13
CHAPTER OUTLINE
13.1
Newton’s Law of Universal
Gravitation
13.2
Measuring the Gravitational
Constant
13.3
FreeFall Acceleration and
the Gravitational Force
13.4
Kepler’s Laws and the
Motion of Planets
13.5
The Gravitational Field
13.6
Gravitational Potential
Energy
13.7
Energy Considerations in
Motion
Planetary and Satellite
Universal Gravitation
ANSWERS TO QUESTIONS
Q13.1
Because
g
is the same for all objects near the Earth’s surface.
The larger mass needs a larger force to give it just the same
acceleration.
Q13.2
To a good first approximation, your bathroom scale reading is
unaffected because you, the Earth, and the scale are all in free
fall in the Sun’s gravitational field, in orbit around the Sun. To
a precise second approximation, you weigh slightly less at
noon and at midnight than you do at sunrise or sunset. The
Sun’s gravitational field is a little weaker at the center of the
Earth than at the surface subsolar point, and a little weaker still
on the far side of the planet. When the Sun is high in your sky,
its gravity pulls up on you a little more strongly than on the
Earth as a whole. At midnight the Sun pulls down on you a
little less strongly than it does on the Earth below you. So you
can have another doughnut with lunch, and your bedsprings
will still last a little longer.
Q13.3
Kepler’s second law states that the angular momentum of the Earth is constant as the Earth orbits
the sun. Since
Lmr
=
ω
, as the orbital radius decreases from June to December, then the orbital speed
must increase accordingly.
Q13.4
Because both the Earth and Moon are moving in orbit about the Sun. As described by
Fm
a
gravitational
centripetal
=
, the gravitational force of the Sun merely keeps the Moon (and Earth) in a
nearly circular orbit of radius 150 million kilometers. Because of its velocity, the Moon is kept in its
orbit about the Earth by the gravitational force of the Earth. There is no imbalance of these forces, at
new moon or full moon.
Q13.5
Air resistance causes a decrease in the energy of the satelliteEarth system. This reduces the diameter
of the orbit, bringing the satellite closer to the surface of the Earth. A satellite in a smaller orbit,
however, must travel faster. Thus, the effect of air resistance is to speed up the satellite!
Q13.6
Kepler’s third law, which applies to all planets, tells us that the period of a planet is proportional to
r
32
. Because Saturn and Jupiter are farther from the Sun than Earth, they have longer periods. The
Sun’s gravitational field is much weaker at a distant Jovian planet. Thus, an outer planet experiences
much smaller centripetal acceleration than Earth and has a correspondingly longer period.
381
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Universal Gravitation
Q13.7
Ten terms are needed in the potential energy:
U
UUUUUUUUUU
=+++++++++
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45
.
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 Spring '08
 Shannon
 Acceleration, Force, General Relativity, kg

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