In the early 1600's, German mathematician and astronomer
Johannes Kepler mathematically analyzed known astronomical
data in order to develop three laws to describe the motion
of planets about the sun. Kepler's three laws emerged from
the analysis of data carefully collected over a span of
several years by his Danish predecessor and teacher, Tycho
Brahe. Kepler's three laws of planetary
motion can be briefly described as follows:

The path of the planets about the sun are elliptical
in shape, with the center of the sun being located at one
focus. (The Law of Ellipses)

An imaginary line drawn from the center of the sun to
the center of the planet will sweep out equal areas in
equal intervals of time. (The Law of Equal Areas)

The ratio of the squares of the periods of any two
planets is equal to the ratio of the cubes of their
average distances from the sun. (The Law of
Harmonies)

(Further discussion of these three laws is given in
Lesson 4.)

While Kepler's laws provided a suitable
framework for describing the motion and paths of planets
about the sun, there was no accepted explanation for why
such paths existed. The cause for how the planets moved as
they did was never stated. Kepler could only suggest that
there was some sort of interaction between the sun and the
planets which provided the driving force for the planet's
motion. To Kepler, the planets were somehow "magnetically"
driven by the sun to orbit in their elliptical trajectories.
There was however no interaction between the planets
themselves.

Newton was troubled by the lack of
explanation for the planet's orbits. To Newton, there must
be some cause for such elliptical motion. Even more
troubling was the circular motion of the moon about the
earth. Newton knew that there must be some sort of force
which governed the heavens; for the motion of the moon in a
circular path and of the planets in an elliptical path
required that there be an inward component of force.
Circular and elliptical motion
were clearly departures from the inertial paths
(straight-line) of objects. And as such, these celestial
motions required a cause in the form of an unbalanced force.
As learned in Lesson 1, circular motion (as well as
elliptical motion) requires a centripetal force. The nature
of such a force - its cause and its origin - bothered Newton
for some time and was the fuel for much mental pondering.
And according to legend, a breakthrough came at age 24 in an
apple orchard in England. Newton never wrote of such an
event, yet it is often claimed that the notion of gravity as
the cause of all heavenly motion was instigated when he was
struck in the head by an apple while lying under a tree in
an orchard in England. Whether it is a myth or a reality,
the fact is certain that it was Newton's ability to relate
the cause for heavenly motion (the orbit of the moon about
the earth) to the cause for Earthly motion (the falling of
an apple to the Earth) which led him to his notion of
universal
gravitation.

A
survey of Newton's writings reveals an illustration similar
to the one shown at the right. The illustration was
accompanied by an extensive discussion of the motion of the
moon as a projectile. Newton's reasoning proceeded as
follows. Suppose a cannonball is fired horizontally from a
very high mountain in a region devoid of air resistance. In
the absence of gravity, the cannonball would travel in a
straight-line, tangential path. Yet in the presence of
gravity, the cannonball would drop below this straight-line
path and eventually fall to Earth (as in
path A). Now suppose
that the cannonball is fired horizontally again, yet with a
greater speed. In this case, the cannonball would still fall
below its straight-line tangential path and eventually drop
to earth. Only this time, the cannonball would travel
further before striking the ground (as in
path B). Now suppose
that there is a speed at which the cannonball could be fired
such that the trajectory of the falling cannonball matched
the curvature of the earth. If such a speed could be
obtained, then the cannonball would fall around the earth
instead of into it. The cannonball would fall towards the
Earth without ever colliding into it and subsequently become
a satellite orbiting in circular motion (as in
path C). And then at
even greater launch speeds, a cannonball would once more
orbit the earth, but in an elliptical path (as in
path D). The motion of
the cannonball orbiting to the earth under the influence of
gravity is analogous to the motion of the moon orbiting the
Earth. And if the orbiting moon can be compared to the
falling cannonball, it can even be compared to a falling
apple. The same force which causes objects on Earth to fall
to the earth also causes objects in the heavens to move
along their circular and elliptical paths. Quite amazingly,
the laws of mechanics which govern the motions of objects on
Earth also govern the movement of objects in the
heavens.

Of course, Newton's
dilemma was to provide reasonable evidence for the extension
of the force of gravity from earth to the heavens. The key
to this extension demanded that he be able to show how the
affect of gravity is diluted with distance. It was known at
the time, that the force of gravity causes earthbound
objects (such as falling apples) to accelerate towards the
earth at a rate of 9.8 m/s^{2}. And it was also
known that the moon accelerated towards the earth at a rate
of 0.00272 m/s^{2}. If the same force which causes
the acceleration of the apple to the earth also causes the
acceleration of the moon towards the earth, then there must
be a plausible explanation for why the acceleration of the
moon is so much smaller than the acceleration of the apple.
What is it about the force of gravity which causes the more
distant moon to accelerate at a rate of acceleration which
is approximately 1/3600-th the acceleration of the
apple?

Newton knew that the force of gravity must somehow be
"diluted" by distance. But how? What mathematical reality is
intrinsic to the force of gravity which causes it to be
inversely dependent upon the distance between the
objects?

The riddle is solved by a comparison
between the distance from the apple to the center of the
earth with the distance from the moon to the center of the
earth. The moon in its orbit about the earth is
approximately 60 times further from the earth's center than
the apple is. The mathematical relationship becomes clear.
The force of gravity between the earth and any object is
inversely proportional to the square of the distance which
separates that object from the earth's center. The moon,
being 60 times further away than the apple, experiences a
force of gravity which is 1/(60)^{2} times that of
the apple. The force of gravity follows an
inverse square law.

The relationship between the force of
gravity
(F_{grav})
between the earth and any other object and the distance
which separates their centers
(d) can be expressed by
the following relationship

Since the distance d is in the
denominator of this relationship, it can be said that the
force of gravity is inversely related to the distance. And
since the distance is raised to the second power, it can be
said that the force of gravity is inversely related to the
square of the distance. This mathematical relationship is
sometimes referred to as an inverse square law since one
quantity depends inversely upon the square of the other
quantity. The inverse square relation between the force of
gravity and the distance of separation provided sufficient
evidence for Newton's explanation of why gravity can be
credited as the cause of both the falling apple's
acceleration and the orbiting moon's acceleration.

Using
Equations as a Guide to Thinking

The inverse square law proposed by Newton suggests that
the force of gravity acting between any two objects is
inversely proportional to the square of the separation
distance between the object's centers. Altering the
separation distance (d) results in an alteration in the
force of gravity acting between the objects. Since the two
quantities are inversely proportional, an increase in one
quantity results in a decrease in the value of the other
quantity. That is, an increase in the separation distance
causes a decrease in the force of gravity and a decrease in
the separation distance causes an increase in the force of
gravity. Furthermore, the
factor by which the force of gravity is changed is the
square of the factor by which the separation distance is
changed. So if the separation distance is doubled (increased
by a factor of 2), then the force of gravity is decreased by
a factor of four (2 raised to the second power). And if the
separation distance is tripled (increased by a factor of 3),
then the force of gravity is decreased by a factor of nine
(3 raised to the second power). Thinking of the
force-distance relationship in this way involves using a
mathematical relationship as a guide to thinking about how
an alteration in one variable affects the other variable.
Equations can be more than recipes for algebraic
problem-solving; they can be guides to thinking.
Check your understanding of the inverse square law as a
guide to thinking by answering the following questions
below. When finished, click the button to check your
answers.

Check Your
Understanding

1 . Suppose that two objects attract each other with a
gravitational force of 16 units. If the distance between the
two objects is doubled, what is the new force of attraction
between the two objects?

2. Suppose that two objects attract each other with a
gravitational force of 16 units. If the distance between the
two objects is tripled, then what is the new force of
attraction between the two objects?

3. Suppose that two objects attract each other with a
gravitational force of 16 units. If the distance between the
two objects is reduced in half, then what is the new force
of attraction between the two objects?

4. Suppose that two objects attract each other with a
gravitational force of 16 units. If the distance between the
two objects is reduced by a factor of 5, then what is the
new force of attraction between the two objects?

5. Having recently completed his first Physics course,
Noah Formula has devised a new business plan based on his
teacher's Physics for Better Living theme. Noah
learned that objects weigh different amounts at different
distances from Earth's center. His plan involves buying gold
by the weight at one altitude and then selling it at another
altitude at the same price per weight. Should Noah buy at a
high altitude and sell at a low altitude or vice versa?