Transcript Gravitation

Gravitation
“On the Shoulders of Giants”
Newton once said that his success was
based on the fact that he “stood on the
shoulders of giants” In other words, his
work was based on the great work done
by his predecessors.
 Tycho Brahe 1546-1601– the original
party animal
 Johannes Kepler 1571-1630—the
original anti-party animal

The Odd Couple

Kepler wanted establish that God created the
universe with a perfect mathematical order;
understanding the mathematics made him feel
closer to God.
 Brahe just wanted to have fun…however,
Brahe (while having a lot of fun) was one of
the best astronomical observers of history and
kept meticulous data!
The Odd Couple Cont’d
Using Brahe’s data, Kepler formulated 3
laws of motion but we are just going to
worry about Kepler’s 3rd Law of Motion
 The squares of the times to complete
one orbit are proportional to the cubes of
the average distances from the sun
T2 ~R3
T= period of orbit, R= radius of orbit

Now for Newton
Newton knows that
FNet  FGravity
For circular orbits, FNet 
mpv2
R
and theperiodof theorbit is T 
2
2R
R
or v 
v
T
R
T 2  mp R
FNet 
R
T2
and from Kepler,Newton knowsT 2  R 3
mp R mp
FNet  3  2
R
R
Using his own 3rd Law, Newton knows that thisforcemust be symmetrici.e. theforce that the
earthexertson themoonmust equal theforce that themoonexertson theearth
mm
mm
FNet  1 2 2  FGravity  k 1 2 2
R
R
where
mp
11
k  G and G  6.62x10
Nm 2
kg 2
Big G, little g
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Big G, is a constant for the entire universe
Little g, only works on earth and is derived
from Big G

g=GmE/R2
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Where mE= mass of earth
R=radius of earth
Little g varies from place to place for 3 reasons

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The earth is not uniform in its composition
The earth is not a uniform sphere (oblate)
The centripetal acceleration due to earth’s rotation
will cause problems
Tunneling Through the Earth
4 3
Let themass of theearthbe M e   r
3
and let 's drill a tunnelthrough he
t earthand drop somethingin.
T hepart icleis at r  distancefromearth's centerand
 is thedensityof theearth(mass/volume)
Gm Me
Gm 4 3
FG  
  2  r
2
r
r
3
4
FG  Gm   r
3
FG  kr (Just like Hooke's Law F  -kx)
T hus,a part icledroppedthrough he
t earth willoscillateback and forth!
Gravitational Potential Energy
Recall
 
 U  W    F  dr
b
a
Mm
Let F  G 2
And U ()  0
r
 U  U (r )  U ()   U (r )
r
r
Mm
Mm
Mm
U (r )   G 2 dr'  G
 G
r'
r' 
r

Escape Speed
A particle escapes from the earth’s gravity
when its kinetic energy is larger than the
earth’s gravitational potential energy
 Escape speed is the speed when these two
energies are equal or


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½ mv2=GMem/R where R=radius of earth at launch
vescape= (2GMe/R)1/2
When a particle reaches escape speed, it will
not orbit but will fly out into space never to
return.
Orbital Velocity

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The speed by which an
object can orbit another
object is called orbital
velocity and depends on
the radius of the orbit.
It is found by matching
the gravitational force at
a particular radius with
its centripetal force
m v2
 FG
r
m v2
Mm
G 2
r
r
GM
2
v 
r
GM
v
r
Period of the Orbit
3
2r
r
2r 2
T
 2r

v
GM
GM
Black Holes


If the escape speed of a object is equal to the speed of
light, the object is called a “black hole”
While you might think that you set v=c and use the
escape speed formula to solve, you would be both
right and wrong

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Wrong from the standpoint that general relativity takes over
and of course, the kinetic energy of a particle is K= mc2-m0c2
Right from the dumb luck that the final form of the equation
does look like what you would expect from the simple
approach…
c=(2GM/Rs)1/2 where M=mass of the object and Rs =
Schwarzchild radius
Mathematical construct of a sphere with radius Rs is called the
“event horizon”.