Newton's Law of Universal Gravitation
Isaac Newton compared the acceleration of the moon to the acceleration of objects on
earth. Believing that gravitational forces were responsible for each, Newton was able to
draw an important conclusion about the dependence of gravity upon distance. This
comparison led him to conclude that the force of gravitational attraction between the
Earth and other objects is inversely proportional to the distance separating the earth's
center from the object's center. But distance is not the only variable effecting the
magnitude of a gravitational force. In accord with Newton's famous equation
Fnet = m*a
Newton knew that the force which caused the apple's acceleration (gravity) must be
dependent upon the mass of the apple. And since the force acting to cause the apple's
downward acceleration also causes the earth's upward acceleration (Newton's third law),
that force must also depend upon the mass of the earth. So for Newton, the force of
gravity acting between the earth and any other object is directly proportional to the mass
of the earth, directly proportional to the mass of the object, and inversely proportional to
the square of the distance which separates the centers of the earth and the object.
But Newton's law of universal gravitation extends gravity beyond earth. Newton's law of
universal gravitation is about the universality of gravity. Newton's place in the Gravity
Hall of Fame is not due to his discovery of gravity, but rather due to his discovery that
gravitation is universal. ALL objects attract each other with a force of gravitational
attraction. This force of gravitational attraction is directly dependent upon the masses of
both objects and inversely proportional to the square of the distance which separates their
centers. Newton's conclusion about the magnitude of gravitational forces is summarized
symbolically as
Since the gravitational force is directly proportional to the mass of both interacting
objects, more massive objects will attract each other with a greater gravitational force. So
as the mass of either object increases, the force of gravitational attraction between them
also increases. If the mass of one of the objects is doubled, then the force of gravity
between them is doubled; if the mass of one of the objects is tripled, then the force of
gravity between them is tripled; if the mass of both of the objects is doubled, then the
force of gravity between them is quadrupled; and so on. You get the point…
Since gravitational force is inversely proportional to the separation distance between the
two interacting objects, more separation distance will result in weaker gravitational
forces. So as two objects are separated from each other, the force of gravitational
attraction between them also decreases. If
the separation distance between two
objects is doubled (increased by a factor
of 2), then the force of gravitational
attraction is decreased by a factor of 4 (2
raised to the second power). If the
separation distance between any two
objects is tripled (increased by a factor of
3), then the force of gravitational
attraction is decreased by a factor of 9 (3
raised to the second power).
The proportionalities expressed by
Newton's universal law of gravitation is
represented graphically by the following
illustration. Observe how the force of
gravity is directly proportional to the
product of the two masses and inversely
proportional to the square of the distance
of separation.
Another means of representing the proportionalities is to express the relationships in the
form of an equation using a constant of proportionality. This equation is shown below.
The constant of proportionality (G) in the above equation is known as the universal
gravitation constant. The precise value of G was determined experimentally by Henry
Cavendish in the century after Newton's death. (This experiment will be discussed later in
Lesson 3.) The value of G is found to be
G = 6.67 x 10-11 N m2 /kg2
The units on G may seem rather odd; nonetheless they are sensible. When the units on G
are substituted into the equation above and multiplied by m1 *m2 units and divided by d2
units, the result will be Newtons - the unit of force.
Knowing the value of G allows us to calculate the force of gravitational attraction
between any two objects of known mass and known separation distance. As a first
example, consider the following problem.
Sample Problem #1
Determine the force of gravitational attraction between the earth (m = 5.98 x
1024 kg) and a 70-kg physics student if the student is standing at sea level, a
distance of 6.37 x 106 m from earth's center.
The solution of the problem involves substituting known values of G (6.67 x 10-11 N
m2 /kg2 ), m1 (5.98 x 1024 kg ), m2 (70 kg) and d (6.37 x 106 m) into the universal
gravitation equation and solving for Fgrav. The solution is as follows:
Sample Problem #2
Determine the force of gravitational attraction between the earth (m = 5.98 x
1024 kg) and a 70-kg physics student if the student is in an airplane at 40000 feet
above earth's surface. This would place the student a distance of 6.38 x 106 m
from earth's center.
The solution of the problem involves substituting known values of G (6.67 x 10-11 N
m2 /kg2 ), m1 (5.98 x 1024 kg ), m2 (70 kg) and d (6.38 x 106 m) into the universal
gravitation equation and solving for Fgrav. The solution is as follows:
Two general conceptual comments can be made about the results of the two sample
calculations above. First, observe that the force of gravity acting upon the student (a.k.a.
the student's weight) is less on an airplane at 40 000 feet than at sea level. This illustrates
the inverse relationship between separation distance and the force of gravity (or in this
case, the weight of the student). The student weighs less at the higher altitude. However,
a mere change of 40 000 feet further from the center of the Earth is virtually negligible.
This altitude change altered the student's weight changed by 3 N which is less than 1% of
the original weight. A distance of 40 000 feet (from the earth's surface to a high altitude
airplane) is not very far when compared to a distance of 6.37 x 106 m (equivalent to
approximately 21 million feet from the center of the earth to the surface of the earth); this
alteration of distance is like a drop in a bucket. As shown in the diagram below, distance
of separation becomes much more influential when a significant variation is made.
The second conceptual comment to be made about the above sample calculations is that
the use of Newton's universal gravitation equation to calculate the force of gravity (or
weight) yields the same result as when calculating it using the quation derived from
Newton’s Second Law:
Fgrav = m*g = (70 kg)*(9.8 m/s 2 ) = 686 N
Both equations accomplish the same result because the value of g is equivalent to the
ratio of (G*Mearth)/(Rearth)2 .
Gravitational interactions do not simply exist between the earth and other objects; and not
simply between the sun and other planets; gravitational interactions exist between all
objects with an intensity which is directly proportional to the product of their masses. So
as you sit in your seat in the physics classroom, you are gravitationally attracted to your
lab partner, to the desk you are working at, and even to your physics book. Newton's
revolutionary idea was that gravity is universal - ALL objects attract in proportion to the
product of their masses. Of course, most gravitational forces are so minimal to be
noticed. Gravitational forces only are recognizable as the masses of objects become large.
To illustrate this, use Newton's universal gravitation equation to calculate the force of
gravity between the following objects presented in class.