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13. (II) Suppose the mass of the Earth were doubled, by
6-1 to 6-3 Law of Universal Gravitation
kept the same density and spherical shape. How would .
21. (Ill) De
weight of objects at the Earth's surface change?
live valu
14 (II) Given that the acceleration of gravity at the surface
Earth is
1. (1) Calculate the force of Earth's gravity on a spacecraft
2:00 Earth radii above the Earth's surface if its mass is 1480 kg.
- 22. (III) It
2. (1) Calculate the acceleration due to gravity on the Moon.
Mars is 0.38 of what it is on Earth, and that Mars' radio
3400 km, determine the mass of Mars.
the force
The Moon's radius is 1.74 x 10'm and its mass is
on the
735 x 10* kg.
IS. (II) At what distance from the Earth will a spacecraft or
Earth, but has the same mass What is the acceleration due
cling directly from the Earth to the Moon experience pek
of grav
A. (1) A hypothetical planet has a radius 23 times that of
net force because the Earth and Moon pull with equal s
cancel
reach
to gravity near its surface?
opposite forces?
16. (II) Determine the mass of the Sun using the known vaj
Appro
4 (1) A hypothetical planet has a mass 180 times that of
Earth, but the same radius What is &amp; near its surface?
(1) If you doubled the mass and tripled the radius of a
for the period of the Earth and its distance from the gin
6-4 Sat
planet, by what factor would g at its surface change?
Ifhint: The force on the Earth due to the Sun is related
6. (II) Calculate the effective value of g. the acceleration of gravity,
the centripetal acceleration of the Earth.] Compare you
23. (1) Th
680 K
movin
at (a) 6400m, and (6) 6400 km, above the Earth's surface
answer to that obtained using Kepler's laws, Example 6
17. (II) Two identical point masses, each of mass M, alway
remain separated by a distance of 2R. A third mass at is g
24. (1)
7. (II) You are explaining to friends why astronauts feel
weightless orbiting in the space shuttle, and they respond that
they thought gravity was just a lot weaker up there, Convince
placed a distance x along the perpendicular bisector of The
Circu
25. (11)
bath
them and yourself that it isn't so by calculating how much
original two masses, as shown in Fig. 6-26. Show that The
weaker gravity is 300 km above the Earth's surface.
gravitational force on the third
15 1
&amp; (II) Every few hundred years most of the planets line up on
mass is directed inward along
OM
the same side of the Sun. Calculate the total force on the Earth
the perpendicular bisector and
26. (11)
ceil
due to Venus, Jupiter, and Saturn, assuming all four planets
has a magnitude of
18
are in a line, Fig. 6-24. The masses are My = 0,815 ME.
My = 318 ME. Meat = 95.1 ME, and the mean distances
F =
2GM MIX
(x2 + RZ)
27
of the four planets from the Sun are 108, 150, 778, and
Earth is this?
1430 million km. What fraction of the Sun's force on the
FIGURE 6-26
Problem 17.
LOM
28.
Earth
Jupiter
Satur
18. (II) A mass M is ring shaped with radius r. A small mass m
Sun
is placed at a distance x along the ring's axis as shown in
29
Fig. 6-27. Show that the gravitational force on the mass m due
FIGURE 6-24 Problem 8 (not to scale).
to the ring is directed inward along the axis and has magnitude
GMmx
9. (II) Four 8.5-kg spheres are located at the corners of a square
30.
of side 0.80 m. Calculate the magnitude and direction of the
gravitational force exerted on one sphere by the other three.
[ Himr: Think of the ring as made up
10. (II) Two objects attract each other gravitationally with a
of many small point masses dM; sum
force of 2.5 x 10 N when they are 0.25 m apart. Their
over the forces due to each dM, and
total mass is 4.00 kg. Find their individual masses.
use symmetry.]
11. (II) Four masses are arranged as shown in Fig. 6-25.
Determine the x and y components of the gravitational
force on the mass at
FIGURE 6-27
the origin (m). Write
Problem 18.
the force in vector
19. (III) (a) Use the binomial expansion
notation (i, j).
(1 + x)&quot; = 1+nxx +
XO
n(n - 1) 2 ...
2
to show that the value of g is altered by approximately
Ag =-28
Yo
FIGURE 6-25
Problem 11.
at a height Ar above the Earth's surface, where ry is the
radius of the Earth, as long as Ar &lt; rE. (b) What is the
12. (II) Estimate the acceleration due to gravity at the surface
of Europa (one of the moons of Jupiter) given that its mass
result to compute the effective value of g at 125 km above
is the same as Earth's.
is 4.9 x 10&quot;kg and making the assumption that its density
the Earth's surface. Compare to a direct use of Eq. 6-1.
20. (III) The center of a 1.00 km diameter spherical pocket of of
is 1.00 km beneath the Earth's surface. Estimate by what
158 CHAPTER 6 Gravitation and Newton's Synthesis
percentage g directly above the pocket of oil would differ
from the expected value of g for a uniform Earth? Assume
the density of oil is 8.0 x 10- kg/m'.

1. The force of Earth's gravity on a spacecraft ( F ) is F = 1 6 1 1 . 4 0 N... View the full answer

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