Gravitation 1

Gravitation

Gravitation 2

Newton’s Law of GravitationNewton proposed that any two masses were attracted by agravitational force (inverse square law).

!

Fg = Gm1m2r 2

G = 6.67 x 10-11 N !m2

kg2"

# $

%

& '

Where:F = gravitational force (N)m = mass of body (kg)r = distance between m1 and m2 (m)

!

v F G =

Gm1m2

r 2

Gravitation 3

Newton’s Law of Gravitation

!

Fg1 ,2 = Fg2 ,1 = Gm1m2r 2

r

!

m1

!

m2

!

v F g2 ,1

!

v F g1 ,2

Gravitation 4

Newton’s Law of Gravitation (Vector Form)

!

v r 1 ,2

!

m1

!

m2

!

v F 1,2 = "G m1m2

r 2ˆ r 1 ,2

!

where ˆ r 1 ,2 is a unit vector directed from m1 to m2

!

ˆ r 1 ,2

!

v F 1,2

Gravitation 5

Newton’s Law of Gravitation (Vector Form)

!

v r 2 ,1

!

m1

!

m2

!

v F 2 ,1 = "G m1m2

r 2ˆ r 2 ,1

!

where ˆ r 2 ,1 is a unit vector directed from m2 to m1

!

v F 2 ,1

!

ˆ r 2 ,1

Gravitation 6

Weight

The acceleration of an object (m) due to a planet ormoon’s (M) gravitation can be found by using theinverse square law and Newton’s second law.

!

Fg = GMmR2

!

=mg

So the acceleration due to gravity at the surface ofthe planet or moon is:

!

g =GMR2

Gravitation 7

Weight

At a point above a planet or moon’s surface a distancer from the center of the planet or moon the weight of abody is:

!

Fg = GMmr 2

R

!

r = R + dM

dm

Gravitation 8

Motion of Satellites (Circular Orbits)

A satellite in an orbit that is always the same heightabove the planet or moon moves with uniform circularmotion. Using Newton’s second law (F = ma):

!

GMmr 2

=mv2

rThe orbital speed is therefore:

!

v =GMr

!

Fg = mac

Gravitation 9

For circular orbits period T for the satellite isrelated to speed v.

!

v =2"rT

Therefore the period is:

!

= 2" r 3

GM

!

T =2"rv

!

= 2"r rGM

Motion of Satellites (Circular Orbits)

Gravitation 10

Gravitational Potential Energy

!

Wgrav = Frdrr1

r2"

!

r1

!

r2

!

M!

m

!

Fr

Because the gravitational force is an attractive force(directed toward the center of the planet or moon) theradial component is negative.

!

Fr = "GMmr 2

Gravitation 11

Gravitational Potential Energy

Therefore Wgrav is given by:

!

Wgrav = "GMmr 2

# $ %

& ' (

r1

r2) dr

!

= "GMm drr 2r1

r2#

!

Wgrav = GMm 1r2"1r1

#

$ %

&

' (

!

= "GMm "1r

# $ %

& ' ( ) * + r1

r2

!

= "GMm "1r2""1r1

#

$ %

&

' (

Gravitation 12

Gravitational Potential Energy

Recalling that

!

Wgrav =GMmr2

"GMmr1

U is zero when the mass m is infinitely far awayfrom the planet or moon.

!

W = "#U

!

= " U2 "U1( )

!

U = "GMmr

the gravitational potential energy is:!

= ""GMmr2

""GMmr1

#

$ %

&

' (

!

UG = "Gm1m2r

Gravitation 13

Gravitational Potential Energy

!

UG

!

r

!

UG = "Gm1m2r

!

R

Gravitation 14

Motion of Satellites

!

E = "GMm2r

!

=12mv2 + "GMm

r# $ %

& ' (

!

E =12m GM

r" # $

% & ' + (GMm

r" # $

% & '

The total mechanical energy E of a satellite in acircular orbit of radius r is:

!

E = K + U

Gravitation 15

Kepler’s Laws of Planetary Motion1.) The paths of the planets are ellipses with the center of

the Sun at one focus.

2.) An imaginary line from the Sun to a planet sweeps outequal areas in equal time intervals. Thus, planets movefastest when closest the Sun, slowest when farthestaway.

3.) The ratio of the squares of the periods of any twoplanets revolving about the Sun is equal to the cubes oftheir average distances from the Sun.

!

TaTb

"

# $

%

& '

2

=rarb

"

# $

%

& '

3

Gravitation 16

Kepler’s Third Law

!

TaTb

"

# $

%

& '

2

=rarb

"

# $

%

& '

3

!

T = 2" r 3

GM

Recall that for circular orbits:

!

T 2 =4" 2

GMr 3

The Law of Gravity 17

Kepler’s Third Law (Newton’s version)

!

T 2 =4" 2

GM#

$ %

&

' ( r 3

!

T12 =

4" 2

GM#

$ %

&

' ( r1

3

!

and T22 =

4" 2

GM#

$ %

&

' ( r2

3

!

T12

T22 =

4" 2

GM#

$ %

&

' ( r1

3

4" 2

GM#

$ %

&

' ( r2

3

!

so T1

T2

"

# $

%

& '

2

=r1r2

"

# $

%

& '

3

Gravitation 18

Satellites in an Elliptical Orbit

The point in the planet’s orbit closest to the Sun isthe perihelion, and the point most distant from theSun is the aphelion.

The longest dimension is the major axis, with half-length a. This half-length is called the semi-majoraxis.

The distance of each focus from the center of theellipse is ea where e is the eccentricity.

a

eaaphelionperihelion

Gravitation 19

Satellites in an Elliptical Orbit

The expressions for period T and total energy Efor satellites in circular orbits of radius r also holdfor elliptical orbits, if r is replaced by a, the lengthof the semi-major axis:

!

E = "GMm2a

!

T = 2" a3

GM

Gravitation 20

Escape Velocity

!

=12mvi

2 + "GMmR

# $ %

& ' ( = "GMm

rmax

!

E = K + U

Energy considerations can be used to determine the minimuminitial speed needed to allow an object to escape a planet ormoon’s gravitational field.At the surface of the planet or moon, v = vi and r = ri = R.When the object reaches its maximum altitude, v = vf = 0 and r= rf = rmax.

Since total energy is constant:

!

vi2 = 2GM 1

R"1rmax

#

$ %

&

' (

!

Letting rmax "# and taking vi = vesc

!

vesc =2GMR

Gravitation 21

Gravity Inside of the Earth

REmE

r!

Mm

O Fg

Gravitation 22

Gravity Inside of the Earth

!

Fr = "G Mmr 2

Assuming a uniform mass density

!

Fr = "GmEmRE3 r

!

M = "VM!

" =mEVE

!

=mE43"RE

3

!

=mE43 "RE

3

#

$ % %

&

' ( ( 43 " ) r

3

!

= mEr 3

RE3

!

= "GmE

r 3

RE3

#

$ %

&

' ( m

r 2

Gravitation 23

Gravity Inside of the Earth

!

Fr

!

r

!

RE

!

"GmEmRE2 !

Fr = "GmEmr 2

!

Fr = "GmEmRE3 r