Universal Gravitation
Student Extras
Visit The Physics Classroom's Flickr Galleries and enjoy a visual tour of the topic of gravitation.
Shockwave StudiosNeed to see it? Try the Gravitation activity from the Shockwave Studios.
PhET Simulation: Gravity Force Lab
Explore the effect of object mass and separation distance on the force of gravity using this interactive Java applet from PhET.
Teacher's Guide
Looking for a lab that coordinates with this page? Try The Great Mass Attraction Lab from The Laboratory.
Curriculum CornerPractice makes perfect with this inverse square law activity from The Curriculum Corner.
Shockwave StudiosGravitation from the Shockwave Studios is an excellent accompanying activity to this page.
PhET Simulation: Gravity Force Lab
Students can explore the effect of object mass and separation distance using this interactive Java applet from PhET.
The Apple, the Moon, and the Inverse Square Law
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 paths 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 that 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
that 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) that 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 that 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 that 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/s2. And it was also known that the moon accelerated towards the earth at a rate of 0.00272 m/s2. If the same force that 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 that causes the more distant moon to accelerate at a rate of acceleration that 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 that causes it to be inversely dependent upon the distance between the objects?
The riddle is solved by a comparison of 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 that separates that object from the earth's center. The moon, being 60 times further away than the apple, experiences a force of gravity that 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 (Fgrav) between the earth and any other object and the distance that 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?