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11/17/09
Gravity
Kepler’s Laws
Newton’s Law of gravitation
Gravitational potential energy
Physic 201
Fall 2009
Kepler’s Laws - historical remarks
Johannes Kepler, (1571 – 1630) was a
German mathematician, astronomer and
astrologer, and key figure in the 17th century
scientific revolution. He is best known for his
eponymous laws of planetary motion, codified
by later astronomers based on his works
Astronomia nova, Harmonices Mundi, and
Epitome of Copernican Astronomy. They also
provided one of the foundations for Isaac
Newton's theory of universal gravitation.
During his career, Kepler was a mathematics
teacher at a seminary school in Graz, Austria,
an assistant to astronomer Tycho Brahe,
the court mathematician to Emperor Rudolf II,
a mathematics teacher in Linz, Austria, and an
adviser to General Wallenstein. He also did
fundamental work in the field of optics,
invented an improved version of the refracting
telescope (the Keplerian Telescope), and
helped to legitimize the telescopic discoveries
of his contemporary Galileo Galilei.
Source: wikipedia
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Kepler’s Laws
•  Kepler’s First Law:
–  All planets move in elliptical orbits with the sun at one focus
•  Kepler’s Second Law:
–  The radius vector drawn from the Sun to a planet sweeps out equal
areas in equal times, (the law of equal areas).
•  Kepler’s Third Law:
–  The square of the orbital period of any planet is proportional to the
cube of the semimajor axis of the elliptical orbit.
2
3
T = Cr
Today we can understand the physical reasons for these
laws …
Let’s remind us first of the geometry of the ellipse and then discuss the
three laws.
Ellipses
e = c/a
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Planetary Orbits
Kepler found that the orbit of Mars was
an ellipse, not a circle.
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2nd law: Equal areas in equal times
dA =
 1 
1
L
r × dr = r × vdt =
dt
2
2
2m
Kepler’s Third Law
Kepler had access to very
good data from the
astronomer Tycho Brahe in
Prague. See table for today’s
data.
After many years of work
Kepler found an intriguing
correlation between the orbital
periods and the length of the
semimajor axis of orbits.
T 2 = Cr 3
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Close-up of Third law relation
Of the satellites shown revolving around Earth, the
one with the greatest speed is
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Of the satellites shown revolving around Earth, the
one with the greatest speed is
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The orbits of two planets orbiting a star are shown. The
semimajor axis of planet A is twice that of planet B. If the
period of planet B is TB, the period of planet A is
A.
2TB
B.
2 2 TB
C. 3TB
D.
E.
3 TB
4TB
A’
rA
A
rB
B
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The orbits of two planets orbiting a star are shown. The
semimajor axis of planet A is twice that of planet B. If the
period of planet B is TB, the period of planet A is
A.
2TB
B.
2 2 TB
C. 3TB
D.
E.
A’
rA
A
rB
3 TB
4TB
B
Newton’s Law of Gravity
•  Newton’s law of gravity will provide a physical theory of Kepler’s laws.
m
M
F12

mM
F 12 = G 2 r̂12
R
F21
r
Magnitude of force
Fg = G
G = 6.67 ⋅10 −11
mM
r2
Nm 2
kg
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The Cavendish experiment
Demo
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Gravitational and inertial mass
•  Gravitation is a force that acts on the gravitational mass
Fg = G
mg M g
r2
•  Newton’s Law acts on the inertial mass
F = mi a
•  One could imagine that the gravitational mass mg is not
the same as m_i, but they are.
Kepler’s Third Law derived from Newton’s Law
Easily derived for a circular orbit
Centripetal force = gravitational force
Fcent = m
v2
= mRω 2
R
Fg = G
mM
R2
mM
R2
4π 2
M
R 2 =G 2
T
R
mRω 2 = G
ω=
2π
T
R 3 GM
=
= const.
T 2 4π 2
Kepler’s Third Law
T2 =
4π 2 3
R
GM S
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Kepler’s Third Law derived from Newton’s Law
Extension for elliptical orbits,
(Without proof R  a)
Kepler’s Third Law
T2 =
T2 =
4π 2 3
R
GM S
4π 2 3
a
GM S
Where a is the semimajor axis
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A woman whose weight on Earth is 500 N is lifted to a
height of two Earth radii above the surface of Earth. Her
weight
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decreases to one-half of the original amount.
decreases to one-quarter of the original amount.
does not change.
decreases to one-third of the original amount.
decreases to one-ninth of the original amount.
A woman whose weight on Earth is 500 N is lifted to a
height of two Earth radii above the surface of Earth. Her
weight
A. 
B. 
C. 
D. 
E. 
decreases to one-half of the original amount.
decreases to one-quarter of the original amount.
does not change.
decreases to one-third of the original amount.
decreases to one-ninth of the original amount.
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Chapter 11 – Gravity
Lecture 2
Physics 201
Fall 2009
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The Cavendish experiment (second try)
Gravitational potential energy
Escape velocity
Gravitational Field
Gravitational Field for mass distributions
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Discrete
Rod
Spherical shell
sphere
Measuring G
•  G was first measured by
Henry Cavendish in 1798
•  The apparatus shown here
allowed the attractive force
between two spheres to
cause the rod to rotate
•  The mirror amplifies the
motion
•  It was repeated for various
masses
Magnitude = G
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mM
r2
G = 6.67 × 10 −11 N m 2 / kg 2
Physics 201, UW-Madison
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