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416 Chapter 10 Gravitation
Questions and Problems
In a few problems, you are given more data than you actually
need; in a few other problems, you are required to supply data
from your general knowledge, outside sources, or informed
­estimate.
Interpret as significant all digits in numerical values that have
trailing zeros and no decimal points. For all problems, use
g = 9.80 m>s2 for the free-fall acceleration due to gravity.
­Neglect friction and air resistance unless instructed to do otherwise. Refer to Appendix B for any required astronomical data
like the masses or orbital radii of the planets.
• Basic, single-concept problem
•• Intermediate-level problem, may require synthesis of concepts and multiple steps
••• Challenging problem
SSM Solution is in Student Solutions Manual
Conceptual Questions
1. •Isaac Newton probably gained no insight by being hit on
the head with a falling apple, but he did believe that “the Moon
is falling.” Explain this statement. SSM
2. •Would the magnitude of the acceleration due to gravity near
Earth’s surface increase more if Earth’s mass were doubled or if
Earth’s radius were cut in half? Justify your answer.
3. •According to Equation 10-2, what happens to the force
­between two objects if (a) the mass of one object is doubled
and (b) if both masses are halved? SSM
4. •Every object in the universe with mass experiences an attractive gravitational force due to every other object with mass.
Why don’t you feel a pull from objects close to you?
5. •The gravitational force acts on all objects in proportion to
their mass. Why don’t heavy objects fall faster than light ones,
if you neglect air resistance?
6. •When powering a trip from Earth to the Moon, the thrusters of a ship don’t have to get the ship all the way to the Moon.
Why do the thrusters only have to get the ship partway there?
7. •The Sun gravitationally attracts the Moon. Does the Moon
orbit the Sun?
8. •Imagine a world where the force of gravity were proportional to the inverse cube of the distance between two ­objects
(Fgravity  1>r 3). How would Kepler’s law of periods be
changed?
9. •Much attention has been devoted to the exact numerical
power in Newton’s law of universal gravitation (Fgravity  1>r 2).
Some theorists have investigated whether the dependence might
be slightly larger or smaller than 2. What would be the significance (what impact would there be) if the power was not
exactly 2?
10. •The fabrication of precision ball bearings may be better
performed on the International Space Station than on the surface of Earth. Explain why.
11. •Why is the gravitational potential energy of two objects
negative?
12. •When a frog is at the bottom of a well, the frog–Earth
system has a negative amount of gravitational potential energy
with respect to the ground level where we choose to set potential energy equal to zero. Can the frog escape from the well if it
Freed_c10_381-419_st_hr1.indd 416
jumps upward with a positive amount of kinetic energy that is
larger than the negative potential energy when the frog is at the
bottom of the well? Explain.
13. •Earth moves faster in its orbit around the Sun during
the winter in the Northern Hemisphere than it does during
the summer in the Northern Hemisphere. Is Earth closer to the
Sun during the Northern Hemisphere’s winter or during the
Northern Hemisphere’s summer? Explain your answer. SSM
14. •A satellite is to be raised from one circular orbit to one
further from Earth’s surface. What will happen to its period?
15. •Describe some ways that Newton’s law of gravity may
have affected human evolution. SSM
16. •Geostationary satellites remain stationary over one point
on Earth. How is this accomplished?
Multiple-Choice Questions
17. •Planet A orbits a star with a period T. Planet B circles the
same star at four times the distance of Planet A. Planet B is
four times as massive as Planet A. Planet B orbits the star with
a period of
A. 4T.
B. 8T.
C. 16T.
D. T.
E. T>4.
18. •According to Newton’s universal law of gravitation,
Gm1 m2
F =
, if both masses are doubled, the force is
r2
A. four times as much as the original value.
B. twice as much as the original value.
C. the same as the original value.
D. one-half of the original value.
E. one-fourth of the original value.
19. •According to Newton’s universal law of gravitation,
Gm1 m2
F =
, if the distance r is doubled, the force is
r2
A. four times as much as the original value.
B. twice as much as the original value.
C. the same as the original value.
D. one-half of the original value.
E. one-fourth of the original value.
20. •Which is larger, the Sun’s pull on Earth or Earth’s pull on
Sun?
A. The Sun’s pull on Earth is larger.
B. Earth’s pull on the Sun is larger.
C. They pull on each other equally.
D. The Sun’s pull on Earth is twice as large as Earth’s
pull on the much larger Sun.
E. There is no pull or force between Earth and the Sun.
21. •Compare the weight of a mountain climber when she is at
the bottom of a mountain with her weight when she is at the
top of the mountain. In which case is her weight larger?
A. She weighs more at the bottom.
B. She weighs more at the top.
C. Both are the same.
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Questions and Problems 417
D. She weighs twice as much at the top.
E. She weighs four times as much at the top. SSM
22. •A satellite and the International Space Station have the
same mass and are going around Earth in concentric orbits. The
distance of the satellite from Earth’s center is twice that of the
International Space Station’s distance. What is the ratio of the
centripetal force acting on the satellite compared to that acting
on the International Space Station?
A. 1>4
B. 1>2
C. 1
D. 2
E. 4
30. •Biology Estimate the difference in height due to compression of the vertebrae when a 160-cm-tall sherpa at the peak of
Mount Everest travels to sea level.
31. •Estimate the impact on the orbit of Earth about the Sun if
Jupiter were not a part of our solar system. SSM
32. ••The eight planets, as well as Pluto and Halley’s comet,
­orbit the Sun according to Newton’s law of universal gravitation.
The table has data for the periods (T) and semimajor axes (a)
for each orbital ellipse. (a) Derive a best-fit constant, similar to
that referred to in Kepler’s law of periods. (b) Predict the value
of a for the elliptical orbit of Halley’s comet.
23. •A satellite is in a circular orbit about Earth moving at a
speed of v. In order to escape the planet, the satellite must attain a speed of
A. v.
B. 12v.
C. 2v.
D. v>2.
E. v> 12. SSM
24. •The escape speed from planet X is v. Planet Y has the same
radius as planet X but is twice as dense. The escape speed from
planet Y is
A. v.
B. 12v.
C. 2v.
D. v>2.
E. v> 12.
25. •Two satellites having equal masses are in circular orbits
around Earth. Satellite A has a smaller orbital radius than satellite B. Which statement is true?
A. Satellite A has more kinetic energy, less potential
energy, and less mechanical energy (kinetic energy
plus potential energy) than satellite B.
B. Satellite A has less kinetic energy, less potential energy,
and less mechanical energy than satellite B.
C. Satellite A has more kinetic energy, more potential
energy, and less mechanical energy than satellite B.
D. Satellite A and satellite B have the same amount of
mechanical energy.
E. Satellite A and satellite B have the same amount of
kinetic energy and no potential energy because they
are in motion. SSM
Estimation/Numerical Analysis
26. •Estimate the ratio between the maximum and minimum
tidal forces on Earth due to the gravitational pull of the Moon.
27. •Estimate how close two 1-kg weights must be in an ordinary science lab in order to measure the gravitational force one
exerts on the other. SSM
28. •Estimate the numerical range of the acceleration constant g between Earth’s highest mountain peaks and its lowest
valleys.
29. •Estimate the inaccuracy in using the acceleration constant g
in kinematic formulas that describe terrestrial free-fall when
you descend from the top of Mount Everest (8850 m elevation)
to sea level.
Freed_c10_381-419_st_hr1.indd 417
Planet
T (days)
a (AU)
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Halley
87.97
224.7
365.2
687.0
4332
10,832
30,799
60,190
90,613.3
27,507
0.3871
0.7233
1.000
1.5234
5.204
9.582
19.23
30.10
9.48
?
Problems
10-1 Gravitation is a force of universal importance
10-2 Newton’s law of universal gravitation explains the orbit
of the Moon
33. •A 500-kg tree stump is located 1000 m from a 12,000-kg
boulder. Determine the magnitude and direction of the gravitational force exerted by the tree stump on the boulder.
34. •Kramer goes bowling and decides to employ the force of
gravity to “pick up a spare.” He rolls the 7.0-kg bowling ball
very slowly so that it comes to rest a center-to-center distance
of 0.2 m from the one remaining 1.5-kg bowling pin. Determine the force of gravity between the ball and the pin and comment on the efficacy of the technique. Treat the ball and pin as
point objects for this problem.
35. •A 150-g baseball is 100 m from a 935-g bat. What is
the force of gravitational attraction between the two objects?
­Ignore the size of the bat and ball (treat them as point objects)
for this problem. SSM
36. •In 1994, the performer Rod Stewart drew over 3 million
people to a concert in Rio de Janeiro, Brazil. (a) If the people in
the group had an average mass of 80 kg, what collective gravitational force would the group have on a 4.5-kg eagle soaring 300 m above the throng? If you treat the group as a point
­object, you will get an upper limit for the gravitational force.
(b) What is the ratio of that force of attraction to the force
­between Earth and the eagle?
37. •Compare the gravitational force on a 1-kg apple that is
on the surface of Earth versus the gravitational force due to the
Moon on the same apple in the same location on the surface of
Earth. Assume that Earth and the Moon are spherical and that
both have their masses concentrated at their respective centers.
38. •Determine the magnitude of the force of gravity between
the Sun and Earth.
39. •Determine the net force of gravity acting on the Moon during an eclipse when it is directly between Earth and the Sun. SSM
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418 Chapter 10 Gravitation
40. •A star that has a mass equal to the mass of our Sun is located
50 AU from another star that has a mass that is one-half of the
Sun’s mass. At what point(s) will the gravitational force from the
two stars on a 100,000-kg space probe be equal to zero?
51. •What was the speed of a space shuttle that orbited Earth
at an altitude of 300 km?
41. •At what point between Earth and the Moon will a
50,000-kg space probe experience no net gravitational force? SSM
53. •A satellite requires 86.5 min to orbit Earth once. Assuming a circular orbit, what is the circumference of the satellite’s
­orbit? SSM
42. •Eight stars, all with the same mass as our Sun, are located
on the vertices of an astronomically sized cube with sides of
100 AU. Determine the net gravitational force on one of the
stars due to the other seven.
43. •Compare the weight of a 5.00-kg object on Earth’s surface
to the gravitational force between a 5.00-kg object that is one
Earth radius from another object of mass equal to 5.98 * 1024 kg.
Use Newton’s universal law of gravitation for the second part
of the question.
10-3 The gravitational potential energy of two objects
is negative and increases toward zero as the objects are
moved farther apart
44. •How much energy would be required to move the Moon
from its present orbit around Earth to a location that is twice
as far away? Assume the Moon’s orbit around Earth is nearly
circular and has a radius of 3.84 * 108 m, and that the Moon’s
orbital period is 27.3 days.
45. •How much work is done by the force of gravity as a 10-kg
object moves from a point that is 6000 m above sea level to a
point that is 1000 m above sea level? SSM
46. •(a) What is the escape velocity of a space probe that is
launched from the surface of Earth? (b) Would the answer
change if the launch occurs on top of a very tall mountain?
Explain your answer.
47. •A small asteroid that has a mass of 100 kg is moving at
200 m>s when it is 1000 km above the Moon. (a) How fast will
the meteorite be traveling when it impacts the lunar surface if
it is heading straight toward the center of the Moon? (b) How
much work does the Moon do in stopping the asteroid if neither the Moon nor the asteroid heats up in the process? The
radius of the Moon is 1.737 * 106 m. SSM
48. ••The volume of water in the Pacific Ocean is about
7 * 108 km3. The density of seawater is about 1030 kg>m3.
(a) Determine the gravitational potential energy of the Moon–
Pacific Ocean system when the Pacific is facing away from the
Moon. (b) Repeat the calculation when Earth has rotated so
that the Pacific Ocean faces toward the Moon. (c) Estimate the
maximum speed of the water in the Pacific Ocean due to the
tidal influence of the Moon. For the sake of the calculations,
treat the Pacific Ocean as a pointlike object (obviously a very
rough approximation).
10-4 Newton’s law of universal gravitation explains Kepler’s
laws for the orbits of planets and satellites
49. •The orbit of Mars around the Sun has a radius that is
1.524 times greater than the radius of Earth’s orbit. Determine
the time required for Mars to complete one revolution.
50. •The space shuttle usually orbited Earth at altitudes of
around 300 km. (a) Determine the time for one orbit of the
shuttle about Earth. (b) How many sunrises per day did the
astronauts witness?
Freed_c10_381-419_st_hr1.indd 418
52. •A satellite orbits Earth at an altitude of 80,000 km. Determine the time for the satellite to orbit Earth once.
54. •The orbital period of Saturn is 29.46 y. Determine the
semimajor axis of its orbit.
55. •Determine the altitude of a geosynchronous satellite. Such
a satellite will have precisely the same orbital period as Earth’s
rotational period.
56. •Given that Earth orbits the Sun with a semimajor axis of
1.000 AU and an approximate orbital period of 365.24 days,
determine the mass of the Sun.
57. •The Moon orbits Earth in a nearly circular orbit that lasts
27.32 days. Determine the distance from the surface of the
Moon to the surface of Earth. SSM
58. •A planet orbits a star with an orbital radius of 1 AU. If the
star has a mass that is 1.75 times our own Sun’s mass, determine the time for one revolution of the planet around the star.
General Problems
59. ••A satellite is launched into an elliptical orbit that has
a period of 702 min. Aphelion of the orbit is 39,200 km and
perihelion is 560 km. (a) Determine the gravitational potential
energy at aphelion and perihelion of the satellite’s orbit. (b) Estimate the energy required to place the satellite in orbit around
the Earth.
60. ••Astronomy The four largest of Jupiter’s moons are listed
in the table below. Using these data, Kepler’s three laws,
and the law of universal gravitation, (a) complete the table and
(b) determine the mass of Jupiter.
Moon
Io
Europa
Ganymede
Callisto
Semimajor Axis (km)
Orbital Period (days)
421,700
671,034
?
?
1.769
?
7.155
16.689
61. ••(a) What speed is needed to launch a rocket due east near
the equator into a low Earth orbit? Assume the rocket skims
along the surface of Earth, so r « RE. (b) Repeat the calculation
for a rocket fired due west. SSM
62. ••The former Soviet Union launched the first artificial Earth
satellite, Sputnik, in 1957. Its mass was 84 kg, and it made one
orbit every 96 min. (a) Determine the altitude of Sputnik’s orbit
above Earth’s surface, assuming circular orbits. (b) What was
Sputnik’s weight in orbit and at Earth’s surface?
63. •Astronomy The 2004 landings of the Mars rovers Spirit
and Opportunity involved many stages, resulting in each
probe having zero vertical velocity about 12 m above the surface of Mars. Determine (a) the time required for the final
free-fall descent of the probes and (b) the vertical velocity at
impact. The mass of Mars is 6.419 * 1023 kg and its radius is
3.397 * 106 m.
64. ••Biology, Astronomy On Earth, froghoppers can jump
­ pward with a takeoff speed of 2.8 m>s. Suppose you took
u
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Questions and Problems 419
some of the insects to an asteroid. If it is small enough, they
can jump free of it and escape into space. (a) What is the
­diameter (in kilometers) of the largest spherical asteroid from
which they could jump free? Assume a typical asteroid density
of 2.0 g>cm3. (b) Suppose that one of the froghoppers jumped
horizontally from a small hill on an asteroid. What would the
diameter (in km) of the asteroid need to be so that the insect
could go into a circular orbit just above the surface?
65. •Astronomy The International Space Station (ISS) ­orbits
Earth in a nearly circular orbit that is 345 km above Earth’s
surface. (a) How many hours does it take for the ISS to
make each orbit? (b) Some of the experiments performed by
­astronauts in the ISS involve the effects of “weightlessness”
on ­objects. What gravitational force does Earth exert on a
­10.0-kg object in the ISS? Express your answer in newtons
and as a fraction of the force that Earth would exert on the
object at Earth’s surface. (c) Considering your answer in part
(b), how can an object be considered weightless in the ISS?
66. ••Astronomy Measurements on the asteroid Apophis
have shown that its aphelion (farthest distance from the Sun)
is 1.099 AU, its perihelion (closest distance from the Sun) is
0.746 AU, and its mass is 2.7 * 1010 kg. (a) Determine the
semimajor axis of Apophis in astronomical units and in meters.
(b) How many days does it take Apophis to orbit the Sun? (c) At
what point in its orbit is Apophis traveling fastest, and at what
point is it traveling slowest? (d) Determine the ratio of its maximum speed to its minimum speed.
67. ••A rack of seven spherical
bowling balls (each 8.00 kg, radius
of 11.0 cm) is positioned along
a line 100 cm from a point P, as
shown in Figure 10-23. Determine
the gravitational force the bowling
balls exert on a ping-pong ball of
mass 2.70 g centered at point P.
Freed_c10_381-419_st_hr1.indd 419
68. •Astronomy The Sun and solar system actually are not at
rest in our Milky Way galaxy. We orbit around the center of the
Milky Way galaxy once every 225,000,000 years, at a distance
of 27,000 light-years. (One light-year is the distance that light
travels in one year: 1 ly = 9.46 * 1012 km = 9.46 * 1015 m.) If
the mass of the Milky Way were concentrated at the center of
the galaxy, what would be the mass of the galaxy?
69. ••Locate the point(s) along the line AB where a small, 1.00-kg
object could rest such that the net gravitational force on it due
to the two objects shown is exactly zero (Figure 10-24). SSM
40 cm
A
B
3 kg
10 kg
Figure 10-24 ​Problem 69
70. •••Find the net gravitational force exerted on a small,
1.00-kg object located at the point P by the three massive
­objects (Figure 10-25). Each of the massive objects lies on a
coordinate axis as shown.
z
m1 = 2.00 × 106 kg
m2 = 6.00 × 106 kg
m3 = 8.00 × 106 kg
z3 = 140,000 m 3
z0
x0 = 100,000 m
y0 = 100,000 m
z0 = 120,000 m
P
P
y0
100 cm
y
2
y2 = 120,000 m
x0
Figure 10-23 ​Problem 67
1
x
x1 = 120,000 m
Figure 10-25 ​Problem 70
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