Transcript File

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Falling Bodies

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
Just as an apple falls to the ground due to
the Earth’s gravity…
The moon is falling towards the Earth!
Why doesn’t the moon crash into the
Earth?
The moon has velocity that keeps it moving
in an orbit. It is “constantly falling around
the Earth.”
Gravitational force between the Earth and
the moon is providing centripetal force to
keep the moon in orbit.
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
Johannes Kepler – astronomer who used
geometry and mathematics with a sun centered
solar system to calculate the motion of the
planets; developed 3 laws that describe the
behavior of every planet and satellite.
Heliocentric: Sun-Centered
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Kepler’s 3 Laws of Planetary Motion
Kepler’s 1st Law of Planetary Motion: the
paths of the planets are ellipses with the
center of the sun at one focus;
 This went against the church’s belief that the
planets followed circular paths!
 Ellipse – oval shape drawn about two foci

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Seasons

So…if the Earth gets closer and farther
from the Sun as it follows its elliptical
path, is this why we have seasons??

It is the tilt of the Earth on its own axis
towards or away from the Sun that
causes the seasons.
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50 years later….Newton takes over
Kepler – Kinematics of planetary
motion (HOW planets move)
Newton – Dynamics of planetary motion
(WHY planets move the way they do)
By Newton’s 1st Law, the Earth
would want to travel through
space in a straight line path.
But it doesn’t!
It travels in an ellipse! So…
Some force must be acting on the Earth
to produce this path.
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Newton continues….
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Newton investigates what variables affect this force. He
first notices that distance is an important variable.
The farther away from the sun the Earth is the smaller
the force. The closer to the sun the larger the force.
Inverse Square Law – any law that states one variable
is inversely proportional to the square of another
variable.
1
F 2
d
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Newton continues….
Other possible variables of this Force:

Volume
To eliminate volume:
 Weight
To eliminate Weight:
-Consider all objects to be a
-Weight depends on mass, we will
single point.
-Generalizes all things in the
universe to be the same size.
account for it through mass
Mass
Can’t eliminate Mass!

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Law of Universal Gravitation

Newton put all of these thoughts together and with a lot of work came up with
the Law of Universal Gravitation.

Law of Universal Gravitation – for any pair of objects, each
object attracts the other with a force directly proportional to
the product of the masses of the objects, and inversely
proportional to the square of the distance between their
centers.
m1m2
F G 2
d
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Law of Universal Gravitation

Universal Gravitational Constant (G) – constant used to
provide the proper units and proportion when using
Newton’s Law of Universal Gravitation.
G  6.67 10
11
N  m2
kg 2
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Finding The Value of G:
The Cavendish Experiment

Newton knew that their had to be a constant G
to convert the final calculation to the proper
magnitude. However he was never able to
develop an experiment to find G.
100 years after Newton’s death….

Henry Cavendish – measured the value of G
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Using Newton and Cavendish’s work
Suppose we send three people to different locations in the world.
The 1st person goes to Washington, D.C. which is located at sea level.
The 2nd person goes to Mt. Whitney in CA which is 4500 m above sea level.
The 3rd person goes to Mt. Everest in China which is 9000 m above sea level.
Based on the work of Newton and Cavendish determine if the acceleration
due to gravity experienced by all three people is the same or if it varies.
Find g for each case.
M EG
g
2
rE  d 
1. g = 9.797 m/s2
2. g = 9.783 m/s2
3. g = 9.769 m/s2
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Some values of g
Miles
(mi)
Equivalent Meters
(m)
Value of g
(m/s2)
0
30
200
300
1000
3000
0
48,000
320,000
480,000
1,600,000
4,800,000
9.80
9.65
8.88
8.47
6.26
3.19
NOTE: Highest altitude on Earth: Mount Everest (8850 m)
Lowest altitude on Earth: Dead Sea (Israel) (-400m)
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Can the Force of Attraction = 0?
m1m2
F G 2
d
For the attractive Force to be zero, either G or one of
the masses must equal zero.
G is a constant and mass is never zero…So,
Attractive Force can NEVER be zero!
Can g = 0?
Since F is never zero,
F  mg
g cannot be zero!
Can g be undefined?
Gm1m2
mg 
2
r
For g to be undefined the denominator must be zero.
Two objects cannot occupy the same space so the
distance between them will never be zero…
So g is never undefined!
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Weightlessness?
Can we ever experience true weightlessness?
Fw = mg
and
g is never zero!
So Weight is never zero!
No such thing as true weightlessness.
support force- force that balances the weight of an object
at rest (normal force, tension, etc)
apparent weightlessness- experiencing the effects of zero
gravity due to the lack of a support force
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Weightlessness?
Objects in orbit are always falling toward
the object they are orbiting. For instance
satellites fall toward the Earth the entire
time they are in orbit.
By Newton’s 1st Law, they want to
continue in a straight line, but the
attraction from the Earth makes them fall
towards the Earth.
Because orbiting objects are always falling
they are lacking a support force and
experience apparent weightlessness.
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Example 3: Two objects have their centers 2.0m apart.
One has a mass of 8.0 kg. The other has a mass of 6.0
kg. What is the gravitational force between them?
8.0 kg
F=?
6.0 kg
2.0 m
m1m2
11 Nm2 (8kg)( 6kg)
F  G 2  (6.67 x10
)
2
kg
d
(2.0m) 2
 8.0 x10
10
N
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Example 4: The Milky Way galaxy contains approximately 4.0 x 1011
stars with an average mass of 2.0 x 1030 kg each. How far away is
the Milky Way from our nearest neighbor, the Andromeda Galaxy, if
Andromeda contains roughly the same number of stars and attracts
the Milky Way with a gravitational force of 2.4 x 1030 N?
Milky Way
F = 2.4 x 1030 N
Andromeda
d=?
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