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Transcript
• Nearly every child knows of the word gravity. Gravity is the
name associated with the mishaps of the milk spilled from the
breakfast table to the kitchen floor and the youngster who
topples to the pavement to end the first bicycle ride. Gravity is
the name associated with the reason for "what goes up, must
come down," whether it be the baseball hit in the
neighborhood sandlot game or the child happily jumping on
the backyard mini-trampoline.
• We all know of the word gravity - it is the thing which causes objects
to fall to Earth. Yet the role of physics is to do more than to
associate words with phenomenon. The role of physics is to explain
phenomenon in terms of underlying principles; in terms of principles
which are so universal that they are capable of explaining more than
a single phenomenon but a wealth of phenomenon in a consistent
manner. Thus, a student's conception of gravity must grow in
sophistication to the point that it becomes more than a mere name
associated with falling phenomenon. Gravity must be understood in
terms of its cause, its source, and its far-reaching implications on
the structure and the motion of the objects in the universe.
• Certainly gravity is a force which exists between the
Earth and the objects which are near it. As you stand
upon the Earth, you experience this force. We have
become accustomed to calling it the force of gravity
and have even represented it by the symbol Fgrav.
Most students of physics progress at least to this
level of sophistication concerning the notion of
gravity.
• This same force of gravity acts upon our bodies as we jump
upwards from the Earth. As we rise upwards after our jump,
the force of gravity slows us down; and as we fall back to
Earth after reaching our peak, the force of gravity speeds us
up. In this sense, the force gravity causes an acceleration of
our bodies during this brief trip away from the earth's surface
and back. In fact, many students of physics have become
accustomed to referring to the actual acceleration of such an
object as the acceleration of gravity.
• Not to be confused with the force of gravity (Fgrav), the
acceleration of gravity (g) is the acceleration experienced by
an object when the only force acting upon it is the force of
gravity. On and near Earth's surface, the value for the
acceleration of gravity is approximately 9.8 m/s/s. It is the
same acceleration value for all objects, regardless of their
mass (and assuming that the only significant force is gravity).
Many students of physics progress this far in their
understanding of the notion of gravity.
• In Lesson 3, we will build on this understanding of gravitation,
making an attempt to understand the nature of this force.
Many questions will be asked: How and by whom was gravity
discovered? What is the cause of this force which we refer to
with the name of gravity? What variables affect the actual
value of the force of gravity? Why does the force of gravity
acting upon an object depend upon the location of the object
relative to the Earth?
• How does gravity affect objects which are far beyond the
surface of the Earth? How far-reaching is gravity's
influence? And is the force of gravity which attracts my
body to the Earth related to the force of gravity between
the planets and the Sun? These are the questions which will
be pursued. And if you can successfully answer them, then
the sophistication of your understanding has extended
beyond the point of merely associating the name "gravity"
with falling phenomenon.
• 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 path 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)
• While Kepler's laws provided a suitable framework for understanding 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
which 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 which 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 fall of an apple to the Earth) which led him to his notion
of universal gravitation.
• A survey of Newton's writings reveals
the illustration at the right 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 in an elliptical path (as in path E).
The motion of the cannonball falling to
the earth 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 which 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 which govern the
motions of objects on Earth also govern
the movement of objects in the heavens.
Satellite Motion
Launch Speed less than 8000 m/s
Projectile falls to Earth
Launch Speed less than 8000 m/s Projectile falls to
Earth
Launch Speed equal to 8000 m/s Projectile orbits
Earth - Circular Path
Launch Speed greater than 8000 m/s Projectile orbits
Earth - Elliptical Path
• 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 effect 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 which 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
which causes the more distant moon to accelerate at a rate of
acceleration which 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 which causes
it to be inversely dependent upon the distance
between the objects?
• The riddle is solved by a comparison between 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
which separates that object from the earth's center.
The moon, being 60 times further away than the apple,
experiences a force of gravity which 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
which 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 effects the
other variable. Equations can be more than merely 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, depress your
mouse on the pop-up menu to check your answers.
Example 1
• 1 . Suppose that two objects attract each other with a
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 1
• F = 4 units
Example 2
• 2. Suppose that two objects attract each other with a
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?
Answer 2
• F = 1.78 units
Example 3
• 3. Suppose that two objects attract each other with a
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 3
• F = 64 units
Example 4
• 4. Suppose that two objects attract each other with a
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?
Answer 4
• F = 400 units
Example 5
• 5. If you wanted to make a profit in buying gold by
weight at one altitude and selling it at another altitude
for the same price per weight, should you buy or sell
at the higher altitude location? What kind of scale
must you use for this work?
Answer 5
• To profit, buy at a high altitude and sell at a lower one.
Gold will weight less at a high altitudes so you will get
more gold for your money by buying at a high altitude.
Then sell at a low altitude where the gold will weigh
more than it was purchased. This illustrates the
inverse relationship between the force of gravity and
distance to the center of the earth.
• 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 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.
• 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.
5.5 Satellites in Circular Orbits
There is only one speed that a satellite can have if the
satellite is to remain in an orbit with a fixed radius.
5.5 Satellites in Circular Orbits
2
mM E
v
Fc  G 2  m
r
r
GM E
v
r
5.5 Satellites in Circular Orbits
Example 9: Orbital Speed of the Hubble Space Telescope
Determine the speed of the Hubble Space Telescope orbiting
at a height of 598 km above the earth’s surface.
v
6.67 10
11

N  m 2 kg 2 5.98 10 24 kg
6
3
6.38 10 m  598 10 m
 7.56 103 m s
16900 mi h 

5.5 Satellites in Circular Orbits
GM E 2 r
v

r
T
2 r
T
GM E
32
5.5 Satellites in Circular Orbits
Global Positioning System
T  24 hours
2 r
T
GM E
32
5.5 Satellites in Circular Orbits
5.6 Apparent Weightlessness and Artificial Gravity
Conceptual Example 12: Apparent Weightlessness and
Free Fall
In each case, what is the weight recorded by the scale?
5.6 Apparent Weightlessness and Artificial Gravity
Example 13: Artificial Gravity
At what speed must the surface of the space station move
so that the astronaut experiences a push on his feet equal to
his weight on earth? The radius is 1700 m.
2
v
Fc  m  mg
r
v  rg

1700 m9.80 m s 2 
5.7 Vertical Circular Motion
2
1
v
FN 1  mg  m
r
FN 2
FN 4
2
2
v
m
r
2
4
v
m
r
2
3
v
FN 3  mg  m
r