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294 CHAPTER 9 Gravitation

GMm
GRAVITATIONAL POTENTIAL ENERGY U (9.20)
r

CONSERVATION OF ENERGY 1 2 GMm (9.21)
E K U mv [constant]
r
2
ENERGY FOR A CIRCULAR ORBIT AROUND GM m (9.24)
THE SUN (Also the energy for an elliptical E S
2r
orbit with semimajor axis r.)

SHAPES OF ORBITS For total E
0 elliptical orbit
E > 0; hyperbola
mechanical energy E, E 0 parabolic trajectory
E 0 hyperbolic trajectory E < 0; ellipse

E = 0; parabola

ESCAPE VELOCITY FROM EARTH 2GM E (9.27)
v
B R
E

QUESTIONS FOR DISCUSSION

1. Can you directly feel the gravitational pull of the Earth with 7. It is easier to launch an Earth satellite into an eastward orbit
your sense organs? (Hint: Would you feel anything if you were than into a westward orbit. Why?
in free fall?) 8. Would it be advantageous to launch rockets into space from a
2. According to a tale told by Professor R. Lichtenstein, some pad at very high altitude on a mountain? Why has this not
apple trees growing in the mountains of Tibet produce apples been done?
of negative mass. In what direction would such an apple fall if 9. Describe how you would play squash on a small, round aster-
it fell off its tree? How would such an apple hang on the tree? oid (with no enclosing wall). What rules of the game would
3. Eclipses of the Moon can occur only at full Moon. Eclipses of you want to lay down?
the Sun can occur only at new Moon. Why? 10. According to an NBC news report of April 5, 1983, a commu-
4. Explain why the sidereal day (the time of rotation of the Earth nications satellite launched from the Space Shuttle went into
relative to the stars, or 23 h 56 min 4 s) is shorter than the an orbit as shown in Fig. 9.27. Is this believable?
mean solar day (the time between successive passages of the
Sun over a given meridian, or 24 h). (Hint: The rotation of the
Earth around its axis and the revolution of the Earth around
the Sun are in the same direction.)
5. Suppose that an airplane flies around the Earth along the
equator. If this airplane flew very fast, it would not need wings
to support itself. Why not?
6. The mass of Pluto was not known until 1978 when a moon of
Pluto was finally discovered. How did the discovery of this FIGURE 9.27 Proposed orbit for
moon help? a communications satellite.