeds, at sea level, from the eqfrom the equator toward either poleuator toward either pole.

g


M

(3) Earth is rotating.Earth is rotating.(3) Earth is rotating.Earth is rotating.

equatorequator

An object located on Earth’son Earth’s surfacesurface anywhere (except at two poles) must rotate in a circlemust rotate in a circle about about the Earth’s rotation axisthe Earth’s rotation axis and thus have a centripital have a centripital accelerationacceleration ( requiring a centripital net forcerequiring a centripital net force ) directed toward the center of the ciecle.

An object located on Earth’son Earth’s surfacesurface anywhere (except at two poles) must rotate in a circlemust rotate in a circle about about the Earth’s rotation axisthe Earth’s rotation axis and thus have a centripital have a centripital accelerationacceleration ( requiring a centripital net forcerequiring a centripital net force ) directed toward the center of the ciecle.

a

Normal force (outward in directionoutward in direction )Normal force (outward in directionoutward in direction )NN

rr

How Earth’s rotationEarth’s rotation causescauses to differ from ? Put a cratecrate of mass on a scaleon a scale at the equatorat the equator and analyze it.

How Earth’s rotationEarth’s rotation causescauses to differ from ? Put a cratecrate of mass on a scaleon a scale at the equatorat the equator and analyze it.

gagm

Free-body diagramFree-body diagramFree-body diagramFree-body diagram

r

gma

Gravitational forceGravitational force (inward in directiondirection )Gravitational forceGravitational force (inward in directiondirection )gma

rCentripital accelerationCentripital acceleration (inward in direction )Centripital accelerationCentripital acceleration (inward in direction )a

a

r rNewton’s secend lawNewton’s secend law for the axisNewton’s secend lawNewton’s secend law for the axis

2( )gN ma ma m R r

(7-10)

N


(3) Earth is rotating.Earth is rotating.(3) Earth is rotating.Earth is rotating.Newton’s secend lawNewton’s secend law for the axis

2( )N gF ma ma m R r

(7-10)

2( )gmg ma m R (7-11)

NF mg Reading on the scale

mearsuremearsure

weightweight = magnitude ofmagnitude of

gravitation forcegravitation force

mass times

centripetal acceleration-

=Free-fallFree-fall

accelerationacceleration

gravitation gravitation

accelerationacceleration

centripetal

acceleration-2 2 6 2(2 (24 3600)) 6.37 10 0.034 /ga g R m s

(7-12) Relation between andRelation between and ga2

gg a R g

7-57-5 Gravitation Inside Earth7-57-5 Gravitation Inside Earth

sellm

Newton’s shell theorem can also be appliedNewton’s shell theorem can also be applied to a particle located Inside a uniform shell:to a particle located Inside a uniform shell:

A uniform spherical shellspherical shell of matter exerts nono net net gravitationalgravitational forforcece on a particle located insidelocated inside itit.If a particle were to move into Earth, the gavitational Force would change :(1) It would tend to increase because the It would tend to increase because the particle would be moving closer to theparticle would be moving closer to the center of Earth.center of Earth.(2) It would tend to decrease because the It would tend to decrease because the thickening shell of material lying outside thickening shell of material lying outside the particle’s radial position would not the particle’s radial position would not exert any net force on the particleexert any net force on the particle. .

The The gravitational potential energygravitational potential energy of a particle-Earth of a particle-Earth systemsystemThe The gravitational potential energygravitational potential energy of a particle-Earth of a particle-Earth systemsystem

7-67-6 Gravitational Potential Energy 7-67-6 Gravitational Potential Energy

( )U y mgy (4-55)

0U r

0U r

However, we now choose a However, we now choose a referance referance configurationconfiguration with equal to with equal to zerozero as as the seperation distancethe seperation distance is large is large enough to be approximated as enough to be approximated as infiniteinfinite..

However, we now choose a However, we now choose a referance referance configurationconfiguration with equal to with equal to zerozero as as the seperation distancethe seperation distance is large is large enough to be approximated as enough to be approximated as infiniteinfinite..

0U ,r

Ur

At finiteAt finite ,r

gravitational potegravitational potential energyntial energygravitational potegravitational potential energyntial energy

GMmU

r

( ( particle particle on Earth’s surface )on Earth’s surface )0( | ) 0yU y (4-5

5)(P101)(P101)


0U r

0U r

(gravitational pot(gravitational potential energy)ential energy)

GMmU

r (7-17)

( ) 0U r ,r For any finite value of , thevalue of is negative. For any finite value of , thevalue of is negative. ( )U r

r

However, for Earth and a apple, However, for Earth and a apple, We often speak of “potential energypotential energy of the apple”, because when a apple moves in the vicinity of Earth, (apple)

However, for Earth and a apple, However, for Earth and a apple, We often speak of “potential energypotential energy of the apple”, because when a apple moves in the vicinity of Earth, (apple)

M m

sysU kE

The gravitational potential energy is aThe gravitational potential energy is aproperty of the system of the two property of the system of the two particles rather than of either particle alongparticles rather than of either particle along

The gravitational potential energy is aThe gravitational potential energy is aproperty of the system of the two property of the system of the two particles rather than of either particle alongparticles rather than of either particle along

U

12 13 23U U U U ( calculating as if the other particle were not there )

ijU

For a system of three particles,For a system of three particles, thegravitational potential energy of gravitational potential energy of the system the system isis the sum ofthe sum of the the gravitational potential energies gravitational potential energies of of all three pairs of particles.all three pairs of particles.


gravitational potegravitational potential energyntial energy

GMmU

r (7-17)

13r

3m

12r

23r

2m1m

1 3 2 31 2

12 13 23

( )Gm m Gm mGm m

Ur r r

(7-18)

( ) ( ) cosF r dr F r dr

(7-20)


dr

R

F

P

r

2

GMmdr

r

0180

Differential displacement

2GMm GMm GMm

r Rr RRW dr

The work done on the ballThe work done on the ball by the gravita- tthe gravitational forceional force as the ball travels fromfrom point P to a great (infinite) distancea great (infinite) distance from Earth is

( )

RW F r dr

(7-19)

m

M

(7-21)

FindFind thethe gravitational gravitational potential energypotential energy of a ball at point Pof a ball at point P, , atat radial distance Rradial distance R fromfrom Earth’s center. Earth’s center.

UProof of (7-17):


dr

R

F

P

r

0180

Differential displDifferential displacementacement

2GMm GMm

rr RRW dr

m

M

(7-21)GMm

R

U U W

From Eq. 4-47 U W

GMmU W

R

0U

(7-17)


Moving a ball from AA to G G along a path :consisting of three radial lengthsthree radial lengths andthree circular arcsthree circular arcs (cented on Earth).

F

E

A

C

G

B

D

Earth

HThe work doneThe work done by the gravitational forcethe gravitational force

on the ball as it moves along ABCDEFG:

A G AB CD EFW W W W AGW

The work done along each circular arcalong each circular arc is zerozero, because at every point. F ds

the gravitational forcethe gravitational force is a conservative forconservative forcece, the work done by it on a particle is indeindependent ofpendent of the actual path taken between pthe actual path taken between points oints AA and and GG.

Path IndependencePath IndependencePath IndependencePath Independence

A G AB BC CD DE EF FGW W W W W W W


From Eq. 4-47:F

E

A

C

G

B

D

Earth

H

f iU U U W (7-22)

The change in gravitational potential energy is also indepenindependent ofdent of the actual path takenthe actual path taken.

U

Since the work done by a coconservative forcenservative force is independent independent ofof the actual path takenthe actual path taken.

W

U W (4-47)

( )U r ( )F r We derived the potential energy functionthe potential energy function

from the force function .the force function .

We derived the potential energy functionthe potential energy function from the force function .the force function .


2( )

dU d GMm GMmF

dr dr r r (7-23)

Now let’s go the other waythe other way: derive the force fthe force functionunction from thethe potential energy functionpotential energy function

Now let’s go the other waythe other way: derive the force fthe force functionunction from thethe potential energy functionpotential energy function

potential energy and forcepotential energy and force potential energy and forcepotential energy and force

( ) ( ) ( )r

GMmU r W r F r dr

r

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

radiallyradially

inwardinward

( DerivationDerivation is the inverse operationthe inverse operation of integrationintegration )( DerivationDerivation is the inverse operationthe inverse operation of integrationintegration )

When the projectile reches infinityinfinity, it stopsstops. Its kinetic energy Its kinetic energy 0K Its Its potential energypotential energy 0U FromFrom the principle of co the principle of conservation of energynservation of energy

Escape Speed:Escape Speed: The minimum initial speedThe minimum initial speed that will cause a projectilea projectile to move up forevermove up forever is called the the (Earth) escape speed(Earth) escape speed..

Escape Speed:Escape Speed: The minimum initial speedThe minimum initial speed that will cause a projectilea projectile to move up forevermove up forever is called the the (Earth) escape speed(Earth) escape speed..


r

M

R

mv

Consider a projectileprojectile ( ) leaving the surface of a planet with escape speedescape speed Consider a projectileprojectile ( ) leaving the surface of a planet with escape speedescape speed

mv

Its Its potential energypotential energy U GMm RIts kinetic energy Its kinetic energy 21

2K mv

mv

212 ( ) 0GMm

RK U mv

2GMv

R (7-24) Escape Speed:Escape Speed:

However, attainingHowever, attaining that speed that speed is is easiereasier if the if the projeprojectilectile is fired is fired in the directionin the direction the launch sitethe launch site is moviis movingng as as the planet rotates about its axisthe planet rotates about its axis . .

However, attainingHowever, attaining that speed that speed is is easiereasier if the if the projeprojectilectile is fired is fired in the directionin the direction the launch sitethe launch site is moviis movingng as as the planet rotates about its axisthe planet rotates about its axis . .


For example, rocketsrockets are launched eastwardlaunched eastward at XiCXiChanghang to take the advantagetake the advantage of the eastward speed othe eastward speed of 1500km/h due to Earth’s rotation. f 1500km/h due to Earth’s rotation.

For example, rocketsrockets are launched eastwardlaunched eastward at XiCXiChanghang to take the advantagetake the advantage of the eastward speed othe eastward speed of 1500km/h due to Earth’s rotation. f 1500km/h due to Earth’s rotation.

eastwardeastwardThe escape speed does escape speed does not depend on the direction not depend on the direction in which a projectile is fired in which a projectile is fired from a planet. from a planet.

The escape speed does escape speed does not depend on the direction not depend on the direction in which a projectile is fired in which a projectile is fired from a planet. from a planet.

v

Escape SpeedEscape Speed Escape SpeedEscape Speed 2GMv

R (7-24)

From Earth: 11.2 /v km s

7-77-7 Planets and Satellites: Kepler's Laws 7-77-7 Planets and Satellites: Kepler's Laws

1 THE LAW OF ORBITSTHE LAW OF ORBITS: All planets move in elli: All planets move in elliptical orbits, with the Sun at one focus.ptical orbits, with the Sun at one focus. 1 THE LAW OF ORBITSTHE LAW OF ORBITS: All planets move in elli: All planets move in elliptical orbits, with the Sun at one focus.ptical orbits, with the Sun at one focus.

The motion of the planets have been a puzzlea puzzle since the dawn of history.

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

M

ea ea

r

a

m

F F

a is the semimajor axisthe semimajor axis of the orbit

e is the eccentricitythe eccentricity of the orbit

ea is the distancethe distance from the center of the ellipse to either focus

the eccentricitythe eccentricity of Earth’s orbitof Earth’s orbit is only 0.0167

e


2 THE LAW OF AREASTHE LAW OF AREAS: A line that connects a p: A line that connects a planet to the Sun lanet to the Sun sweeps outsweeps out equal areasequal areas in the pl in the plane of the planet’s orbit ane of the planet’s orbit in equal timesin equal times; that is, th; that is, the rate e rate dA/dtdA/dt at which it sweeps out area A is at which it sweeps out area A is conconstantstant. .

2 THE LAW OF AREASTHE LAW OF AREAS: A line that connects a p: A line that connects a planet to the Sun lanet to the Sun sweeps outsweeps out equal areasequal areas in the pl in the plane of the planet’s orbit ane of the planet’s orbit in equal timesin equal times; that is, th; that is, the rate e rate dA/dtdA/dt at which it sweeps out area A is at which it sweeps out area A is conconstantstant. .

This second law tell us that that tthe planet will move he planet will move most slomost slowlywly when when it is farthest from tit is farthest from the Sunhe Sun and and most rapidlymost rapidly wh when en it is nearest to the Sun. it is nearest to the Sun.

Sun

M

Proof of Kepler’s second law is totally equivalent to the law of conservation of angular momentumthe law of conservation of angular momentum. Proof of Kepler’s second law is totally equivalent to the law of conservation of angular momentumthe law of conservation of angular momentum.


Sun

M

r r

Sun

M

r

A

rp

The areaThe area of the wedgethe wedgeThe areaThe area of the wedgethe wedge

m

21 12 2A r r r

The instantaneousThe instantaneous raterate at which area is been sweept outarea is been sweept out is The instantaneousThe instantaneous raterate at which area is been sweept outarea is been sweept out is

pp

The magnitudeThe magnitude of the angular momen-the angular momen- tumtum of the planetthe planet about the Sunthe Sun isThe magnitudeThe magnitude of the angular momen-the angular momen- tumtum of the planetthe planet about the Sunthe Sun is

( ) ( )L rp r mv r m r

2 21 1

2 2

dA dr r

dt dt

(7-26)

2mr (7-27)

2

dA L

dt m

dA dt constantconstantconstantconstant

L constantconstantconstantconstant


3 THE LAW OF PERIODSTHE LAW OF PERIODS: : The square of theThe square of the periperiodod of any planet is of any planet is proportional toproportional to the cube of the the cube of the semimajor axis of its orbitsemimajor axis of its orbit. .

3 THE LAW OF PERIODSTHE LAW OF PERIODS: : The square of theThe square of the periperiodod of any planet is of any planet is proportional toproportional to the cube of the the cube of the semimajor axis of its orbitsemimajor axis of its orbit. .

Applying Newton’s second law to Applying Newton’s second law to the orbiting planet :the orbiting planet :Applying Newton’s second law to Applying Newton’s second law to the orbiting planet :the orbiting planet :

Mr

m

F ma 22

( )GMm

m rr

(7-29)

22 34

T rGM

(7-30)

From Eq. 11-20

2T

2

T

The quantity in parenthesesin parentheses is a constantconstant that depends only the mass M of the central bodythe central body about which the planetthe planet orbits.

The quantity in parenthesesin parentheses is a constantconstant that depends only the mass M of the central bodythe central body about which the planetthe planet orbits.


3 THE LAW OF PERIODSTHE LAW OF PERIODS:: 3 THE LAW OF PERIODSTHE LAW OF PERIODS:: 2 2

3

4T

a GM

(7-30)

（水星）

（金星）

（火星）

（木星）

（土星）

（天王星）

（海王星）

（冥王星）

（地球）

7-87-8 Satellites: Orbits and Energy 7-87-8 Satellites: Orbits and Energy

As a satellitea satellite orbits Earth on its elliptical path, its speedits speed and the distance from the center of Earth fluctuatefluctuate with fixed periods periods. However, the mechanical energy E of the satellite remains constantremains constant.

As a satellitea satellite orbits Earth on its elliptical path, its speedits speed and the distance from the center of Earth fluctuatefluctuate with fixed periods periods. However, the mechanical energy E of the satellite remains constantremains constant.

The potential energyThe potential energy of the

system (or the satellitethe satellite) is

GMmU

r

212 2

GMmK mv

r

To find the kinetic energythe kinetic energy of the satellitethe satellite, use Newton’s second law

F ma2

2

GMm vm

r r

(7-33)

(7-32)Compare Compare UU and and KK

(7-34)2

UK

( Circular orbit( Circular orbit ) )

For a satellite in an elliptical orbitelliptical orbit of semimajor axis a

Compare Compare EE and and KK

2

GMmK

r


The total mechanical energy E of the satellite isisThe total mechanical energy E of the satellite isis

2

GMm GMmE K U

r r

(7-36)E K ( circular orbit )( circular orbit )

(7-35)2

GMmE

r ( circular orbit )( circular orbit )

(7-37)( elliptical orbit( elliptical orbit ) )2

GMmE

a


7-15 7-16

GMmU

r

GMmU

r

2

GMmK

r2

GMmK

r

2

GMmE

r

2

GMmE

r

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制作 张昆实 Yangtze University

制作张昆实 Yangtze University