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Gravity
Geocentric vs. Heliocentric Model
The Geocentric Model
Arguments For:
• Parallax not seen
• Almagest says so
• Fits with “heavenly” perfection
Arguments Against:
• Complicated system -- wheels
within (off-centered) wheels
• Not very accurate
• No “why”
The Heliocentric Model
Arguments For:
• Simple system
• Almagest is wrong
• Not everything revolves about
the Earth (Jupiter’s Moons)
• Heavens aren’t perfect
Arguments Against:
• Parallax not seen
• Not very accurate
• No “why”
• Earth no longer special
Tycho Brahe Measurements
Tycho Brahe (without a telescope) made extremely accurate
measurements of the positions of the stars and planets.
What Tycho Brahe Observed
• A nova was outside the Earth’s atmosphere
What Tycho Brahe Observed
• A nova was outside the Earth’s atmosphere
• 20 years of extremely accurate measurements of
the positions of the planets (~ 2 arcmin precision)
Tycho Brahe’s Model
• Earth at the center of the
“Universe” (because
parallax is not seen)
• The Sun travels about the
Earth in a perfect circle
• The planets move around
the Sun in perfect circles
Kepler’s Analysis
By working for Tycho
Brahe (and then stealing the
data), Johannes Kepler had
access to the most precise
data on planetary positions
in history.
He fit the data in every way
imaginable.
Kepler’s Laws
First define the distance from the Earth to the Sun to be
1 Astronomical Unit (1 A.U.)
Kepler’s Laws
1) Planets move in ellipses
with the Sun at one focus
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Kepler’s Laws
First define the distance from the Earth to the Sun to be
1 Astronomical Unit (1 A.U.)
Kepler’s Laws
1) Planets move in ellipses
with the Sun at one focus
2) Planets in their orbits
sweep out equal areas in
equal times
Kepler’s Laws
First define the distance from the Earth to the Sun to be
1 Astronomical Unit (1 A.U.)
Kepler’s Laws
1) Planets move in ellipses
with the Sun at one focus
QuickTime™ and a
Video decompressor
are needed to see this picture.
2) Planets in their orbits
sweep out equal areas in
equal times
3) The period, P, of an orbit (in years) squared is equal to
a, the semi-major axis of the orbit (in A.U.) cubed. In
terms of mathematics, P2 = a3
These laws are empirical – Kepler knew no reason for them
State of Physics
By now the world knew:
• Bodies of different weights fall
at the same speed
• Bodies in motion did not
necessarily come to rest
• Moons could orbit different
planets
• Planets moved around the Sun in
ellipses with the Sun at one focus
• The orbital speeds of the planets
obeyed Kepler’s 2nd and 3rd law
But why??? Isaac Newton put it all together.
Newton’s Concepts
1)
2)
3)
4)
m (mass): How much stuff something contains
v (velocity): A body’s speed and direction
a (acceleration): The change in a body’s velocity
F (force): What is needed to change a body’s velocity
Newton’s Laws of Motion
1)
A body’s velocity will remain constant, unless acted upon by
an outside force.
2)
A body’s acceleration depends on the force acting upon it, and
will be in the direction of that force. Its resistance to
acceleration depends on its mass. In equation form, this is
F=ma
3)
For every force, there is an equal and opposite force.
Newton’s Law of Gravity
There is an attractive force between
two bodies called gravity. The force
of gravity depends on the masses of
the two bodies, and their separation
(squared); the larger the mass, the
greater the attraction; the larger the
separation, the smaller the attraction.
Note that the word “separation”
means the distance between the
centers of the two bodies.
G M 1 M2
F = 
r2
Example of Gravity – a Thrown Ball
When you throw a ball, there are
two motions: sideways, and up.
The sideways motion obeys
Newton’s first law (bodies in
motion will stay in motion). The
attractive force of gravity causes
the upward motion to decelerate,
and then change direction. You
see the composite of the two
behaviors.
Example of Gravity – a Planetary Orbit
Imagine a planet moving
sideways with respect to the
Sun. Newton’s first law says
that it will continue to move
sideways. But the law of
gravity says that it will also
be pulled towards the Sun.
The result is a combination
motion, in which the planet
falls towards the Sun, but
misses. This is an orbit.
Example of Gravity – a Planetary Orbit
Imagine a planet moving
sideways with respect to the
Sun. Newton’s first law says
that it will continue to move
sideways. But the law of
gravity says that it will also
be pulled towards the Sun.
The result is a combination
motion, in which the planet
falls towards the Sun, but
misses. This is an orbit.
If the initial sideways motion exactly balances the attractive
motion, the orbit will be circular. Otherwise, it will be elliptical.
Example of Gravity – Weightlessness
You feel weight because of
Newton’s third law. Gravity is
pulling you down, but the ground
is not allowing you to fall. It
must therefore be exerting a
force on you to keep you from
falling. That force is the weight
that you feel.
If you were allowed to fall, you
would not feel any weight.
Example of Gravity – Binary Stars
According to Newton’s third law, just
as the Sun exerts a force on the Earth,
the Earth exerts a force on the Sun.
However, since the Sun is so much
more massive, it doesn’t move much.
But if the Earth were more massive,
the Sun would move. In fact, both
the Sun and the Earth circle a
common center of mass.
Thus, Kepler’s 3rd law is not quite
complete. The true law is
(M1 + M2) P2 = a3
where the orbital period (P) is in years,
the semi-major axis (a) is in A.U.s, and
the masses (M) are in solar units.
Example of Gravity – Binary Stars
According to Newton’s third law, just
as the Sun exerts a force on the Earth,
the Earth exerts a force on the Sun.
However, since the Sun is so much
more massive, it doesn’t move much.
But if the Earth were more massive,
the Sun would move. In fact, both
the Sun and the Earth circle a
common center of mass.
Thus, Kepler’s 3rd law is not quite
complete. The true law is
(M1 + M2) P2 = a3
where the orbital period (P) is in years,
the semi-major axis (a) is in A.U.s, and
the masses (M) are in solar units.
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Example of Gravity – Binary Stars
According to Newton’s third law, just
as the Sun exerts a force on the Earth,
the Earth exerts a force on the Sun.
However, since the Sun is so much
more massive, it doesn’t move much.
But if the Earth were more massive,
the Sun would move. In fact, both
the Sun and the Earth circle a
common center of mass.
Thus, Kepler’s 3rd law is not quite
complete. The true law is
(M1 + M2) P2 = a3
where the orbital period (P) is in years,
the semi-major axis (a) is in A.U.s, and
the masses (M) are in solar units.
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Example of Gravity – Tides
The effects of gravity do not depend
on the composition of a body, just its
mass and distance. The Moon exerts
a force on the Earth, but since the
Earth has a finite size, this force is
different from one side of the Earth to
the other. The side of the Earth near
the Moon gets pulled most, the center
of the Earth less, and the backside
least of all. Since most of the Earth is
solid, it doesn’t move much, but water
reacts to this difference. So we have
tides.
Note that due to the Earth’s rotation,
there are 2 high tides and 2 low tides
per day.
d
d
The difference in gravity over
on a body with size s is
G M1
F = 
s
3
r
Example of Gravity – Tides
The effects of gravity do not depend
on the composition of a body, just its
mass and distance. The Moon exerts
a force on the Earth, but since the
Earth has a finite size, this force is
different from one side of the Earth to
the other. The side of the Earth near
the Moon gets pulled most, the center
of the Earth less, and the backside
least of all. Since most of the Earth is
solid, it doesn’t move much, but water
reacts to this difference. So we have
tides.
Note that due to the Earth’s rotation,
there are 2 high tides and 2 low tides
per day.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Spring and Neap Tides
The Sun also produces tides on
the Earth, but due to its much
larger distance, the solar tides
are only half as strong as the
lunar tides. When the Sun,
Earth, and Moon are aligned,
you get “spring’’ or extrastrong tides. When the Sun,
Earth, and Moon are at right
angles, the solar tide partially
cancels the lunar tide. This
produces “neap” tides.
Note that since you can tell time from the Moon, you also can
approximate the time of the local high and low tides.
Spring and Neap Tides
The Sun also produces tides on
the Earth, but due to its much
larger distance, the solar tides
are only half as strong as the
lunar tides. When the Sun,
Earth, and Moon are aligned,
you get “spring’’ or extrastrong tides. When the Sun,
Earth, and Moon are at right
angles, the solar tide partially
cancels the lunar tide. This
produces “neap” tides.
Note that since you can tell time from the Moon, you also can
approximate the time of the local high and low tides.
Spring and Neap Tides
In reality, the times of high and low tides are usually an hour
or so off the simple prediction, due to the rotation of the
Earth. The water wants to align itself with the Moon, but the
rotation of the Earth moves it away.
Next time -- Light