Physics For Scientists And Engineers 6E - part 105
Problems
417
transmit data easily. It moves in a near-circle around the Sun
that is smaller than the Earth’s circular orbit. Its period,
however, is just equal to 1 yr. It is always located between the
Earth and the Sun along the line joining them. Both objects
exert gravitational forces on the observatory. Show that its
distance from the Earth must be between 1.47 # 10
9
m and
1.48 # 10
9
m. In 1772 Joseph Louis Lagrange determined
theoretically the special location allowing this orbit. The
SOHO spacecraft took this position on February 14, 1996.
Suggestion: Use data that are precise to four digits. The mass
of the Earth is 5.983 # 10
24
kg.
48.
Let ,g
M
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
,
g
M
/g, where g is the Earth’s gravitational field. (This dif-
ference is responsible for the occurrence of the lunar tides
on the Earth.)
49.
Review problem. Two identical hard spheres, each of mass
m and radius r, are released from rest in otherwise empty
space with their centers separated by the distance R. They
are allowed to collide under the influence of their gravita-
tional attraction. (a) Show that the magnitude of the im-
pulse received by each sphere before they make contact is
given by [Gm
3
(1/2r " 1/R)]
1/2
. (b) What If? Find the
magnitude of the impulse each receives if they collide elas-
tically.
50.
Two spheres having masses M and 2M and radii R and 3R,
respectively, are released from rest when the distance be-
tween their centers is 12R. How fast will each sphere be
moving when they collide? Assume that the two spheres in-
teract only with each other.
51.
In Larry Niven’s science-fiction novel Ringworld, a rigid
ring of material rotates about a star (Fig. P13.51). The tan-
gential speed of the ring is 1.25 # 10
6
m/s, and its radius
is 1.53 # 10
11
m. (a) Show that the centripetal accelera-
tion of the inhabitants is 10.2 m/s
2
. (b) The inhabitants of
this ring world live on the starlit inner surface of the ring.
Each person experiences a normal contact force n. Acting
alone, this normal force would produce an inward acceler-
ation of 9.90 m/s
2
. Additionally, the star at the center of
the ring exerts a gravitational force on the ring and its in-
habitants. The difference between the total acceleration
and the acceleration provided by the normal force is due
to the gravitational attraction of the central star. Show that
the mass of the star is approximately 10
32
kg.
54. Voyagers 1 and 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
n
F
g
Star
Figure P13.51
A
B
Figure P13.53
NASA
52.
(a) Show that the rate of change of the free-fall accelera-
tion with distance above the Earth’s surface is
This rate of change over distance is called a gradient. (b) If
h is small in comparison to the radius of the Earth, show
that the difference in free-fall acceleration between two
points separated by vertical distance h is
(c) Evaluate this difference for h ! 6.00 m, a typical height
for a two-story building.
53.
A ring of matter is a familiar structure in planetary and stel-
lar astronomy. Examples include Saturn’s rings and a ring
nebula. Consider a uniform ring of mass 2.36 # 10
20
kg and
radius 1.00 # 10
8
m. An object of mass 1 000 kg is placed at
a point A on the axis of the ring, 2.00 # 10
8
m from the cen-
ter of the ring (Figure P13.53). When the object is released,
the attraction of the ring makes the object move along the
axis toward the center of the ring (point B). (a) Calculate
the gravitational potential energy of the object–ring system
when the object is at A. (b) Calculate the gravitational poten-
tial energy of the system when the object is at B. (c) Calcu-
late the speed of the object as it passes through B.
# ,g # !
2GM
E
h
R
E
3
dg
dr
! "
2GM
E
R
E
3