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 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. 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 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 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
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 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 = 6.673 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.
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 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
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.
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.
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.
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?
Answer: F = 4 units
If the distance is increased by a factor of 2, then force will
be decreased by a factor of 4 (22). The new force is
then 1/4 of the original 16 units.
F =
(16 units ) / 4 = 4 units
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?
Answer: F = 64 units
If the distance is decreased 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) • 4 = 64 units
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?
Answer: F = 64 units
If each mass is increased by a factor of 2, then force will be
increased by a factor of 4 (2*2). The new force is then 4 times the original 16
units.
F =
(16 units ) • 4 = 64 units
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?
Answer: F = 16 units
If each mass is increased by a factor of 2, then force will be
increased by a factor of 4 (2*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) • 4 / 4 = 16 units
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?
Answer: F = 36 units
If each mass is increased by a factor of 3, then force will be
increased by a factor of 9 (3*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) * 9 / 4 = 36 units
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?
Answer: 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) • 2 / 9 = 3.56 units
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.)
Answer: 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 the mass of the sun nor
the distance between the Earth's and sun's centers would
change.
8. Having recently completed her first Physics course, Dawn Well
has devised a new business plan based on her teacher's Physics for Better Living theme. Dawn learned that objects
weigh different amounts at different distances from Earth's center. Her plan involves buying gold by the
weight at one altitude and then selling it at another altitude at the same
price per weight. Should Dawn buy at a high altitude and sell at a low altitude
or vice versa?
Answer: Buy high and sell low
The mass of
the purchased gold would be the same at both altitudes. Yet it would weight less at higher altitudes. So to make a profit,
Dawn should buy at high altitudes and sell at low altitudes. She would have
more gold (by weight) to sell at the lower altitudes.
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 effect 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?
Answer:
"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. 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
weighs 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 1500 N on the surface of Jupiter. Explain how this can be.
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.