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Newton's Law of Universal Gravitation
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Circular Motion and Satellite Motion ­ Lesson 3 ­ Universal Gravitation
Newton's Law of Universal Gravitation
Gravity is More Than a Name
The Apple, the Moon, and the Inverse Square Law
Newton's Law of Universal Gravitation
Cavendish and the Value of G
The Value of g
As discussed earlier in Lesson 3, 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 affecting the magnitude of a gravitational force. Consider Newton's famous equation
Fnet = m • a
Newton knew that the force that 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 that separates the centers of the earth and the object.
The UNIVERSAL Gravitation Equation
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. ALLobjects attract
each other with a force of gravitational attraction. Gravity is universal. This force of gravitational attraction
is directly dependent upon the masses of both objects and inversely proportional to the square of the
distance that 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
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doubled, then the force of gravity between them is doubled. If the mass of one of the objects is tripled,
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Newton's Law of Universal Gravitation
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.
Since gravitational force is inversely proportional to the square of 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).
Thinking Proportionally About Newton's Equation
The proportionalities expressed by Newton's universal law of gravitation are 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 theuniversal 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 = 6.673 x 10­11 N m2/kg2
The units on G may seem rather odd; nonetheless they are sensible. When the units on G are substituted
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into the equation above and multiplied bym1• m2 units and divided by d2 units, the result will be Newtons
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Newton's Law of Universal Gravitation
­ the unit of force.
Using Newton's Gravitation Equation to Solve Problems
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
70­kg physics student if the student is standing at sea level, a distance of 6.38 x 106 m from
earth's center.
The solution of the problem involves substituting known values of G (6.673 x 10­11 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:
Sample Problem #2
Determine the force of gravitational attraction between the earth (m = 5.98 x 1024 kg) and a
70­kg physics student if the student is in an airplane at 40000 feet above earth's surface. This
would place the student a distance of 6.39 x 106 m from earth's center.
The solution of the problem involves substituting known values of G (6.673 x 10­11 N m2/kg2), m1 (5.98 x
1024 kg), m2 (70 kg) and d (6.39 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 2 N that is much 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 6.38 x 106 m (equivalent to nearly 20 000 000 feet from the
center of the earth to the surface of the earth). This alteration of distance is like a drop in a bucket when
compared to the large radius of the Earth. As shown in the diagram below, distance of separation becomes
much more influential when a significant variation is made.
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Newton's Law of Universal Gravitation
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 equation presented in Unit 2:
Fgrav = m•g = (70 kg)•(9.8 m/s2) = 686 N
Both equations accomplish the same result because (as we will study later in Lesson 3) the value of g is
equivalent to the ratio of (G•Mearth)/(Rearth)2.
The Universality of Gravity
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 that 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. Gravity is universal. Of course, most gravitational forces are so minimal to be
noticed. Gravitational forces are only 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 familiar
objects. Click the buttons to check answers.
a.
b.
Mass of Object 1 Mass of Object 2 Separation Distance Force of Gravity
(kg)
(kg)
(m)
(N)
Football Player
100 kg
Earth
6.38 x 106 m
5.98 x1024 kg
(on surface)
Ballerina
40 kg
Earth
6.38 x 106 m
5.98 x1024 kg
(on surface)
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c. Physics Student
Earth
6.60 x 106 m
See Answer
See Answer
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Newton's Law of Universal Gravitation
70 kg
5.98 x1024 kg
(low­height orbit)
Physics Student Physics Student
70 kg
70 kg
1 m
Physics Student Physics Student
70 kg
70 kg
0.2 m
f.
Physics Student Physics Book
70 kg
1 kg
1 m
g.
Physics Student
70 kg
Moon
7.34 x 1022 kg
1.71 x 106 m
(on surface)
h.
Physics Student
70 kg
Jupiter
1.901 x 1027 kg
6.98 x 107 m
(on surface)
d.
e.
See Answer
See Answer
See Answer
See Answer
See Answer
See Answer
Today, Newton's law of universal gravitation is a widely accepted theory. It guides the efforts of scientists
in their study of planetary orbits. Knowing that all objects exert gravitational influences on each other, the
small perturbations in a planet's elliptical motion can be easily explained. As the planet Jupiter approaches
the planet Saturn in its orbit, it tends to deviate from its otherwise smooth path; this deviation,
or perturbation, is easily explained when considering the effect of the gravitational pull between Saturn
and Jupiter. Newton's comparison of the acceleration of the apple to that of the moon led to a surprisingly
simple conclusion about the nature of gravity that is woven into the entire universe. All objects attract each
other with a force that is directly proportional to the product of their masses and inversely proportional to
their distance of separation.
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