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296 CHAPTER 9 Gravitation

your weight. Why don’t you feel these forces? (Hint: You and the near and far surfaces of Io (due to Jupiter) with the
the Earth are in free fall toward the Sun and the Moon.) difference in accelerations on the near and far side of
6. Calculate the gravitational force between our Galaxy and the the Earth (due to the Moon), both as absolute accelera-
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Andromeda galaxy.Their masses are 2.0 10 and 3.0 10 11 tions and as a fraction of the surface g. Io has a mass of 3
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times the mass of the Sun, respectively, and their separation is 8.9 10 kg and a radius of 1820 km, and is 422 10
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2.2 10 light-years. Treat both galaxies as point masses. km from the center of Jupiter.
7. The nearest star is Alpha Centauri, at a distance of 4.4 light- *18. Suppose that the Earth, Sun, and Moon are located at the ver-
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years from us. The mass of this star is 2.0 10 kg. Compare tices of a right triangle, with the Moon located at the right
the gravitational force exerted by Alpha Centauri on the Sun angle (at first or last quarter moon; see Fig. 9.29). Find the
with the gravitational force that the Earth exerts on the Sun. magnitude and direction of the sum of the gravitational forces
Which force is stronger? exerted by the Earth and the Sun on the Moon.
8. What is the magnitude of the gravitational attraction the Sun
exerts on the Moon? What is the magnitude of the gravitational Moon
attraction the Earth exerts on the Moon? Suppose that the
three bodies are aligned, with the Earth between the Sun and
the Moon (at full moon). What is the direction of the net force
acting on the Moon? Suppose that the three bodies are aligned
with the Moon between the Earth and the Sun (at new moon). Earth
What is the direction of the net force acting on the Moon?
9. Calculate the value of the acceleration due to gravity at the
surfaces of Jupiter, Saturn, and Uranus. Use the values of the Sun
planetary masses and radii given in the table printed inside the
book cover.
FIGURE 9.29 Earth, Moon, and Sun.
10. Somewhere between the Earth and the Sun is a point where
the gravitational attraction of the Earth exactly balances that
of the Sun. At what fraction of the Earth–Sun distance does
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this occur? *19. Mimas, a small moon of Saturn, has a mass of 3.8 10 kg
11. Compare the weight of a 1-kg mass at the Earth’s surface with and a diameter of 500 km. What is the maximum equatorial
the gravitational force between our Sun and another star of velocity with which we can make this moon rotate about its
the same mass located at the far end of our galaxy, about axis if pieces of loose rock sitting on its surface at its equator
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5 10 m away. are not to fly off?
12. Each of two adjacent 1.5-kg spheres hangs from a ceiling
by a string. The center-to-center distance of the spheres is
8.0 cm. What (small) angle does each string make with the 9.3 Circular Orbits †
vertical?
20. The Midas II spy satellite was launched into a circular orbit at
13. A 7.0-kg mass is on the x axis at x 3.0 m, and a 4.0-kg mass
a height of 500 km above the surface of the Earth. Calculate
is on the y axis at y 2.0 m. What is the resultant gravita-
the orbital period and the orbital speed of this satellite.
tional force (magnitude and direction) due to these two
masses on a third mass of 3.0 kg located at the origin? 21. Consider the communications satellite described in Example 6.
What is the speed of this satellite?
14. Three equal masses m are located at the vertices of an equilat-
eral triangle of side a. What is the magnitude of the net gravi- 22. Calculate the orbital speed of Venus from the data given in
tational force on each mass due to the other two? Example 5.
15. Find the acceleration of the Moon due to the pull of the 23. The Sun is moving in a circular orbit around the center of our
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Earth. Express your result in units of the standard g. Galaxy. The radius of this orbit is 3 10 light-years.
Calculate the period of the orbital motion and calculate the
16. If a “tower to the sky” of height 2000 km above the Earth’s
orbital speed of the Sun. The mass of our Galaxy is 4
surface could be built, what would be your weight when 41
10 kg, and all of this mass can be regarded as concentrated
standing at the top? Assume the tower is located at the
at the center of the Galaxy.
South Pole. Express your answer in terms of your weight at
the Earth’s surface. 24. Table 9.2 lists some of the moons of Saturn. Their orbits are
circular.
17. It has been suggested that strong tidal forces on Io, a
moon of Jupiter, could be responsible for the dramatic (a) From the information given, calculate the periods and
volcanic activity observed there by Voyager spacecraft. orbital speeds of all these moons.
Compare the difference in gravitational accelerations on (b) Calculate the mass of Saturn.

† For help see Online Concept Tutorial 11 at www wwnorton com/physics