CHAPTER 5 | GRAVITY 81
distance is the distance from Earth’s center, not the distance from
Earth’s surface. Because the moon is about 60 Earth radii away,
Earth’s gravity at the distance of the moon should be about 60^2
times less than at Earth’s surface. Instead of being 9.8 m/s^2 at
Earth’s surface, it should be about 0.0027 m/s^2 at the distance of
the moon.
Now, Newton wondered, was this enough acceleration to
keep the moon in orbit? He knew the moon’s distance and its
orbital period, so he could calculate the actual acceleration
needed to keep it in its curved path. Th e answer turned out to be
0.0027 m/s^2 , as his inverse-square-law calculations predicted.
Th e moon is held in its orbit by gravity, and gravity obeys the
inverse square law.
Newton’s third law says that forces always occur in pairs, so if
Earth pulls on the moon, then the moon must pull on Earth. Th is
is called mutual gravitation and is a general property of the uni-
verse. Th e sun, the planets, and all their moons must also attract
each other by mutual gravitation. In fact, every particle of mass in
the universe must attract every other particle, which is why
Newtonian gravity is often called universal mutual gravitation.
Clearly the force of gravity depends on mass. Your body is
made of matter, and you have your own personal gravitational
fi eld. But your gravity is weak and does not attract personal satel-
lites orbiting around you. Larger masses have stronger gravity.
From an analysis of the third law of motion, Newton realized
that the mass that resists acceleration in the fi rst law is identical
to the mass associated with causing gravity. Newton performed
precise experiments with pendulums and confi rmed this equiva-
lence between the mass that resists acceleration and the mass that
produces gravity.
From this, combined with the inverse square law, he was able
to write the famous formula for the gravitational force between
two masses, M and m:
F _______ GMm
r^2
Th e constant G is the gravitational constant; it is the constant
that connects mass to gravity. In the equation, r is the distance
between the masses. Th e negative sign means that the force is
attractive, pulling the masses together and making r decrease. In
plain English, Newton’s law of gravitation says: Th e force of grav-
ity between two masses M and m is proportional to the product
of the masses and inversely proportional to the square of the
distance between them.
Newton’s description of gravity was a diffi cult idea for physi-
cists of his time to accept because it is an example of action at a
distance. Earth and moon exert forces on each other even though
there is no physical connection between them. Modern scientists
resolve this problem by referring to gravity as a fi eld. Earth’s pres-
ence produces a gravitational fi eld directed toward Earth’s center.
Th e strength of the fi eld decreases according to the inverse square
law. Any particle of mass in that fi eld experiences a force that
depends on the mass of the particle and the strength of the fi eldMutual Gravitation
Once Newton understood the three laws of motion, he was able
to consider the force that causes objects to fall. Th e fi rst and
second laws tell you that falling bodies accelerate downward
because some force must be pulling downward on them. Newton
realized that some force has to act on the moon. Th e moon fol-
lows a curved path around Earth, and motion along a curved
path is accelerated motion. Th e second law of motion says that
an acceleration requires a force, so a force must be making the
moon follow that curved path.
Newton wondered if the force that holds the moon in its
orbit could be the same force that causes apples to fall—gravity.
He was aware that gravity extends at least as high as the tops of
mountains, but he did not know if it could extend all the way to
the moon. He believed that it could, but he thought it would be
weaker at greater distances, and he guessed that its strength
would decrease as the square of the distance increased.
Th is relationship, the inverse square law, was familiar to
Newton from his work on optics, where it applied to the intensity
of light. A screen set up 1 meter from a candle fl ame receives a
certain amount of light on each square meter. However, if that
screen is moved to a distance of 2 meters, the light that originally
illuminated 1 square meter must now cover 4 square meters
(■ Figure 5-6). Consequently, the intensity of the light is inversely
proportional to the square of the distance to the screen.
Newton made a second assumption that enabled him to
predict the strength of Earth’s gravity at the distance of the
moon. Not only did he assume that the strength of gravity fol-
lows the inverse square law, but he also assumed that the critical
2The inverse square law1■ Figure 5-6
As light radiates away from a source, it spreads out and becomes less in-
tense. Here the light falling on one square meter on the inner sphere must
cover four square meters on a sphere twice as big. This shows how the inten-
sity of light is inversely proportional to the square of the distance.