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Lecture 21:
Chapter 11 – Gravity
Kepler’s (empirical) Laws
Describe the features of planetary motions,
and led to the Newton’s law of gravity.
Kepler’s Laws
Newton’s Universal Law of gravitation
Johannes Kepler (1571--1630): a German mathematician, astronomer and astrologer;
an assistant to astronomer Tycho Brahe who took a lot of data from telescopes.
Gravitational and inertia mass
Gravitational potential energy
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.
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Kepler’s 1st Law:
Ellipses
e = c/a
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Kepler found that the orbit of Mars was
an ellipse, not a circle.
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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.
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Of the satellites shown revolving around Earth, the
one with the greatest speed is
• 
• 
• 
• 
• 
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
1
2
3
4
5
The constant-area law.
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Newton’s Universal Law of Gravity
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Newton’s Law of Gravity
•  Newton’s law of gravity will provide a physical theory of Kepler’s laws.
m
M
F12
Magnitude of force
F21
r
m
M
F12
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F21
r
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Measuring G
The Cavendish experiment
•  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
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Gravitational and inertial mass
Gravitational torsional
balance used in student labs
for the measurement of G. A
tiny angular deflection of the
balance results in a large
angular deflection of the laser
beam that reflects from a
mirror on the balance.
•  Gravitation is a force that acts on the gravitational mass
(the masses are the source)
•  Newton’s Law of motion acts on the inertial mass
•  In principle, they are not necessarily related, that the
gravitational mass mg is not the same as mi
But they are, up to the current experimental accuracy:
Equivalence principle: gravity is equivalent to acceleration.
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Kepler’s Third Law derived from Newton’s Law
Easily derived for a circular orbit
Extension for elliptical orbits,
(Without proof R  a)
Centripetal force = gravitational force
Kepler’s Third Law
Independent of m!
Where a is the semimajor axis
Kepler’s Third Law
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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
A. 
B. 
C. 
D. 
E. 
Msun = 1.99 x 1030 kg
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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.
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Potential Energy
From work to
gravitational
potential energy.
m
2
M1
In the example before,
it does not matter on
what path the person is
elevated to 2 Earth radii
above.
Only the final height (or
distance) matters for the
total amount of work
performed.
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r
Work done to bring mass m from initial to final position.
Zero point is arbitrary. Choose zero at infinity.
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