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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 affecting the magnitude of a gravitational force. Consider 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. 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 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.
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 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 the universal gravitation
constant. The precise value of G was determined experimentally by Henry Cavendish in the century
after Newton's death. The value of G is found to be
G = 6.673 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 6.38 x
106 m from earth's center.
The solution of the problem involves substituting known values of G (6.673 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:
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 6.39 x 106 m from earth's center.
The solution of the problem involves substituting known values of G (6.673 x 10-11 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 which 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.
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 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 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. 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.
Mass of Object 1
(kg)
Mass of Object 2 (kg)
Football Player 100
Earth
kg
Separation Distance
Force of Gravity
(m)
(N)
5.98 x1024 kg
6.38 x 106 m (on surface)
5.98 x1024 kg
6.38 x 106 m (on surface)
a.
b.
Ballerina 40 kg
c. Physics Student
Earth
Earth
5.98 x1024 kg
70 kg
d.
e.
f.
6.60 x 106 m (low-height
orbit)
Physics Student
70 kg
Physics Student
70 kg
Physics Student
70 kg
g. Physics Student
Physics Student
Physics Student
Physics Book
70 kg
70 kg
1 kg
1m
0.2 m
1m
Moon 7.34 x 1022 kg
1.71 x 106 m (on surface)
Jupiter 1.901 x 1027 kg
6.98 x 107 m (on surface)
70 kg
h.
Physics Student
70 kg
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 affect 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 which is
woven into the entire universe. All objects attract each other with a force which is directly
proportional to the product of their masses and inversely proportional to their distance of
separation.
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 reduced in half, 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 mass of
both objects was doubled, and if the distance between the objects remained the same, then what
would be 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 mass of
both objects was doubled, and if the distance between the objects was doubled, then what would be
the new force of attraction between the two objects?
5. Suppose that two objects attract each other with a gravitational force of 16 units. If the mass of
both objects was tripled, and if the distance between the objects was doubled, then what would be
the new force of attraction between the two objects?
6. Suppose that two objects attract each other with a gravitational force of 16 units. If the mass of
object 1 was doubled, and if the distance between the objects was tripled, then what would be the
new force of attraction between the two objects?
7. As a star ages, it is believed to undergo a variety of changes. One of the last phases of a star's
life is to gravitationally collapse into a black hole. What will happen to the orbit of the planets of the
solar system if our star (the Sun shrinks into a black hole)? (And of course, this assumes that the
planets are unaffected by prior stages of the Sun's evolving stages.)
8. 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?
9. Anita Diet is very concerned about her weight but seldom does anything about it. After learning
about Newton's law of universal gravitation in Physics class, she becomes all concerned about the
possible affect of a change in Earth's mass upon her weight. During a (rare) free moment at the
lunch table, she speaks up "How would my weight change if the mass of the Earth increased by
10%?" How would you answer Anita?
10. When comparing mass and size data for the planets Earth and Jupiter, it is observed that Jupiter
is about 300 times more massive than Earth. One might quickly conclude that an object on the
surface of Jupiter would weigh 300 times more than on the surface of the Earth. For instance, one
might expect a person who weights 500 N on Earth would weigh 150000 N on the surface of Jupiter.
Yet this is not the case. In fact, a 500-N person on Earth weighs about 150 N on the surface of
Jupiter. Explain how this can be.
Key
Answers for the table:
a. Fgrav = 980 N
b. Fgrav = 392 N
c. Fgrav = 686 N
d. Fgrav = 3.27 X 10-7 N
e. Fgrav = 8.17 X 10-6 N
f. Fgrav = 4.67 X 10-9 N
g. Fgrav = 117 N
h. Fgrav = 1823 N
Answers for Check Your Understanding
1. F = 4 units If the distance is increase by a factor of 2, then force
will be decreased by a factor of 4 (22). The new force is then ¼ of the
original 16 units
F = 16 units/4 = 4 units
2. F = 64 units If the distance is decrease by a factor of 2, then
force will be increased by a factor of 4 (22). The new force is then 4
times the original 16 units
F = 16 units X 4 = 64 units
3. F = 64 units If each mass is increased by a factor of 2, then force
will be increased by a factor of 4 (2 X 2). The new force is then 4
times the original 16 units.
F = 16 units X 4 = 64 units
4. F = 16 units. If each mass is increased by a factor of 2, then force
will be increased by a factor of 4 (2 X 2). But this affect is offset by
the doubling of the distance. Doubling the distance would cause the
force to be decreased by a factor of 4 (22); the result is that there is
no net affect on force.
F = 16 units X 4/4 = 16 units
5. F = 36 units If each mass is increase by a factor of 3, then force
will be increased by a factor of 9 (3 X 3). But this affect is partly
offset by the doubling of the distance. Doubling the distance would
cause the force to be decreased by a factor of 4 (22). The net affect
on force is that it increased by 9/4.
F = 16 units X 9/4 = 36 units
6. F = 3.56 units If the mass of one object is doubled, then the force
of attraction will be doubled as well. But this affect is more than offset
by the tripling of the separation distance. Tripling the distance would
cause the force to be decreased by a factor of 9 (32). The net affect
on force is that it decreased by a factor of 2/9.
F = 16 units X 2/9 = 3.56 units
7. No affect. The shrinking of the sun into a black hole would not
influence the amount of force with which the sun attracted the Earth
since neither mass of the sun nor the distance between the Earth’s and
sun’s centers would change.
8. Buy high and sell low. The mass of the purchased fold would be
the same at both altitudes. Yet it would weigh less at higher altitudes.
So to make a profit, Noah should buy at high altitudes and sell at low
altitudes. He would have more gold (by weight) to sell at the lower
altitudes.
9. Anita-that’s a great question. Since your weight is directly
dependent upon the mass of the Earth, you would weigh 10% more.
But don’t worry honey, you wouldn’t look any different than you do
now since your mass would remain as is.
10. The affect of the greater mass of Jupiter is partly offset by the
fact that the radius of Jupiter is larger. An object on Jupiter’s surface
is 10 times farther from Jupiter’s center than it would be if on Earth’s
surface. So the 300-fold increase in force (due to the greater mass)
must be divided by 100 since the separation distance is 10 times
greater.