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Early Observations of Planetary Motion
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Just looking at the sky, one would assume that
the earth is the center of the universe, and that
everything we see is in orbit around us.
Early models of the solar system were
geocentric, but some of the planets were
observed to exhibit retrograde motion:
Kepler’s Laws:
Early Observations of Planetary Motion
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The geocentric model
was made more
elaborate, with epicycles
upon epicycles, but
ultimately was unable to
explain retrograde
motion.
Kepler’s Laws and Newton’s Synthesis
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Planets move in planar elliptical
paths with the Sun at one focus
of the ellipse.
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During equal time intervals the
radius vector from the Sun to a
planet sweeps out equal areas.
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If T is the time that it takes for a
planet to make one full revolution
around the Sun, and if S is half
the major axis of the ellipse (S
reduces to the radius of the
planet’s orbit if that orbit is
circular), then:
where C is a constant whose value
T2
ΣFn = G
but
S3
=C
∴G
m1mSun
v12
m
=
1
r1
r12
2πr1
v1 =
T1
mSun 4π 2 r1
= 2
r12
T1
⇒
T12
4π 2
=
r13 GmSun
⇒ C=
4π 2
GmSun
is the same for all planets.
1
The asteroid belt between Mars and Jupiter consists of
many fragments, once hypothesized to have been a planet.
a) If the center of mass of the asteroid belt is about 3.0
times farther from the Sun than the Earth is, how long would
have it taken for the hypothetical planet to orbit the Sun?
b) Can we use this data to deduce the mass of that planet?
Perturbations
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Kepler’s Laws are not exactly correct
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Perturbations in orbits were observed
Newton expected this
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A body’s motion is influenced by the gravitational
attraction of ALL other bodies in the universe.
⎛ m
⎞
m
m
F Earth = GmEarth ⎜⎜ 2 Sun rˆSun − Earth + 2 Mars rˆMars − Earth + 2 Venus rˆVenus − Earth + ... ⎟⎟
r
r
r
Mars − Earth
Venus − Earth
⎝ Sun − Earth
⎠
Newton’s Idea
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Force at a distance
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‰
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People were familiar with “contact” forces.
This was new (and NOT universally accepted initially)
Implications
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Causality
Deterministic Universe
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Gravitational force per unit mass at point.
“Caused” by a mass
Taken from Electricity and Magnetism
Gravity
Electroweak
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Formally, electromagnetic and weak nuclear
Strong nuclear
Lead to “Modern Physics”
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Predominate view for about 200 years
Force at a distance and causality caused problems
Concept of a gravitational field
„
Forces in Nature
Relativity
Quantum Mechanics
Non-deterministic
Causality is difficult
New Ideas
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GUT – Grand Unified Theory
TOE – Theory of Everything
Sting Theory
Holographic theory
MORE SLIDES TO FOLLOW!!!
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ConcepTest 12.1a Earth and Moon
Which is stronger,
Earth’s pull on the
Moon, or the
Moon’s pull on
Earth?
ConcepTest 12.2 Fly Me Away
1) the Earth pulls harder on the Moon
You weigh yourself on a scale inside
2) the Moon pulls harder on the Earth
an airplane that is flying with constant
3) they pull on each other equally
speed at an altitude of 20,000 feet.
4) there is no force between the Earth and
the Moon
How does your measured weight in the
5) it depends upon where the Moon is in its
orbit at that time
measured on the surface of the Earth?
airplane compare with your weight as
1) greater than
2) less than
3) same
At a high altitude, you are farther away from the
By Newton’s 3rd Law, the forces are
center of Earth. Therefore, the gravitational force in
equal and opposite.
the airplane will be less than the force that you
would experience on the surface of the Earth.
ConcepTest 12.3 Two Satellites
Two satellites A and B of the same mass
are going around Earth in concentric
orbits. The distance of satellite B from
Earth’s center is twice that of satellite A.
What is the ratio of the centripetal force
acting on B compared to that acting on A?
ConcepTest 12.4 Averting Disaster
1) It’s in Earth’s gravitational field
1) 1/8
2) 1/4
The Moon does not
3) 1/2
crash into Earth
4) it’s the same
because:
5) 2
2) The net force on it is zero
3) It is beyond the main pull of Earth’s
gravity
4) It’s being pulled by the Sun as well as by
Earth
5) none of the above
Using the Law of Gravitation:
Mm
F =G 2
R
The Moon does not crash into Earth because of its high
Note the
1/r2 factor
speed. If it stopped moving, it would, of course, fall
directly into Earth. With its high speed, the Moon
would fly off into space if it weren’t for gravity
we find that the ratio is 1/4.
providing the centripetal force.
Follow-up: What happens to a satellite orbiting Earth as it slows?
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ConcepTest 12.5 In the Space Shuttle
ConcepTest 12.6 Guess my Weight
1) They are so far from Earth that Earth’s gravity
doesn’t act any more.
If you weigh yourself at the equator
Astronauts in the 2) Gravity’s force pulling them inward is cancelled
by the centripetal force pushing them outward.
space shuttle
3) While gravity is trying to pull them inward, they
float because:
are trying to continue on a straight-line path.
of Earth, would you get a bigger,
smaller or similar value than if you
1) bigger value
2) smaller value
3) same value
weigh yourself at one of the poles?
4) Their weight is reduced in space so the force of
gravity is much weaker.
Astronauts in the space shuttle float because
The weight that a scale reads is the normal force exerted by the
floor (or the scale). At the equator, you are in circular motion, so
they are in “free fall” around Earth, just like a
satellite or the Moon. Again, it is gravity that
there must be a net inward force toward Earth’s center. This
means that the normal force must be slightly less than mg. So
the scale would register something less than your actual weight.
provides the centripetal force that keeps them
in circular motion.
Follow-up: How weak is the value of g at an altitude of 300 km?
ConcepTest 12.7 Force Vectors
A planet of mass m is a
distance d from Earth.
Another planet of mass 2m
is a distance 2d from Earth.
Which force vector best
represents the direction of
the total gravitation force
on Earth?
2d
5
4
d
1
The force of gravity on the
Earth due to m is greater than
the force due to 2m, which
means
that
the
force
component pointing down in
the figure is greater than the
component pointing to the
right.
2m
2
3
m
F2m = GME(2m) / (2d)2 = 1/2 GMm / d2
Fm = GME m / d2 = GMm / d2
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