Gravity and the Expanding Universe

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Gravity and the Expanding Universe

• Thursday, January 31

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Isaac Newton (1642-1727) Discovered 3 Laws
of Motion, Law of Gravity
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Newtons First Law of Motion
An object remains at rest, or moves in a
straight line at constant speed, unless acted on
by an outside force.
Mathematical laws require precise definitions of
terms.
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SPEED rate at which an object changes its
position.
Example 65 miles per hour.
VELOCITY speed plus direction of travel
Example 65 miles per hour to the north.
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ACCELERATION rate at which an object changes
its velocity.

• Acceleration can involve
• increase in speed
• 2) decrease in speed
• 3) change in direction.

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Example of acceleration an apple falls from a
tree.
Acceleration 9.8 meters/second/second.
After 1 sec, speed 9.8 meters/sec, After 2
sec, speed 19.6 meters/sec, etc
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FORCE a push or pull acting to accelerate an
object.
Examples Gravity pull Electrostatic attractio
n pull
Electrostatic repulsion push
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Restatement of First Law In the absence
of outside forces, velocity is constant.
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Second Law of Motion The
acceleration of an object is directly
proportional to the force acting on it, and
inversely proportional to its mass.
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Example a package of cookies has mass m
0.453 kilograms.
It experiences a gravitational acceleration a
9.8 meters/sec2.
How large is the force acting on the cookies?
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F (0.453 kg) (9.8 m/sec2)
F 4.4 kg m / s2
F 4.4 Newtons
F 1 pound
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Third Law of Motion For
every action, there is an equal and opposite
reaction.
If A exerts a force on B, then B exerts a force
on A thats equal in magnitude and opposite in
direction.
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Example I balance a package of cookies on my
hand.
Cookies push on hand F 1 pound,
downward. Hand pushes on cookies F 1 pound, up
ward.
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I remove my hand.
Earth pulls on cookies F 1 pound,
downward. Cookies pull on Earth F 1 pound, u
pward.
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Third Law states force on
Earth force on cookies.
Second Law states acceleration force
divided by mass.
Mass of Earth 1025 mass of cookies Therefore,
acceleration of cookies 1025
acceleration of Earth.
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Newtons Law of Gravity
Gravity is an attractive force between all pairs
of massive objects.
How big is the force? Thats given by a (fairly)
simple formula.
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Newtons Law of Gravity
F force
m1 mass of one object
m2 mass of other object r
distance between centers of objects G
universal constant of gravitation (G 6.7
10-11 Newton meter2 / kg2)
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Gravity makes apples fall it also keeps the Moon
on its orbit around the Earth, the Earth on its
orbit around the Sun, the Sun on its orbit
around the Galactic center.
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The universe is full of objects attracting each
other are these attractive forces enough to stop
the expansion?
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Lets start with a related problem
A boy standing on the Earth throws an apple
upward initially, the distance between apple
Earth is increasing.
Is the attractive force between apple Earth
enough to stop the apple from rising?
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What goes up must come down.
unless its traveling faster than the escape
speed.
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Small initial speed short distance upward.
Larger initial speed long distance upward.
Speed escape speed to infinity!!
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Escape speed from a planet (or star) depends on
its density (?) radius (r).
Escape speed from Earth 11 km/sec
25,000 mph
Escape speed from Sun 620 km/sec
1,400,000 mph
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Suppose a sphere of gas (radius r) is expanding
outward at a speed v.
v
r
v
v
v
If expansion speed is greater than escape speed
(v vesc), sphere will expand forever.
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Higher density ? leads to a higher escape speed
vesc.
v
r
v
v
v
For given values of v and r, there is a critical
density ?crit at which vesc v.
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Offered without proof critical density below
which the sphere expands forever is
(Small, rapidly expanding spheres need a higher
density to recollapse them.)
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Suppose our sphere of gas is part of the
expanding universe, so that v H0r
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This critical density depends only on the
universal constant of gravitation G and on the
Hubble constant H0.
We know the values of G and H0!
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With H0 70 km/sec/Mpc, the critical density for
the universe is
?crit 9 10-27 kg/m3
Yes, this is a very low density! Water 1000
kg/m3 Air 1 kg/m3
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Most of the universe consists of very low density
intergalactic voids.
Not immediately obvious that ? ?crit
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Newton says fate of the universe depends on the
ratio of its density to the critical density.
Omega (O) is also called the density parameter.
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O O 1
Distance between two galaxies
O 1
Time
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(No Transcript)
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O1 (density greater than critical) The Big
Crunch
(recollapse, becoming hotter)
O1 (density less than or equal to critical) The
Big Chill
(perpetual expansion, becoming cooler)
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Amusing speculation of the day perhaps a Big
Crunch would lead to a Big Bounce.
Or here
Or maybe here
You are here
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Thursdays Lecture
Einsteins Universe
Reading
Chapter 6