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272 CHAPTER 9 Gravitation

? The Syncom satellite was carried by the Space Shuttle to a low-altitude orbit, and
then it used its own booster rocket to lift itself to the high-altitude geosynchronous
orbit. What is the increase of mechanical energy (kinetic and gravitational) of the
satellite during this transfer from one orbit to another? (Example 9, page 290)

ithin the Solar System, planets orbit around the Sun, and satellites orbit around
W the planets.These circular, or nearly circular, motions require a centripetal force
pulling the planets toward the Sun and the satellites toward the planets. It was Newton’s
great discovery that this interplanetary force holding the celestial bodies in their orbits
is of the same kind as the force of gravity that causes apples,and other things,to fall down-
ward near the surface of the Earth. Newton found that a single formula, his Law of
Universal Gravitation, encompasses both the gravitational forces acting between celes-
tial bodies and the gravitational force acting on bodies near the surface of the Earth.
By the nineteenth century, Newton’s theory of gravitation had proved itself so trust-
worthy that when astronomers noticed an irregularity in the motion of Uranus, they
could not bring themselves to believe that the theory was at fault.Instead,they suspected
that a new,unknown planet caused these irregularities by its gravitational pull on Uranus.
The astronomers J.C.Adams and U. J. J. Leverrier proceeded to calculate the expected
position of this hypothetical planet—and the new planet was immediately found at just
about the expected position.This discovery of a new planet, later named Neptune, was
a spectacular success of Newton’s theory of gravitation. Newton’s theory remains one of
the most accurate and successful theories in all of physics, and in all of science.
In this chapter, we will examine Newton’s Law of Universal Gravitation; we will
see how it includes the familiar gravitational force near the Earth’s surface. We will
also examine circular and elliptical orbits of planets and satellites, and we will become
familiar with Kepler’s laws describing these orbits. Finally, we will discuss gravitational
potential energy and apply energy conservation to orbital motion.

Online 9.1 NEWTON’S LAW OF
11
Concept UNIVERSAL GRAVITATION
Tutorial

Newton proposed that just as the Earth gravitationally attracts bodies placed near its
surface and causes them to fall downward, the Earth also attracts more distant bodies,
such as the Moon, or the Sun, or other planets. In turn, the Earth is gravitationally
attracted by all these bodies. More generally, every body in the Universe attracts every
other body with a gravitational force that depends on their masses and on their distances.
The gravitational force that two bodies exert on each other is large if their masses are
large, and small if their masses are small.The gravitational force decreases if we increase
the distance between the bodies. The Law of Universal Gravitation formulated by
Newton can be stated most easily for the case of particles:
Every particle attracts every other particle with a force directly proportional to the
product of their masses and inversely proportional to the square of the distance
between them.
Expressed mathematically, the magnitude of the gravitational force that two par-
ticles of masses M and m separated by a distance r exert on each other is
GMm
Law of Universal Gravitation F (9.1)
2
r
where G is a universal constant of proportionality, the same for all pairs of particles.