Download GRAVITY

'I
GRAVITY
THE BIG
IDEA
The same force that pulls apples off
trees and pulls you toward Earth's
center-gravity-underlies the motion
of satellites.
.6. Earth's gravity holds the space shuttle in orbit.
If it didn't, the space shuttle would fly off in a
straight line.
I
s the space shuttle in the grip of Earth
gravity, or is it beyond it? Was gravity
discovered by Isaac Newton? Or was
gravity discovered by earlier people who fell
from trees or from their caves? If Newton
didn't discover gravity, what did he discover
about gravity? Does gravity reach the Moon?
Does it reach to the planets? To the stars?
How far does gravity reach? Why are there
ocean tides? Is it true that if a ba ll is thrown
fast enough, and above the atmosphere, it
becomes a satellite? Why doesn't Earth's
gravitational pull make a satellite crash into
Earth? To answer t hese questions, we need
to learn the physics of gravity and satellite
motion.
DISCOVER!
What Makes Ink Flow in a Ballpoint Pen?
Observe and Record
1. Put a piece of paper flat on your desk. Write
your name on it with a ballpoint pen. (A ballpoint pen has a hole in the barrel through which
ink flows to the pen's point.)
2. Tape the paper to a book. Hold it over your
head.
3. With the pen upside down, try to write your
name on the paper.
Analyze and Conclude
1. Observing Were you able to write your name
with the pen upside down?
2. Predicting Would you be able to write with an
upside-down pencil? Try it and see.
3. Analyzing Why is there a difference between
the pen's and pencil's ability to write while
upside down?
4 . Applying Would an astronaut be able to use a
ballpoint pen in Earth orbit? Why or why not?
113
114
PART ONE
Physics
'1.1 The Falling Apple and the
Falling Moon
FIGURE 7.1 ...
Newton realizes that Earth's
gravity affects both the Moon
and the apple.
Uh-oh, the Moon is falling!
G
...
)o
Legend tells us that when Newton was a young man sitting under an
apple tree, he made a connection that changed the way we see the world.
He saw an apple fall. Perhaps he looked up through the tree branches
toward the origin of the falling apple and noticed the Moon. In any
event, .I Newton had the insight to realize that the force pulling on
a falling apple is the same force that pulls on the Moon. Newton
realized that Earth's gravity reaches to the Moon.
Why doesn't the Moon fall toward Earth, as an apple falls from a
tree? If the apple or anything else drops from rest, it falls in a vertical
straight-line path. To get a better idea of this, consider a tree in the
back of a truck (Figure 7.2). If the truck is at rest when the apple falls,
we see that the apple's path is vertical. But if the truck is moving when
the apple begins its fall, the apple follows a curved path. Can you see
that the faster the truck moves, the wider the curved path of the falling
apple? Later in this chapter we'll see that if the apple or anything else
moves fast enough so that its curved path matches Earth's curvature, it
will not fall to Earth-it will fall around Earth. The falling object orbits
Earth as an Earth satellite.
v
0
FIGURE 7.3 ...
The tangential velocity
of the Moon allows it to fall
around Earth rather than
directly into it.
FIGURE 7.2 ...
If an apple falls from a tree at rest, it falls straight downward. But if it falls
from a moving tree, it falls in a curved path.
The Moon is an Earth satellite. As the Moon revolves around Earth,
it maintains a tangential velocity-a velocity parallel to Earth's surface.
Newton realized that the Moon's tangential velocity keeps it continually
falling around Earth instead of directly into it. Newton further realized
that the Moon's path around Earth is similar to the paths of the planets
around the Sun.
CHECK YOUR THINKING
In Figure 7.3, we see that the Moon falls around Earth rather than
straight into it. If the tangential velocity were zero, how would the
Moon move?
What was Newton's great
insight that related a falling
apple to the Moon?
Answer
If the Moon's tangential velocity were zero, it would fall straight down and crash into Earth!
(Compare this idea with Figure 7.2.)
..
CHAPTER 7
GRAVITY
115
'1.2 Newton's Law of Universal
Gravitation
Newton realized that everything pulls on everything else-everywhere
and all the time-through the force of gravity. .I Further, Newton
discovered that that force of gravity is simple--it involves only mass
and distance. According to Newton, every mass attracts every other
mass with a force that is directly proportional to the product of the two
interacting masses. And the force is inversely proportional to the square
of the distance separating them. This is the law of universal gravitation .
mass
1 X_mass
Force,....., _
_.:...._
_--=.2
distance 2
Expressed in symbol shorthand,
where m 1 and m2 are the masses, and dis the distance between their
centers. Thus, the greater the masses m 1 and m 2, the greater the force of
attraction between them. The greater the distance of separation d, the
weaker the force of attraction-weaker as the inverse square of the
distance between their centers.
Just as sheet music guides
a musician playing music,
equations guide an
integrated science student
to see how concepts are
connected.
CHECK YOUR THINKING
1. According to the equation for gravity, what happens to the force
between two bodies if the mass of one body is doubled?
2. What happens if instead the mass of the other body is doubled?
3. What happens if the masses of both bodies are doubled?
4. What happens if the mass of one body is doubled, and the other
tripled?
Answers
I. When one mass is doubled, the force between them doubles.
2. The force is still doubled, because it doesn't make any difference which mass doubles.
(2 X 1 =I X 2; same product either way!)
3. The force is four times as much.
4. Double X triple = six:. So the force is six times as much. (If you don't see why, discuss
this with a friend before going further. )
DING
What two factors does the
force of gravity depend on?
'1.3 Gravity, Distance, and the
Inverse-Square Law
Gravity gets weaker with distance the same way a light gets dimmer as
you move farther away from it. Consider the candle flame in Figure 7.4.
Light from the flame travels in all directions in straight lines. A patch is
shown 1 m from the flame. Notice that at a distance of 2 m away, the
PhysicsPiace.com
Video
Inverse-Square Law
116
PART ONE
Physics
FIGURE 7.4 A
Light from the flame spreads
in all directions. At twice the
distance, the same light is
spread over four times the
area; at three times the
distance it is spread over
nine times the area.
Saying that F is inversely
proportional to the square
of d means, for example,
that if d gets bigger by 3,
F gets smaller by 9.
FIGURE 7.5 ...
The inverse-square law. Paint
spray travels in straight lines
away from the nozzle of the
can. Like gravity, the "strengt h"
of the spray obeys the inversesquare law. Fill in the blanks.
ttC:H
C:K
YOUR READI G
What kinds of natural
phenomena does the
inverse-square law apply to?
light rays that fall on the patch spread to fill a patch twice as
tall and twice as wide. The same light falls on a patch with four
times the area. The same light 3 m away spreads to fill a patch
three times as tall and three times as wide. The light would fill a
patch with nine times the area.
As the light spreads out, its brightness decreases. When
you're twice as far away, can you see that it appears one-fo urth
as bright? And can you see that when you're three times as far
away, it appears one-ninth as bright? There is a rule here: The intensity
of t he light decreases as the inverse square of the distance. We therefore
say the inverse-square law describes the intensity of light.
Paint from a paint sprayer also follows an inverse-square law. Pretend you hold a paint gun at the center of a sphere with a radius of l m
(Figure 7.5). Suppose that a burst of paint produces a square patch of
paint 1 mm thick. How thick would the patch be if the experiment were
done in a sphere with twice the radius-that is, with the spray gun twice
as far away? The answer is not half as thick, because the paint would
spread to a patch twice as tall and twice as wide. It would spread over an
area four times as big, and its thickness would be only-~ mm. Can you see
that for a sphere of radius 3 m, the thickness of the paint patch would be
only~ m m? Do you see that the thickness of paint decreases as the
square of t he distance? The inverse-square law holds for light, for paint
spray, and for gravity. .! The inverse-square law applies to all things
that uniformly spread away from a local source throughout the
surrounding space. We'll see this to be true of the electric field about an
electron, light from a match, radiation from a piece of uranium, and
sound from a cricket.
1 area unit
Paint spray
1 layer
thick
4 area units
1;4
layer
thick
( ) area units
( ) area units
( ) layer
thick
( ) layer
thick
The greater the distance from Earth's center, the less the gravitational
force on an object. In using Newton's equation for gravity, the distance
term d is the distance between the centers of the masses of objects attracted
to each other. Note in Figure 7.6 that the girl at the top of the ladder weighs
only one-fourth as much as she weighs at Earth's surface. That's because
she is twice the distance from Earth's center.
CHAPTER 7
GRAVITY
117
CHECK YOUR THINKING
1. How much does the force of gravity change between Earth and
a receding rocket when the distance between them is doubled?
Tripled? Ten times as much?
2. Consider an apple at the top of a tree. The apple is pulled by
Earth's gravity with a force of 1 N. If the tree were twice as tall,
would the force of gravity be on1y one-fourth as strong? Defend
your answer.
Answers
1
I. When the distance is doubled, the force is as much. \t\'hen tripled,~ as much. When
10 times, 1 ~ as much.
2. No, because the twice-as-tall apple tree is not twice
as far from Earth's center. The taller tree would have
to be 6,370 km tall (Earth's radius) for the apple's
weight to reduce to~ N. For a decrease in weight by
I%, an object must be raised 32 km-nearly four
times the height of Mt. E\·crest. So as a practical
matter we disregard the effects of everyday changes
in elevation for gravity. The apple has practically
the same weight at the top of the tree as at the
bottom.
FIGURE 7.6 A
At the top of the ladder,
the girl is twice as far from
Earth's center and weighs
only one-fourth as much as
at the bottom of the ladder.
So gravity gets weaker with increasing distance. But no matter how
far away, Earth's gravitational force approaches, but never reaches, zero.
Even if you traveled to the far reaches of the universe, the gravitational
influence of home would still be with you. It may be overwhelmed by
the gravitational influences of nearer and/or more massive bodies, but it
is there. The gravitational influence of every material object, however
small or far, is exerted through all of space.
FIGURE 7 .7 A
As the rocket gets farther from
Earth, gravitation between the
rocket and Earth decreases.
Distance
Apple weighs
( )N here
FIGURE 7.8 A
!NTERAC TIVE FIGURE,
i
If an apple weighs 1 N at Earth's surface, it weighs only N twice as far
from Earth's center. At three times the distance, it weighs only~ N. What
would it weigh at four times the distance? Five times?
118
PART ONE
Physics
CHECK YOUR THINKING
1. Light from the Sun, like gravity, obeys the inverse-square law.
If you were on a planet twice as far from the Sun, how bright
would the Sun look?
2. How bright would the Sun look if you were on a planet half as
far from the Sun?
Answers
Just as rc is the constant for
the circumference of a circle,
C = reD, G is the constant
for the force of gravity,
m1 m2
F=GdZ.
I. One-quarter as bright.
2. Four times as bright.
'1.4 The Universal Constant
of Gravitation, G
II The law of universal gravitation can be written as an exact equation
when the universal constant of gravitation, G, is used. Then we have
F = Gmlm2
d2
flicHECH
0
What constant do we
need to know to convert
the relationship between
gravitational force, mass,
and distance to an exact
equation?
The units of G make the force come out in newtons. The magnitude
of G is the sam e as the gravitation al force between two 1-kg masses that
are 1 m apart: 0.0000000000667 N.
G = 6.67 X 10-ll N · m 2/kg 2
The universal constant G is an extremely small number. It shows that
gravity is a very weak force compared with electrical forces. The large net
gravitational force we feel as weight is because of the enormity of atoms
in planet Earth that are pulling on us.
Comparing Gravitational Attractions
How do different gravitational pulls compare? Try
a few calculations with Newton's law of universal
gravitation.
Solutions
(a) Mars:
m1m2
F = G---;]2
( 6.67
Problems
A 3-kg newborn baby at Earth's surface is attracted by
gravity to Earth with a force of about 30 N.
(a) Calculate the force of gravity with which the baby
on Earth is attracted to the planet Mars, when
Mars is closest to Earth. (The mass of the Mars is
6.4 X 10 23 kg, and its closest distance from Earth is
5.6 X 10 10 m .)
(b) Calculate the force of gravity between the baby and
the physician who delivers her. Assume that the
physician has a mass of 100 kg and is 0.5 m from
the baby.
(c) How do these forces compare?
X 10-l l
N · m 2/kg 2)(3 kg)(6.4 X 1023 kg)
(5.6 X J0 10 m) 2
= 4.1 X
10- 8 N
(b) Physician:
mLm2
F=G--
d2
=
( 6.67 X 10-
11
= 8.0 X 10- 8 N
N · m 2/kg2)(3 kg)(10 2 kg)
(0.52 m )
(c) The baby is pulled toward the physician with about
twice the gravitational force with which she is
pulled toward Mars.
..
CHAPTER 7
_ll
GRAVITY
119
,..,. INTEGRATED SCIENCE
-.., EARTH SCIENCE AND ASTRONOMY
-~
Ocean Tides
Sailors have always known there is a connection between ocean tides and
the Moon. .! Newton was the first to show that tides are caused by
diHerences in the gravitational pull by the Moon on Earth's opposite
sides. Because gravitational force gets weaker with distance, the gravitational force between Earth and the Moon is stronger on the side of Earth
nearer to the Moon than on the opposite side of Earth.
_. FIGURE 7.9
Low tide; high tide.
Low tide
High tide
To understand why these different pulls produce tides, let's look at a
ball ofJell-0 (Figure 7.10). If you exerted the same force on every part
of the ball, the ball would remain perfectly round as it accelerated. But
if you pull harder on one side than the other, the different pulls would
stretch the ball. That's what's happening to this big ball on which we live.
Different pulls of the Moon stretch Earth, most notably in its oceans.
The stretch produces an average ocean bulge of nearly 1 m on each side
of Earth. That's why the oceans on opposite sides of Earth bulge about
1 m above the ocean's average surface level. Earth rotates once per day,
so a given point on Earth passes beneath both of these bulges each day.
This produces two sets of ocean tides per day-two high tides and two
low tides.
The Moon pulls more on the
side of Earth closest to it, and
less on the opposite and
farther side. This difference in
pulls stretches Earth to
produce ocean bulgesexperienced as tides.
.. FIGURE 7.10
A ball of Jell-0 stays spherical
when all parts are pulled
equally in the same direction.
When one side is pulled more
than the other, its shape is
distorted.
While Earth rotates, the Moon also moves in its orbit. The Moon
appears at the same position in our sky every 24 hours and 50 minutes,
so the two-high-tide cycle is actually at 24-hour-and-50-minute intervals. This means that tides do not occur at the same time every day.
The Sun also contributes to ocean tides, but it's about half as
effective as the Moon. Interestingly, the Sun pulls 180 times as hard on
Earth as on the Moon. Why aren't tides due to the Sun 180 times as
large as tides due to the Moon? Because of the Sun's great distance, the
120
PART ONE
Physics
FIGURE 7.11 lill>
Two tidal bulges are produced
by differences in gravitational
pulls by the Moon.
1tcHECH
EADING
What causes tides?
0
difference in gravitational pulls on opposite sides of Earth is very small. In
other words, the Sun pulls almost as hard on the far side of Earth as it does
on the near side. You'll understand tides more when you tackle "Ocean
Tides" in the Conceptual Integrated Science Explorations Practice Book.
CHECK YOUR THINKING
1. If you pull a blob of Jell-0 equally on all parts, it will keep its
shape as it moves. But if you pull harder on one end than the
other, it will stretch. How does this relate to tides?
2. If the Moon didn't exist, would Earth still have ocean tides? If so,
how often?
3. We know that both the Moon and the Sun produce our ocean
tides. And we know that the Moon plays the greater role because
it is closer. Does its closeness mean it pulls the oceans with more
gravitational force than the Sun?
Answers
I. Just as differences in pulls on the Jell-0 will distort it, differences in pulls on the oceans
distort the ocean and produce tides.
2. Yes, Earth's tides would be due only to the Sun. They'd occur twice per day (every
I 2 hours instead of every I 2.4 hours) because of Earth's daily rotation.
3. No, the Sun's pull is much stronger. But tides arc not caused by gravitational pulls.
Tides are caused by differences in pulls across a body. Differences in pulls, not pulling
strength, is the key to tides.
UNift'ING
CONCEPT
Newton's Laws of Motion
SECTION
2.5
PhysicsPiace.com
Videos
Weight and Weightlessness;
Apparent Weightlessness
'1.5 Weight and Weightlessness
The force of gravity, like any force, can produce acceleration. Objects
under the influence of gravity accelerate toward each other. Because we
are almost always in contact with Earth, we think of gravity as something
that presses us against Earth rather than as something that accelerates us.
The pressing against Earth is the sensation we interpret as weight.
Stand on a bathroom scale that is supported on a stationary floor.
The gravitational force between you and Earth pulls you against the supporting floor and the scale. By Newton's third law, at the same time, the
floor and scale push upward on you. Located in between you and the
supporting floor are springs inside the bathroom scale. Compression of
the springs is read as your weight.
CHAPTER 7
INTERACTIVE fiGURl
h
Normal
weight
~~
Qlj
Less than
normal weight
l
121
~ FIGURE 7.12
CJ
Greater than
normal weight
GRAVITY
The sensation of weight is
equal to the force that you
exert against the supporting
floor. If the floor accelerates
up or down, your weight
seems to vary. You feel
weightless when you lose
your support in free fall.
I
Zero weight
If you repeat this weighing procedure in a moving elevator, your
weight reading would vary-not during steady motion, but during accelerated motion. If the elevator accelerates upward, the bathroom scale
and floor push harder against your feet. So the springs inside the scale
are compressed even more. The scale shows an increase in your weight.
If the elevator accelerates downward, the scale shows a decrease in
your weight. The support force of the floor is now less. If the elevator
cable breaks and the elevator falls freely, the scale registers zero pounds.
You are "weightless." Similarly, astronauts in Earth orbit are weightless
because they have no support force as they fall freely around Earth.
In Chapter 2, we defined weight as the gravitational force acting
on an object. We can now refine this definition and say that the
weight of an object is the force it exerts against a supporting floor
(or weighing scale). According to this definition, you are as heavy as
~ FIGURE 7.13
These astronauts in training
feel weightless because they
are freely falling, like a tossed
projectile. They are not pressing against anything that
would provide a support
force.
124
PART ONE
Physics
CHECK YOUR THINKING
At the instant a horizontal cannon fires a
cannonball from atop a high cliff, another
cannonball is simply dropped from the
same height. Which hits the ground below
first, the one fired downrange, or the one
that drops straight down?
Answer
Both cannonballs hit the ground at the same time, because
both fall the snme vertical distance.
Tutorial
Projectile Motion
Videos
Projectile Motion Demo; More
Projectile Motion
In Figure 7.19, we consider a stone thrown upward at an angle. If
there were no gravity, the path would be along the dashed line with the
arrow. Positions of the stone at 1-s intervals along the line are shown by
red dots. Because of gravity, the actual positions (dark dots) are below
these points. How far below? The answer is, the same distance an object
would fall if it were dropped from the red-dot positions. When we connect the dark dots to plot the path, we get a different parabola.
In Figure 7.20, we consider a stone thrown at a downward angle. The
physics is the same. If there were no gravity, it would follow the dashed
line with the arrow. Because of gravity, it falls beneath this line, just as in
the previous cases. The path is a somewhat different parabola.
.. e '"'
--' .'
FIGURE 7.19 ....
I
A stone thrown at an upward angle
would follow the dashed line in the
absence of gravity. Because of
gravity, it falls beneath this line
and describes the parabola shown
by the solid curve.
FIGURE 7.20 .A
.-,'
A stone thrown at a downward
angle follows a somewhat different
parabola.
CHAPTER 7
DISCOVER!
GRAVITY
3
4
125
5
Make your own model of projectile paths. On a
ruler or a stick, at position 1, hang a bead from
a string 1 em long as shown. At position 2, hang
a bead from a string 4 em long. At position 3, do
the same with a 9-cm length of string. At position 4,
use 16 em of string; for position 5, use 25 em of
string. Hold the stick horizontally and you have a
version of Figure 7 .18. Hold it at a slight upward
angle to show a version of Figure 7.19. Hold it at
a downward angle, and you have Figure 7.20.
~ The curved path of any projectile is a combination of horizontal
and vertical motions. This can be shown graphically. A vector represents
the projectile's overall velocity. Its length is the projectile's speed, and its
direction is the direction the projectile is moving at that point in time. For
example, consider the girl throwing the stone in Figure 7.21. The velocity
she gives the stone is shown by the diagonal vector. Notice that this vector
has horizontal and vertical components. (Read more about drawing
vector components in Appendix C.) These velocity components are independent of each other. Each of them acts as if the other didn't exist. When
combined, they produce the curved path.
FIGURE 7.22 A
FIGURE 7.21 A
The velocity of the ball
(diagonal vector) has vertical
and horizontal components.
The vertical component relates to how high the ball will
go. The horizontal component
relates to the horizontal range
of the ball.
INTERACTIVE FIGURE
The velocity of a projectile at various points. Note that the vertical component
changes, while the horizontal component is the same everywhere.
CHECK YOUR THINKING
1. At what part of its trajectory does a projectile have minimum
speed?
2. (Challenge Question) A tossed ball changes speed along its
parabolic path. When the Sun is directly overhead, does the
shadow of the ball across the field also change speed?
Answers
1. The speed of a projectile is a minimum at the top of its path. If it is launched vertically,
its speed at the top is zero. If it is projected at an angle, the vertical component of
speed is zero at the top, leaving only the horizontal component. So the speed at the top
is equal to the horizontal component of the projectile's velocity at any point.
2. No, because the shadow moves at constant velocity across the field, showing exactly the
motion due to the horizontal component of the ball's velocity.
tfc:HEC:K
What two motions add
together to make a
projectile's curved path?
126
PART ONE
/
L/
Physics
.. ,--- ....
/
.....
'
' "'" Ideal
' path
\
/
Actual path__..
;:s::;;;:szpz=
FIGURE 7 .23
\
..
£2!!
===
\
\
\
We have considered projectile motion without air drag.
You can neglect air drag for a ball you toss back and fo rth
with your friends because the speed is small. But higher
speed makes a difference. Air drag is a factor for high-speed
projectiles. The result of air drag is that both range (horizontal distance traveled) and altitude (height) are less
(Figure 7.23).
~
INTERACTIVE FIGURE.
In the presence of air drag, a high-speed projectile falls short of a
parabolic path. The dashed line shows an ideal path with no air drag. The
solid line is the actual path.
UNIJYING
CONCEPT
'1. 'I Fast-Moving ProjectilesSatellites
Suppose a cannon fires a cannonball so fast that its curved path matches
the curvature of Earth. Then, without air drag, it would be an Earth
SECTION 3.6
satellite! The same would be true if you could throw a stone fast
enough. Any satellite is simply a projectile moving fast
------ ----- f enough to fall continually around Earth.
5m
In Figure 7.24, we see the curved paths of a stone
~ thrown horizontally at different speeds. Whatever the pitching speed, in each case the stone drops the same vertical
distance in the same time. For a 1-s drop, that distance is
FIGURE 7.24 ~
5 m (perhaps by now you have made use of this fact in Lab).
Throw a stone at any speed and 1 slater it falls
So if you simply drop a stone from rest, it will fall 5 m in 1 s
5 m below where it would have been if there
of fall. Toss the stone sideways, and in 1 s it will be 5 m
were no Earth gravity.
below where it would have been without gravity. v' To be
an Earth satellite, the stone's horizontal velocity must be great enough
for its falling distance to match Earth's curvature.
Friction
What horizontal velocity
must a tossed stone possess
to become an Earth satellite?
,.... INTEGRATED SCIENCE
w..J ASTRONOMY
Earth Satellites
It is a geometrical fact that the surface of Earth drops a vertical distance
Do es 8000 m/s (or 8 km/s)
seem fast to you? If not ,
convert this speed to
kilomete rs per hou r. You get
an impressive 29,000 km/ h
(18,000 milh). Fast, indeed!
of 5 m for every 8000 m tangent to the surface. (A tangent to a circle or
to Earth's surface is a straight line that touches the circle or surface at
only one place.) What do we call a projectile that moves fast enough to
travel a horizontal distance of 8000 m during 1 s? We call it a satellite.
Neglecting air drag, this 8000-m/s projectile would follow the curvature
of Earth. Eight thousand meters per second, or 8 km/s, is very fast.
At this speed, atmospheric friction would burn up the projectile.
This happens to grains of sand and other meteors that graze Earth's
atmosphere, burn up, and appear as "falling stars." That is why satellites
like the space shuttles are launched to altitudes higher than 150 km-to
be above the atmosphere.
=====
CHAPTER 7
8,000
GRAVITY
?::;:;:.9_
/
'
;
I
I
FIGURE 7.25 A
Earth's curvature drops a vertical distance of
5 m for each 8000 m tangent (not to scale).
It is a common misconception that satellites orbiting at high altitudes are free from gravity. Nothing could be further from the truth.
The force of gravity on a satellite 150 km above Earth's surface is nearly
as great as at the surface. If there were no gravity, motion would be along
a straight-line path instead of curving around Earth. High altitude puts
the satellite beyond Earth's atmosphere, but not beyond Earth's gravity.
As mentioned previously, Earth gravity goes on forever. It gets weaker
with distance, but it never reaches zero.
Satellite motion was understood by Isaac Newton. He reasoned
that the Moon is simply a projectile circling Earth under gravitational
attraction. This concept is illustrated in Figure 7.27, which is an
actual drawing by Newton. He compared the Moon's motion to a
cannonball fired from the top of a high mountain. He imagined that
the mountaintop was above Earth's atmosphere, so that air drag
would not slow the motion of the cannonball. If a cannonball were
fired with a low horizontal speed, it would follow a curved path and
soon hit Earth below. If it were fired faster, its path would be wider
and it would hit a place on Earth farther away. If the cannonball
were fired fast enough, Newton reasoned, the curved path would
become a circle and the cannonball would circle Earth indefinitely. It
would be in orbit.
Newton calculated the speed for circular orbit about Earth. However,
because such a cannon-muzzle velocity was dearly impossible, he did
not foresee humans launching satellites. And quite likely he didn't foresee multistage rockets.
.! Both the cannonball and the Moon have a tangential ("sideways")
velocity, parallel to Earth's surface. This velocity is enough to ensure
motion around Earth rather than into it. Without air drag to reduce
speed, the Moon or any Earth satellite "falls" around and around Earth
indefinitely. Similarly, the planets continually fall arow1d the Sun in
closed paths.
Why don't the planets crash into the Sun? They don't because of
their tangential velocities. What would happen if their tangential
velocities were reduced to zero? The answer is simple enough: their
motion would be straight toward the Sun and they would indeed
crash into it. Any objects in the solar system without enough tangential velocity have long ago crashed into the Sun. What remains is the
harmony we observe.
II I
127
I
'
\
'
._
__
FIGURE 7.26 ._
If the stone is thrown fast
enough that its curve matches
Earth's curvature, it will be a
satellite.
UNI~NG
CONCEPT
Newton's Laws of Motions
SECTION
2.5
FIGURE 7.27 A
A drawing by Newton
showing how a faster and
faster projectile could circle
Earth and become a satellite.
-.ticHECK
Why do the Moon and other
Earth satellites fall around
Earth rather than into it?
PhysicsPiace.com
Video
Circular Orbits
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PART ONE
Physics
DISCOVER!
Swing a bucket of water in a vertical circle, as shown
by physics teacher Marshall Ellenstein. If you swing it
sufficiently fast, the water won't spill. The explanation is similar to why satellites don't "fall" to Earth.
Actually, both the water in the bucket and satellites
are falling. The water doesn't spill at the top of the
swing because the bucket swings downward at least
as fast as the water falls. Similarly, a satellite doesn't
get closer to Earth because it falls a distance that
matches Earth's curvature. Analogies are the way to
understand concepts!
CHECK YOUR THINKING
1. Can we also say that a satellite stays in orbit because it is above
Earth's main pull of gravity?
2. What would happen to a rocket launched vertically that remains
vertical as it rises?
Answer
l. No, no, no! No satellite is completcly"above" Earth's gravity.lfthe satellite were not in the
grip of Earth's gravity, it would not orbit and would follow instead a straight line path.
2. After the rocket reaches its highest point it would fall back to its launching site-not
a good idea!
Satellite TV
The time it takes an Earth satellite to make a complete
trip around our planet depends on its distance from
Earth. At a distance of about 5.5 Earth radii, a satellite
orbits the planet in 24 h. This is also the amount of
time it takes for Earth to make one full spin on its axis.
That's why a satellite orbiting around the equator once
every 24 h stays above the same point on the ground.
We say this satellite is in geosynchronous orbit-it
travels along with Earth's surface below.
Satellite television technology uses communications satellites that are in geosynchronous orbit to
bounce television signals from space to Earth. Because
a communications satellite is in geosynchronous orbit,
its position relative to the receiving dish is fixed. It's
always above the location it serves. Therefore, you
don't need to constantly adjust your dish to stay with
your satellite-just grab the remote.
'I REVIEW
WORDS TO KNOW AND USE
Inverse-square law A law relating the intensity of
an effect to the inverse square of the distance from the
cause:
.
Intensity
~
l
.
1
distance-
Law of universal gravitation Every body in the universe
attracts every other body with a mutually attracting
force. For two bodies, this force is directly proportional
to the product of their masses and inversely proportional
to the square of the distance separating them:
5. How does the brightness of light change when a
point source of light is brought twice as far away?
6. At what distance from Earth is the gravitational
force on an object zero?
7.4 The Universal Constant
of Gravitation, G
7. What is the magnitude of gravitational force
between two 1-kg bodies that are l m apart?
8. What do we call the gravitational force between
Earth and your body?
7.5 Weight and Weightlessness
9. Why do you feel weightless when falling?
10. Are astronauts weightless because they are beyond
Parabola The curved path followed by a projectile
near Earth under the influence of gravity only.
Projectile Any object that moves through the air or
through space under the influence of gravity.
Satellite A projectile or small body that orbits a larger
body.
Universal constant of gravitation, G The proportionality constant in Newton's law of universal gravitation.
REVIEW QUESTIONS
7.1 The Falling Apple and the Falling Moon
Earth's gravitational influence? Defend your
answer.
7.6 Projectile Motion
11 . With no gravity, a horizontally moving projectile
follows a straight-line path. With gravity, how far
below the straight-line path does it fall compared
with the distance of free fall?
12. As an object moves horizontally through air (without air drag), how much speed does it gain moving
horizontally? How much vertically?
13. A ball is batted upward at an angle. What happens
to the vertical component of its velocity as it rises?
As it falls?
1 . What connection did Newton make between a
7.7 Fast-Moving Projectiles-Satellites
falling apple and the Moon?
2. In what sense does the Moon "fall"?
14. How can a projectile "fall around Earth"?
7.2 Newton's Law of Universal Gravitation
3. State Newton's law of universal gravitation in
?.
words. Then do the same with one equation.
7.3 Gravity, Distance, and the
Inverse-Square Law
4. How does the force of gravity between two bodies
change when the distance between them is
doubled?
129
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PART ONE
Physics
,...,_ INTEGRATED SCIENCE
.,_, THINK AND LINK
Earth Science and AstronomyOcean Tides
1. Do tides depend more on the strength of
gravitational pull or on the difference in strengths?
Explain.
2. Why do both the Sun and the Moon exert a greater
gravitational force on one side of Earth than the
other?
3. Which pulls with greater force on Earth's
oceans, the Sun or the Moon? Which is more
effective in raising tides? Why are your answers
different?
Astronomy-Earth Satellites
1. Why will a projectile that moves horizontally at
8 km/s follow a curve that matches the curvature
of Earth?
2. Why is it important that the projectile in Question 1
be above Earth's atmosphere?
3. Arc the planets of the solar system simply
projectiles falling around and around the Sun?
THINK AND DO
1. Hold your hands outstretched with one hand twice
as far from your eyes as the other. Make a casual
judgment about which hand looks bigger. Most
people see them to be about the same size, while
many see the nearer hand as slightly bigger. Very
few people see the nearer hand as four times as big.
But by the inverse-square law, the nearer hand
should appear twice as tall and twice as wide. Twice
times twice means four times as big! That's four
times as much of your visual field as the farther
hand. Your belief that your hands are the same size
is so strong that you likely overrule this information. Try it again, only this time overlap your hands
slightly and view them with one
eye closed. Aha! Do you now
more clearly see that the nearer
hand appears bigger? This raises
an interesting question: what
other illusions do you have that
are not so easily checked?
2. Repeat the eyeballing experiment, only this time
use two dollar bills--one flat, and the other folded
in half lengthwise and again widthwise, so it has
one-fourth the area. Now hold the two in front of
your eyes. Where do you bold the folded bill so
that it looks the same size as the unfolded one?
Share this with your friends!
THINK AND EXPLAIN
1. Comment on whether this label on a consumer
product should be cause for concern. CAUTION:
The mass of this product pulls on every other mass
in the universe, witlz an attracting force that is proportional to the product of the masses and inversely
proportional to the square of the distance between
them.
2. Gravitational force acts on all objects in propor-
tion to their masses. Why, then, doesn' t a heavy
object fall faster than a light object?
3. What would be the path of the Moon if somehow
all gravitational forces on it vanished to zero?
4. Is the force of gravity stronger on a piece of iron
than a piece of wood if both have the same mass?
Defend your answer.
5. What is the magnitude and direction of the gravitational force that acts on a teacher who weighs
1000 N at the surface of Earth?
6. Earth and the Moon are attracted to each other by
gravitational force. Does the more massive Earth
attract the less massive Moon with a force that is
greater, smaller, or the same as the force with
which the Moon attracts Earth?
7. Would ocean tides exist if the gravitational pull of the
Moon (and Sun) were somehow equal on all parts of
the world? Explain.
8 . A heavy crate accidentally falls from a high-flying
airplane just as it flies directly above a shiny red
sports car parked in a car
~
lot. Relative to the car,
~
where will the crate crash?
9. How does the vertical
component of motion
for a ball kicked off a
high cliff compare with
the motion of vertical
free fall?
CHAPTER 7
1 o.
In the absence of air drag, why does the horizontal
component of the ball's motion not change, while
the vertical component does?
11. A park ranger shoots a monkey hanging from a
branch of a tree with a tranquilizing dart. The
ranger aims directly at the monkey, not realizing
that the dart will follow a parabolic path and thus
fall below the monkey. The monkey, however, sees
the dart leave the gun and lets go of the branch to
avoid being hit. Will the monkey be hit anyway?
Defend your answer.
GRAVITY
131
are doubled but the distance between them stays
the same.
3. Show that there is no change in the force of gravity
between two objects when their masses arc doubled
and the distance between them is also doubled.
4. The mass of Earth is 6 X 1024 kg and the mass of
the Sun is 2 X I 030 kg. The average distance
between the two is 1.5 X 10 11 m. Show that the
force of gravity between them is 3.6 X 1022 N.
5. Students in a lab roll a steel ball off the edge of a
table. They measure the speed of the horizontally
launched ball to be 4.0 m/s. They also know that
when the ball is simply dropped from rest off the
edge of the table, it takes 0.5 s to hit the floor. Show
that they should place a small piece of paper 2.0 m
from the bottom of the table so that the rolled ball
will hit it when it lands.
12. Because the Moon is gravitationally attracted to
Earth, why doesn't it simply crash into Earth?
13. What is the effect of air drag on the height and
range of a batted baseball?
THINK AND COMPARE
The planet and its moon gravitationally attract each
other. Rank the gravitational attraction between them
from most to least.
A@. ~
2d
THINK AND SOLVE
1. Suppose you stood atop a ladder that was so tall
that you were three times as far from Earth's center
as you presently are. Show that your weight would
be one-ninth of its present value.
2. Show that the gravitational force between two
planets is doubled if the masses of both planets
MULTIPLE CHOICE QUESTIONS
Choose the best answer to the following questions.
Check your answers with your teacher.
1. The force of gravity between two planets depends
on their
(a) masses and distance apart.
(b) planetary atmospheres.
(c) rotational motions.
(d) All of the above
2. When the distance between two stars is reduced
by!, the force between them
(a) decreases by
(b) decreases by~(c) increases twice times as much.
(d) increases two times as much.
3. If the Sun were twice as massive, its pull on Mars
would be
(a) unchanged.
(b) twice.
(c) half.
(d) four times as much.
1-
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PART ONE
Physics
4. The body most responsible for ocean tides on
Earth is
the Earth itself.
(b) the Sun.
(c) the Moon.
(d) Jupiter and other massive planets.
5. The highest ocean tides occur when Earth and
Moon are
(a) lined up with the Sun.
(b) at right angles to the Sun.
(c) at any angle to the Sun.
(d) lined up during spring.
6. When an astronaut in orbit is weightless, he or
she is
(a) beyond the pull of Earth's gravity.
(b) still in the grips of Earth's gravity.
(c) in the grips of interstellar gravity.
(d) None of the above
7. When no air resistance acts on a fast-moving
baseball, its acceleration is
(a) zero.
(b) due to a combination of constant horizontal
motion and accelerated downward motion.
(c) opposite to the force of gravity.
(d) downward, g.
(a)
8. When you toss a projectile sideways, it curves as it
falls. lt will be an Earth satellite if the curve it
makes
(a) matches the curve of Earth's surface.
(b) results in a straight line.
(c) spirals out indefinitely.
(d) None of the above
9. A satellite in geosynchronous orbit
(a) takes 24 hours to complete each orbit.
(b) travels at a speed of 8 km/s.
(c) travels at a speed slightly greater than 8 km/s.
(d) has zero orbital speed and doesn't move.
10. A satellite in Earth orbit is above Earth's
(a) atmosphere.
(b) gravitational field.
(c) Both (a) and (b)
(d) Neither (a) nor (b)