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298 CHAPTER 9 Gravitation

**34. A binary star system consists of two stars of masses m and m
1 2
orbiting about each other. Suppose that the orbits of the stars
Cape Canaveral
are circles of radii r and r centered on the center of mass
1 2
(Fig. 9.33). The center of mass is a point between the stars
such that the radii r and r are in the ratio r /r m /m .
1 2 1 2 2 1
Show that the period of the orbital motion is given by 28°
4p 2 3
2
T (r r )
2
1
G (m m )
1 2
m 2 FIGURE 9.35 Orbit of
a satellite launched from
Cape Canaveral.
r 2
westward speed must the satellite be given if it is to travel
r 1 along the same orbit in the opposite direction? For the pur-
pose of this problem, pretend that “low altitude” means
m 1
FIGURE 9.33 A binary
essentially “zero altitude.”
star system. The orbits are
(b) Suppose that the satellite has a mass of 14.0 kg. What
circles about the center of
mass. kinetic energy must the launch vehicle give to the satellite
for an eastward orbit? For a westward orbit?
*35. The binary system Cygnus X-1 consists of two stars orbiting
about their center of mass under the influence of their mutual 9.4 Elliptical Orbits; Kepler’s Laws †
gravitational forces.The orbital period of the motion is 5.6 days.
One of the stars is a supergiant with a mass 25 times the mass 38. Halley’s comet (Fig. 9.36) orbits the Sun in an elliptical orbit
of the Sun.The other star is believed to be a black hole with a (the comet reached perihelion in 1986). When the comet is at
10
mass about 10 times the mass of the Sun. From the information perihelion, its distance from the Sun is 8.78 10 m, and its
4
given, determine the distance between the stars; assume that the speed is 5.45 10 m/s. When the comet is at aphelion, its
12
orbits of both stars are circular. (Hint: See Problem 34.) distance is 5.28 10 m. What is the speed at aphelion?
**36. A hypothetical triple star system consists of three stars orbit-
ing about each other. For the sake of simplicity, assume that
all three stars have equal masses and that they move along a
common circular orbit maintaining an angular separation of
120 (Fig. 9.34). In terms of the mass M of each star and the
orbital radius R, what is the period of the motion?

120° 120°
120°
FIGURE 9.36 Halley’s comet photographed in 1986.
FIGURE 9.34 Three
identical stars orbiting
39. Explorer I, the first American artificial satellite, had an elliptical
about their center of mass. 6
orbit around the Earth with a perigee distance of 6.74 10 m
6
and an apogee distance of 8.91 10 m.The speed of this
3
**37. Take into account the rotation of the Earth in the following satellite was 6.21 10 m/s at apogee. Calculate the speed at
problem: perigee.
(a) Cape Canaveral is at a latitude of 28 north. What eastward 40. The Explorer X satellite had an orbit with perigee 175 km and
speed (relative to the ground) must a satellite be given if it apogee 181,200 km above the surface of the Earth. What was
is to achieve a low-altitude circular orbit (Fig. 9.35)? What the period of this satellite?

† For help see Online Concept Tutorial 12 at www wwnorton com/physics