204
16 S. Some communications satellites are in orbits called geosynchronous:
the satellite takes one day to orbit the earth from west to east, so that as
the earth spins, the satellite remains above the same point on the equator.
What is such a satellite’s altitude above the surface of the earth.
17 S. As is discussed in more detail in section 5.1 of book 2, tidal interac-
tions with the earth are causing the moon’s orbit to grow gradually larger.
Laser beams bounced off of a mirror left on the moon by astronauts have
allowed a measurement of the moon’s rate of recession, which is about 1
cm per year. This means that the gravitational force acting between earth
and moon is decreasing. By what fraction does the force decrease with
each 27-day orbit. [Hint: If you try to calculate the two forces and
subtract, your calculator will probably give a result of zero due to round-
ing. Instead, reason about the fractional amount by which the quantity 1/
r
2
will change. As a warm-up, you may wish to observe the percentage
change in 1/r
2
that results from changing r from 1 to 1.01. Based on a
problem by Arnold Arons.]
18. Suppose that we inhabited a universe in which, instead of Newton’s
law of gravity, we had
F=km
1
m
2
/r
2
, where k is some constant with
different units than G. (The force is still attractive.) However, we assume
that a=F/m and the rest of Newtonian physics remains true, and we use
a=F/m to define our mass scale, so that, e.g., a mass of 2 kg is one which
exhibits half the acceleration when the same force is applied to it as to a 1
kg mass. (a) Is this new law of gravity consistent with Newton’s third law.
(b) Suppose you lived in such a universe, and you dropped two unequal
masses side by side. What would happen. (c) Numerically, suppose a 1.0-
kg object falls with an acceleration of 10 m/s
2
. What would be the accel-
eration of a rain drop with a mass of 0.1 g. Would you want to go out in
the rain. (d) If a falling object broke into two unequal pieces while it fell,
what would happen. (e) Invent a law of gravity that results in behavior
that is the opposite of what you found in part b. [Based on a problem by
Arnold Arons.]
19 S. (a) A certain vile alien gangster lives on the surface of an asteroid,
where his weight is 0.20 N. He decides he needs to lose weight without
reducing his consumption of princesses, so he’s going to move to a differ-
ent asteroid where his weight will be 0.10 N. The real estate agent’s
database has asteroids listed by mass, however, not by surface gravity.
Assuming that all asteroids are spherical and have the same density, how
should the mass of his new asteroid compare with that of his old one. (b)
Jupiter’s mass is 318 times the Earth’s, and its gravity is about twice
Earth’s. Is this consistent with the results of part a. If not, how do you
explain the discrepancy.
20. Where would an object have to be located so that it would experience
zero total gravitational force from the earth and moon.
21. The planet Uranus has a mass of 8.68
x
10
25
kg and a radius of
2.56
x
10
4
km. The figure shows the relative sizes of Uranus and Earth. (a)
Compute the ratio g
U
/g
E
, where g
U
is the strength of the gravitational field
at the surface of Uranus and g
E
is the corresponding quantity at the surface
of the Earth. (b) What is surprising about this result. How do you explain
it.
Problem 21. The sizes of Uranus and
Earth are compared. The image of
Uranus is from the Voyager 2 probe,
and the photo of the earth was taken
by the Apollo 11 astronauts.
Chapter 10Gravity