Download - 12 Lecture Lam - University of Hawaii Systemplam/ph170_summer_2011/L12/12_Lecture...3 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Gravitational potential

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Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

PowerPoint® Lectures for University Physics, Twelfth Edition – Hugh D. Young and Roger A. Freedman

Lectures by James Pazun

Chapter 12

Gravitation

Modified by P. Lam 6_13_2011 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

Goals for Chapter 12

•  To study Newton’s Law of Gravitation

•  Use Newton’s Law of Gravitation and Newton’s Second Law of Motion (F=ma) to calculate orbital speed of satellites and planets

•  Use energy method to determine “escape velocity”

•  Use Newton’s Law of Gravitation and Newton’s Second Law of Motion to explain Kepler’s Laws of planetary motion


Newton’s Law of Gravitation

•  The gravitational force is always attractive and its magnitude depends on both the masses of the bodies involved and their separations.

!

! F g =

Gm1m2

r2

G = 6.6742x10"11

N • m2/kg

2


Superposition principle

•  The total force on m1= force on m1 by m2 + force on m1 by m3

m1 m2

m3

!

Find ! F 1,! F 2and the net force

! F

Express the forces in ˆ i , ̂ j .

2


Compare little “g” with big “G”

Consider the gravitational force on a mass (m) by the Earth:

F =GmME

r2=ma

! a =GME

r2

Near Earth's surface ! r~RE

! a =GME

RE2" 9.8

m

s2

#

$%&

'(= g (free) fall acelleration)

The magnitude of "g" decreases as the object gets

further away from the Earth.


Satellite motion

Orbital motion = projectile motion with very large initial velocity How large an initial velocity is required? Answer on next slide


Calculate satellite orbits - assume circular orbit

!

This example consider satellite around the Earth

(but the same method applies to motion of satellite around

any large mass even for a planet around the Sun)

(1) Draw free diagram on the satellite

(2) Set up Newton's 2nd Law F = ma equation (a = v2 /r)

" F = mv

2

r

(3) Apply Newton's Law of Gravitaion : F =GmM

E

r2

(4) Combine "GmM

E

r2

= mv

2

r

(5) Solve for v in terms of r

" v =GME

r

For an orbit near the surface of the Earth

(r # RE = 6.4x106m),

v # 8x103m /s ~ 18,000mi /h

m

ME


Calculate planetary orbits - assume circular orbit

mp

MS

!

GmpMS

r2

= mpv

2

r

v2 =GMS

r

v =2"r

T; T = orbital period

#2"r

T

$

% &

'

( )

2

=GMS

r

#r

3

T2

=GMS

4" 2= same number for all

planets in the solar system.

(This called Kepler's 3rd Law)

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Gravitational potential energy

!

The expression for the gravitaitonal

potential energy for a mass at height

h from the surface of the Earth :

U = mgh is only valid for an object

near the surface of the Earth.

The more general expression is :

U = " GmME

r= potential energy for the

mass - Earth system separated by a distance of r.

Near the surface of the Earth r = RE + h

U(RE + h) "U(RE ) = "GmME

RE + h

#

$ %

&

' ( " "

GmME

RE

#

$ %

&

' ( ) m

GME

RE2h = mgh


Calculate escape velocity

•  What is the minimum initial velocity of a rocket for it to escape from the gravitational effect of the Earth?

!

Ki +Ui = K f +Uf

1

2mvi

2+1

2MEVi

2"GmME

RE= 0 +

1

2MEVf

2"GmME

#

Vf $Vi

%1

2mvi

2"GmME

RE= 0

% vi =2GME

RE& 25,300mi /h

!

Note : If a star is very dense such that it has a large mass

but a small radius, then vescape > speed of light!!

Such a star is a "black hole".

However, correct treatment of black hole requires

Einstein's theory of Gravitation (General Relativity).

Newton's theory of graviation is only approximate.


Newton’s Law of gravitation

How did Newton come up with the Law of Gravitation?

!

! F =

Gm1m2

r2

Newton relied on the astronomical data of Kepler.


Kepler’s laws for planetary motion

•  1st Law: Each planet moves in an elliptical orbit with the sun at one of the foci

•  2nd Law: A line connecting the sun to a given planet sweeps out equal areas in equal times.

•  3rd Law: The periods of the planets are proportional to the 3/2 powers of the major axis lengths in their orbits. (Planets further away from the Sun takes longer to go around the Sun)

4


Kepler’s 1st Law - elliptical orbits

Elliptical orbits are consequence of the 1/r2 dependence of the gravitational force.

The proof requires advanced calculus.

Circular orbit is a special case of elliptical orbit.


Kepler’s 2rd Law - Conservation of angular momentum • 2nd Law: A line connecting the sun to a given planet sweeps out equal areas in equal times => a planet moves faster when it is closer to the sun.

!

Kepler's 2nd Law : dA

dt= constant

Relate dA

dt to angular momentum (

! L ) of the planet about the Sun

dA

dt=

1

2

! r "! v =

1

2m

! r "m

! v =

! L

2m

dA

dt= constant #

! L = constant # small r means larger v.

! L = constant when torque = 0.

Newtons concluded that the gravitaional force must be directed

along the "radius" from the Sun to the planet.

Since ! F and

! r are along the same direction,

! r "! F = 0.


Kepler’s 3rd Law and Newton’s Law of Gravitation

!

Derived in an earlier slide, repeat here :

Newton's 2nd Law : F = ma = mpv2

r

Newton's Law of gravitation : F =GmpMS

r2

= mpv2

r

" v2 =GM

S

r

Assume circular orbit " v =2#r

T, T = orbital period

"2#r

T

$

% &

'

( )

2

=GM

S

r

" T =(2# )r3 / 2

GMS

* r3 / 2 (Kepler's 3rd Law)

Newton did this backward, he started with Kepler's 3rd Law and

deduce his law of gravitaiton.

Note : We can use Kepler's 3rd law to calculate the mass of the Sun

based on the radius and the period of the Earth's orbit around the Sun.


Sun and planet revolve about the center of mass.

We have assumed the Sun was stationary while the planet orbits around the Sun.

In fact, a better way to view their motion is that they both revolve about their center of mass.

Download - 12 Lecture Lam - University of Hawaii Systemplam/ph170_summer_2011/L12/12_Lecture...3 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Gravitational potential

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