September (autumnal) equinox (which contains the
summer solstice) longer than the interval from the
September to the March equinox rather than being
equal to that interval? Choose one of the following
reasons. (a)They are really the same, but the Earth
spins faster during the “summer” interval, so the
days are shorter. (b) Over the “summer” interval, the
Earth moves slower because it is farther from the Sun.
(c) Over the March-to-September interval, the Earth
moves slower because it is closer to the Sun. (d)The
Earth has less kinetic energy when it is warmer.
(e)The Earth has less orbital angular momentum
when it is warmer.
9. Rank the magnitudes of the following gravitational
forces from largest to smallest. If two forces are equal,
show their equality in your list. (a) the force exerted by
a 2-kg object on a 3-kg object 1 m away (b) the force
exerted by a 2-kg object on a 9-kg object 1 m away
(c) the force exerted by a 2-kg object on a 9-kg object
2 m away (d) the force exerted by a 9-kg object on a
2-kg object 2 m away (e) the force exerted by a 4-kg
object on another 4-kg object 2 m away
10. The gravitational force exerted on an astronaut on
the Earth’s surface is 650 N directed downward. When
she is in the space station in orbit around the Earth,
is the gravitational force on her (a) larger, (b) exactly
the same, (c)smaller, (d) nearly but not exactly zero, or
(e) exactly zero?
11. Halley’s comet has a period of approximately 76 years,
and it moves in an elliptical orbit in which its distance
from the Sun at closest approach is a small fraction of
its maximum distance. Estimate the comet’s maximum
distance from the Sun in astronomical units (AUs)
(the distance from the Earth to the Sun). (a) 6 AU
(b) 12 AU (c) 20 AU (d) 28 AU (e) 35 AU
5. Imagine that nitrogen and other atmospheric gases
were more soluble in water so that the atmosphere of
the Earth is entirely absorbed by the oceans. Atmo-
spheric pressure would then be zero, and outer space
would start at the planet’s surface. Would the Earth
then have a gravitational field? (a) Yes, and at the sur-
face it would be larger in magnitude than 9.8 N/kg.
(b) Yes, and it would be essentially the same as the
current value. (c) Yes, and it would be somewhat less
than 9.8 N/kg. (d) Yes, and it would be much less than
9.8 N/kg. (e) No, it would not.
6. An object of mass m is located on the surface of a
spherical planet of mass M and radius R. The escape
speed from the planet does not depend on which
of the following? (a) M (b) m (c) the density of the
planet (d) R (e) the acceleration due to gravity on
7. A satellite originally moves in a circular orbit of radius
R around the Earth. Suppose it is moved into a circu-
lar orbit of radius 4R. (i) What does the force exerted
on the satellite then become? (a) eight times larger
(b) four times larger (c) one-half as large (d) one-
eighth as large (e)one-sixteenth as large (ii) What
happens to the satellite’s speed? Choose from the
same possibilities (a) through (e). (iii)What hap-
pens to its period? Choose from the same possibilities
(a) through (e).
8. The vernal equinox and the autumnal equinox are
associated with two points 180° apart in the Earth’s
orbit. That is, the Earth is on precisely opposite sides
of the Sun when it passes through these two points.
From the vernal equinox, 185.4 days elapse before
the autumnal equinox. Only 179.8days elapse from
the autumnal equinox until the next vernal equinox.
Why is the interval from the March (vernal) to the
denotes answer available in Student Solutions Manual/Study Guide
1. Each Voyager spacecraft was accelerated toward escape
speed from the Sun by the gravitational force exerted by
Jupiter on the spacecraft. (a) Is the gravitational force
a conservative or a nonconservative force? (b) Does the
interaction of the spacecraft with Jupiter meet the defi-
nition of an elastic collision? (c) How could the space-
craft be moving faster after the collision?
2. In his 1798 experiment, Cavendish was said to have
“weighed the Earth.” Explain this statement.
3. Why don’t we put a geosynchronous weather satellite in
orbit around the 45th parallel? Wouldn’t such a satel-
lite be more useful in the United States than one in
orbit around the equator?
4. (a) Explain why the force exerted on a particle by a
uniform sphere must be directed toward the center
of the sphere. (b) Would this statement be true if the
mass distribution of the sphere were not spherically
5. (a) At what position in its elliptical orbit is the speed of
a planet a maximum? (b) At what position is the speed
6. You are given the mass and radius of planet X. How
would you calculate the free-fall acceleration on this
7. (a) If a hole could be dug to the center of the Earth,
would the force on an object of mass m still obey Equa-
tion 13.1 there? (b) What do you think the force on m
would be at the center of the Earth?
8. Explain why it takes more fuel for a spacecraft to travel
from the Earth to the Moon than for the return trip.
Estimate the difference.
9. A satellite in low-Earth orbit is not truly traveling
through a vacuum. Rather, it moves through very thin
air. Does the resulting air friction cause the satellite to
chapter 13 Universal Gravitation
magnitude of the gravitational force exerted by one
particle on the other?
8. Why is the following situation impossible? The centers of two
homogeneous spheres are 1.00 m apart. The spheres
are each made of the same element from the peri-
odic table. The gravitational force between the spheres
9. Two objects attract each other with a gravitational
force of magnitude 1.00 3 1028 N when separated by
20.0 cm. If the total mass of the two objects is 5.00 kg,
what is the mass of each?
10. Review. A student proposes to study the gravita-
tional force by suspending two 100.0-kg spherical
objects at the lower ends of cables from the ceiling
of a tall cathedral and measuring the deflection of
the cables from the vertical. The 45.00-m-long cables
are attached to the ceiling 1.000 m apart. The first
object is suspended, and its position is carefully mea-
sured. The second object is suspended, and the two
objects attract each other gravitationally. By what dis-
tance has the first object moved horizontally from its
initial position due to the gravitational attraction to
the other object? Suggestion: Keep in mind that this
distance will be very small and make appropriate
Section 13.2 Free-Fall acceleration and
the Gravitational Force
11. When a falling meteoroid is at a distance above the
Earth’s surface of 3.00 times the Earth’s radius, what is
its acceleration due to the Earth’s gravitation?
12. The free-fall acceleration on the surface of the Moon
is about one-sixth that on the surface of the Earth.
The radius of the Moon is about 0.250R
radius5 6.373 106 m). Find the ratio of their average
13. Review. Miranda, a satellite of Uranus, is shown in Fig-
ure P13.13a. It can be modeled as a sphere of radius
242 km and mass 6.68 3 1019 kg. (a) Find the free-fall
acceleration on its surface. (b) A cliff on Miranda is
5.00 km high. It appears on the limb at the 11 o’clock
position in Figure P13.13a and is magnified in Figure
P13.13b. If a devotee of extreme sports runs horizon-
tally off the top of the cliff at 8.50 m/s, for what time
interval is he in flight? (c) How far from the base of the
vertical cliff does he strike the icy surface of Miranda?
(d) What will be his vector impact velocity?
Section 13.1 newton’s law of universal Gravitation
Problem 12 in Chapter 1 can also be assigned with this
1. In introductory physics laboratories, a typical Caven-
dish balance for measuring the gravitational constant
G uses lead spheres with masses of 1.50 kg and 15.0 g
whose centers are separated by about 4.50 cm. Calcu-
late the gravitational force between these spheres, treat-
ing each as a particle located at the sphere’s center.
2. Determine the order of magnitude of the gravitational
force that you exert on another person 2 m away. In
your solution, state the quantities you measure or esti-
mate and their values.
3. A 200-kg object and a 500-kg object are separated by
4.00m. (a) Find the net gravitational force exerted
by these objects on a 50.0-kg object placed midway
between them. (b) At what position (other than an infi-
nitely remote one) can the 50.0-kg object be placed so
as to experience a net force of zero from the other two
4. During a solar eclipse, the Moon, the Earth, and the
Sun all lie on the same line, with the Moon between
the Earth and the Sun. (a) What force is exerted by
the Sun on the Moon? (b) What force is exerted by the
Earth on the Moon? (c) What force is exerted by the
Sun on the Earth? (d) Compare the answers to parts
(a) and (b). Why doesn’t the Sun capture the Moon
away from the Earth?
5. Two ocean liners, each with a mass of 40 000 metric
tons, are moving on parallel courses 100 m apart. What
is the magnitude of the acceleration of one of the lin-
ers toward the other due to their mutual gravitational
attraction? Model the ships as particles.
6. Three uniform spheres of
5 2.00 kg, m
4.00kg, and m
5 6.00 kg
are placed at the corners of
a right triangle as shown in
Figure P13.6. Calculate the
resultant gravitational force
on the object of mass m
assuming the spheres are
isolated from the rest of the
7. Two identical isolated particles, each of mass 2.00 kg,
are separated by a distance of 30.0 cm. What is the
(0, 3.00) m
4.00, 0) m
The problems found in this
chapter may be assigned
online in Enhanced WebAssign
full solution available in the Student
Solutions Manual/Study Guide
Analysis Model tutorial available in
Master It tutorial available in Enhanced
Watch It video solution available in
tional fields acting on the occupants in the nose of the
ship and on those in the rear of the ship, farthest from
the black hole? (This difference in accelerations grows
rapidly as the ship approaches the black hole. It puts
the body of the ship under extreme tension and even-
tually tears it apart.)
Section 13.4 Kepler’s laws and the Motion of Planets
17. An artificial satellite circles the Earth in a circular orbit
at a location where the acceleration due to gravity is
9.00 m/s2. Determine the orbital period of the satellite.
18. Io, a satellite of Jupiter, has an orbital period of 1.77 days
and an orbital radius of 4.22 3 105 km. From these
data, determine the mass of Jupiter.
19. A minimum-energy transfer orbit to an outer planet
consists of putting a spacecraft on an elliptical trajec-
tory with the departure planet corresponding to the
perihelion of the ellipse, or the closest point to the Sun,
and the arrival planet at the aphelion, or the farthest
point from the Sun. (a) Use Kepler’s third law to calcu-
late how long it would take to go from Earth to Mars on
such an orbit as shown in Figure P13.19. (b) Can such
an orbit be undertaken at any time? Explain.
in the x direction, a distance b from
the x axis (Fig. P13.20). (a) Does the particle possess any
angular momentum about the origin? (b) Explain why
the amount of its angular momentum should change or
should stay constant. (c) Show that Kepler’s second law
is satisfied by showing that the two shaded triangles in
the figure have the same area when t
Section 13.3 analysis Model: Particle in a Field (Gravitational)
14. (a) Compute the vector gravitational field at a point P
on the perpendicular bisector of the line joining two
objects of equal mass separated by a distance 2a as
shown in Figure P13.14. (b) Explain physically why the
field should approach zero as r S 0. (c) Prove math-
ematically that the answer to part (a) behaves in this
way. (d)Explain physically why the magnitude of the
field should approach 2GM/r2 as r S `. (e) Prove math-
ematically that the answer to part (a) behaves correctly
in this limit.
15. Three objects of equal mass are located at three cor-
ners of a square of edge length , as shown in Figure
P13.15. Find the magnitude and direction of the gravi-
tational field at the fourth corner due to these objects.
16. A spacecraft in the shape of a long cylinder has a length
of 100 m, and its mass with occupants is 1 000 kg.
It has strayed too close to a black hole having a mass
100 times that of the Sun (Fig. P13.16). The nose of
the spacecraft points toward the black hole, and the
distance between the nose and the center of the black
hole is 10.0 km. (a) Determine the total force on the
spacecraft. (b) What is the difference in the gravita-
chapter 13 Universal Gravitation
21. Plaskett’s binary system consists of two stars that revolve
in a circular orbit about a center of mass midway between
them. This statement implies that the masses of the two
stars are equal (Fig. P13.21). Assume the orbital speed
of each star is 0v
0 5 220 km/s and the orbital period
of each is 14.4 days. Find the mass M of each star. (For
comparison, the mass of our Sun is 1.99 3 1030 kg.)
22. Two planets X and Y travel counterclockwise in circu-
lar orbits about a star as shown in Figure P13.22. The
radii of their orbits are in the ratio 3:1. At one moment,
they are aligned as shown in Figure P13.22a, making a
straight line with the star. During the next five years,
the angular displacement of planet X is 90.0° as shown
in Figure P13.22b. What is the angular displacement of
planet Y at this moment?
23. Comet Halley (Fig. P13.23) approaches the Sun to
within 0.570 AU, and its orbital period is 75.6 yr. (AU is
the symbol for astronomical unit, where 1 AU 5 1.50 3
1011 m is the mean Earth–Sun distance.) How far from
the Sun will Halley’s comet travel before it starts its
(Orbit is not drawn
24. The Explorer VIII satellite, placed into orbit November 3,
1960, to investigate the ionosphere, had the following
orbit parameters: perigee, 459 km; apogee, 2 289 km
(both distances above the Earth’s surface); period,
112.7 min. Find the ratio v
of the speed at perigee to
that at apogee.
25. Use Kepler’s third law to determine how many days it
takes a spacecraft to travel in an elliptical orbit from a
point 6 670 km from the Earth’s center to the Moon,
385 000 km from the Earth’s center.
26. Neutron stars are extremely dense objects formed from
the remnants of supernova explosions. Many rotate
very rapidly. Suppose the mass of a certain spherical
neutron star is twice the mass of the Sun and its radius
is 10.0 km. Determine the greatest possible angular
speed it can have so that the matter at the surface of
the star on its equator is just held in orbit by the gravi-
27. A synchronous satellite, which always remains above
the same point on a planet’s equator, is put in orbit
around Jupiter to study that planet’s famous red spot.
Jupiter rotates once every 9.84 h. Use the data of Table
13.2 to find the altitude of the satellite above the sur-
face of the planet.
28. (a) Given that the period of the Moon’s orbit about the
Earth is 27.32 days and the nearly constant distance
between the center of the Earth and the center of the
Moon is 3.84 3 108 m, use Equation 13.11 to calculate
the mass of the Earth. (b) Why is the value you calcu-
late a bit too large?
29. Suppose the Sun’s gravity were switched off. The plan-
ets would leave their orbits and fly away in straight lines
as described by Newton’s first law. (a) Would Mercury
ever be farther from the Sun than Pluto? (b) If so, find
how long it would take Mercury to achieve this passage.
If not, give a convincing argument that Pluto is always
farther from the Sun than is Mercury.
Section 13.5 Gravitational Potential Energy
Note: In Problems 30 through 50, assume U 5 0 at r 5 `.
30. A satellite in Earth orbit has a mass of 100 kg and is
at an altitude of 2.00 3 106 m. (a) What is the poten-
tial energy of the satellite–Earth system? (b) What is
the magnitude of the gravitational force exerted by the
Earth on the satellite? (c) What If? What force, if any,
does the satellite exert on the Earth?
31. How much work is done by the Moon’s gravitational
field on a 1 000-kg meteor as it comes in from outer
space and impacts on the Moon’s surface?
32. How much energy is required to move a 1 000-kg
object from the Earth’s surface to an altitude twice the
33. After the Sun exhausts its nuclear fuel, its ultimate fate
will be to collapse to a white dwarf state. In this state,
it would have approximately the same mass as it has
now, but its radius would be equal to the radius of the
Earth. Calculate (a) the average density of the white
dwarf, (b) the surface free-fall acceleration, and (c) the
sphere will produce only a beautiful meteor shower. The
astronaut finds that the density of the spherical asteroid
is equal to the average density of the Earth. To ensure its
pulverization, she incorporates into the explosives the
rocket fuel and oxidizer intended for her return journey.
What maximum radius can the asteroid have for her to
be able to leave it entirely simply by jumping straight up?
On Earth she can jump to a height of 0.500 m.
42. Derive an expression for the work required to move an
Earth satellite of mass m from a circular orbit of radius
to one of radius 3R
43. (a) Determine the amount of work that must be done
on a 100-kg payload to elevate it to a height of 1 000 km
above the Earth’s surface. (b) Determine the amount
of additional work that is required to put the payload
into circular orbit at this elevation.
44. (a) What is the minimum speed, relative to the Sun,
necessary for a spacecraft to escape the solar system if
it starts at the Earth’s orbit? (b) Voyager 1 achieved a
maximum speed of 125 000 km/h on its way to pho-
tograph Jupiter. Beyond what distance from the Sun is
this speed sufficient to escape the solar system?
45. A satellite of mass 200 kg is placed into Earth orbit
at a height of 200 km above the surface. (a) Assum-
ing a circular orbit, how long does the satellite take to
complete one orbit? (b) What is the satellite’s speed?
(c) Starting from the satellite on the Earth’s surface,
what is the minimum energy input necessary to place
this satellite in orbit? Ignore air resistance but include
the effect of the planet’s daily rotation.
46. A satellite of mass m, originally on the surface of
the Earth, is placed into Earth orbit at an altitude h.
(a) Assuming a circular orbit, how long does the sat-
ellite take to complete one orbit? (b) What is the sat-
ellite’s speed? (c) What is the minimum energy input
necessary to place this satellite in orbit? Ignore air
resistance but include the effect of the planet’s daily
rotation. Represent the mass and radius of the Earth as
47. Ganymede is the largest of Jupiter’s moons. Consider
a rocket on the surface of Ganymede, at the point far-
thest from the planet (Fig. P13.47). Model the rocket as
a particle. (a) Does the presence of Ganymede make
Jupiter exert a larger, smaller, or same size force on the
rocket compared with the force it would exert if Gany-
mede were not interposed? (b) Determine the escape
speed for the rocket from the planet–satellite system.
The radius of Ganymede is 2.643 106m, and its mass
gravitational potential energy associated with a 1.00-kg
object at the surface of the white dwarf.
34. An object is released from rest at an altitude h above the
surface of the Earth. (a) Show that its speed at a distance
r from the Earth’s center, where R
# r # R
1 h, is
(b) Assume the release altitude is 500 km. Perform the
to find the time of fall as the object moves from the
release point to the Earth’s surface. The negative sign
appears because the object is moving opposite to the
radial direction, so its speed is v 5 2dr/dt. Perform the
35. A system consists of three particles, each of mass 5.00g,
located at the corners of an equilateral triangle with
sides of 30.0 cm. (a) Calculate the potential energy
of the system. (b) Assume the particles are released
simultaneously. Describe the subsequent motion of
each. Will any collisions take place? Explain.
Section 13.6 Energy Considerations in Planetary
and Satellite Motion
36. A space probe is fired as a projectile from the Earth’s
surface with an initial speed of 2.00 3 104 m/s. What will
its speed be when it is very far from the Earth? Ignore
atmospheric friction and the rotation of the Earth.
37. A 500-kg satellite is in a circular orbit at an altitude of
500 km above the Earth’s surface. Because of air fric-
tion, the satellite eventually falls to the Earth’s surface,
where it hits the ground with a speed of 2.00 km/s. How
much energy was transformed into internal energy by
means of air friction?
38. A “treetop satellite” moves in a circular orbit just above
the surface of a planet, assumed to offer no air resis-
tance. Show that its orbital speed v and the escape speed
from the planet are related by the expression v
39. A 1 000-kg satellite orbits the Earth at a constant alti-
tude of 100 km. (a) How much energy must be added
to the system to move the satellite into a circular orbit
with altitude 200 km? What are the changes in the sys-
tem’s (b)kinetic energy and (c) potential energy?
40. A comet of mass 1.20 3 1010 kg moves in an elliptical
orbit around the Sun. Its distance from the Sun ranges
between 0.500 AU and 50.0 AU. (a) What is the eccen-
tricity of its orbit? (b) What is its period? (c) At aphelion,
what is the potential energy of the comet–Sun system?
Note: 1 AU 5 one astronomical unit 5 the average dis-
tance from the Sun to the Earth 5 1.496 3 1011 m.
41. An asteroid is on a collision course with Earth. An astro-
naut lands on the rock to bury explosive charges that
will blow the asteroid apart. Most of the small fragments
will miss the Earth, and those that fall into the atmo-
57. (a) A space vehicle is launched vertically upward from
the Earth’s surface with an initial speed of 8.76 km/s,
which is less than the escape speed of 11.2 km/s. What
maximum height does it attain? (b) A meteoroid falls
toward the Earth. It is essentially at rest with respect to
the Earth when it is at a height of 2.51 3 107 m above
the Earth’s surface. With what speed does the meteor-
ite (a meteoroid that survives to impact the Earth’s sur-
face) strike the Earth?
58. (a) A space vehicle is launched vertically upward from
the Earth’s surface with an initial speed of v
comparable to but less than the escape speed v
maximum height does it attain? (b) A meteoroid falls
toward the Earth. It is essentially at rest with respect
to the Earth when it is at a height h above the Earth’s
surface. With what speed does the meteorite (a meteor-
oid that survives to impact the Earth’s surface) strike
the Earth? (c) What If? Assume a baseball is tossed up
with an initial speed that is very small compared to the
escape speed. Show that the result from part (a) is con-
sistent with Equation 4.12.
59. Assume you are agile enough to run across a horizon-
tal surface at 8.50 m/s, independently of the value of
the gravitational field. What would be (a) the radius
and (b)the mass of an airless spherical asteroid of
uniform density 1.10 3 103 kg/m3 on which you could
launch yourself into orbit by running? (c) What would
be your period? (d) Would your running significantly
affect the rotation of the asteroid? Explain.
60. Two spheres having masses M and 2M and radii R and
3R, respectively, are simultaneously released from
rest when the distance between their centers is 12R.
Assume the two spheres interact only with each other
and we wish to find the speeds with which they collide.
(a) What two isolated system models are appropriate for
this system? (b) Write an equation from one of the mod-
els and solve it for v
, the velocity of the sphere of mass
M at any time after release in terms of v
, the veloc-
is 1.495 3 1023 kg. The distance between Jupiter and
Ganymede is 1.071 3 109 m, and the mass of Jupiter is
1.90 3 1027 kg. Ignore the motion of Jupiter and Gany-
mede as they revolve about their center of mass.
48. A satellite moves around the Earth in a circular orbit
of radius r. (a) What is the speed v
of the satellite?
(b) Suddenly, an explosion breaks the satellite into
two pieces, with masses m and 4m. Immediately after
the explosion, the smaller piece of mass m is stationary
with respect to the Earth and falls directly toward the
Earth. What is the speed v of the larger piece immedi-
ately after the explosion? (c)Because of the increase in
its speed, this larger piece now moves in a new ellipti-
cal orbit. Find its distance away from the center of the
Earth when it reaches the other end of the ellipse.
49. At the Earth’s surface, a projectile is launched straight
up at a speed of 10.0 km/s. To what height will it rise?
Ignore air resistance.
50. A rocket is fired straight up through the atmosphere
from the South Pole, burning out at an altitude of
250 km when traveling at 6.00 km/s. (a) What maxi-
mum distance from the Earth’s surface does it travel
before falling back to the Earth? (b) Would its maxi-
mum distance from the surface be larger if the same
rocket were fired with the same fuel load from a launch
site on the equator? Why or why not?
51. Review. A cylindrical habitat in space 6.00 km in diam-
eter and 30.0 km long has been proposed (by G. K.
O’Neill, 1974). Such a habitat would have cities, land,
and lakes on the inside surface and air and clouds in
the center. They would all be held in place by rotation
of the cylinder about its long axis. How fast would the
cylinder have to rotate to imitate the Earth’s gravita-
tional field at the walls of the cylinder?
52. Voyager 1 and Voyager 2 surveyed the surface of Jupiter’s
moon Io and photographed active volcanoes spewing
liquid sulfur to heights of 70 km above the surface of
this moon. Find the speed with which the liquid sul-
fur left the volcano. Io’s mass is 8.9 3 1022 kg, and its
radius is 1 820 km.
53. A satellite is in a circular orbit around the Earth at an
altitude of 2.80 3 106 m. Find (a) the period of the
orbit, (b) the speed of the satellite, and (c) the accel-
eration of the satellite.
54. Why is the following situation impossible? A spacecraft is
launched into a circular orbit around the Earth and
circles the Earth once an hour.
55. Let Dg
represent the difference in the gravitational
fields produced by the Moon at the points on the
Earth’s surface nearest to and farthest from the Moon.
Find the fraction Dg
/g, where g is the Earth’s gravi-
tational field. (This difference is responsible for the
occurrence of the lunar tides on the Earth.)
56. A sleeping area for a long space voyage consists of two
cabins each connected by a cable to a central hub as
shown in Figure P13.56. The cabins are set spinning
Documents you may be interested
Documents you may be interested