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# Conceptual Physics Fundamentals Author: Lillian Hewitt Created Date: 12/7/2012 8:26:20 PM

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Conceptual Physics
FundamentalsChapter 6:

GRAVITY, PROJECTILES, AND
SATELLITES

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understand:â¢
The Universal Law of Gravity

â¢
The Universal Gravitational Constant,
G

â¢
Gravity and Distance: The Inverse
-
Square Law

â¢
Weight and Weightlessness

â¢
Universal Gravitation

â¢
Projectile Motion

â¢
Fast
-
Moving Projectiles
â
Satellites

â¢
Circular Satellite Orbits

â¢
Elliptical Orbits

â¢
Energy Conservation and Satellite Motion

â¢
Escape Speed

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Gravity, Projectiles, and
SatellitesâThe greater the velocityâ¦with (a stone) is
projected, the farther it goes before it falls
to the Earth. We may therefore suppose
the velocity to be so increased, that it
would describe an arc of 1, 2, 5, 10, 100,
1000 miles before it arrived at the Earth,
till at last, exceeding the limits of the Earth,
it should pass into space without
touching.â

â
Isaac Newton

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The Universal Law of Gravity

â¢
Newton was not the first to discover
gravity. Newton discovered that gravity is
universal.

â¢
Legend
â
Newton, sitting

under an apple tree, realized

that the force between Earth

and the apple is the same as

that between moons and

planets and everything else.
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The Universal Law of Gravity

Law of universal gravitation

â¢
everything pulls on everything else

â¢
every body attracts every other body with a force
that is directly proportional to the product of their
masses and inversely proportional to the square
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The Universal Law of Gravity

â¢
in equation form: where
m

is mass of object and
d

is the distance
between their centersexamples:

â¢
the greater the masses
m
1

and
m
2

of two bodies,
the greater the force of attraction between them

â¢
the greater the distance of separation
d
, the
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Newtonâs most celebrated synthesis was and is of

A.

earthly and heavenly laws.

B.
weight on Earth and weightlessness in outer space.C.
masses and distances.

D.
the paths of tossed rocks and the paths of satellites.The Universal Law of Gravity

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Newtonâs most celebrated synthesis was and is of

A.

earthly and heavenly laws.

B.
weight on Earth and weightlessness in outer space.C.
masses and distances.

D.
the paths of tossed rocks and the paths of satellites.
Comment
:This synthesis provided hope that other natural phenomena
followed universal laws, and ushered in the âAge of
Enlightenment.âThe Universal Law of Gravity

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The Universal Gravitational
Constant,
G

â¢
Gravity is the weakest of four known
fundamental forces

â¢
With the gravitational constant
G,

we have the
equation:â¢
Universal gravitational constant, G

= 6.67

10
-
11

Nm
2
/kg
2

â¢
Once value was known, mass of Earth was
calculated as 6

10
24

kg
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The universal gravitational constant,
G
mass and distance, is similar to the familiar constant

A.

.

B.
g
.C.
acceleration due to gravity.

D.
speed of uniform motion.The Universal Gravitational Constant,
G

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The universal gravitational constant,
G
mass and distance, is similar to the familiar constant

A.

.

B.
g
.C.
acceleration due to gravity.

D.
speed of uniform motion.
Explanation
:Just as

relates the circumference of a circle to its diameter,
G

relates force to mass and distance.

The Universal Gravitational Constant,
G

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Gravity and Distance: The
Inverse
-
Square Law

Inverse
-
square law

â¢
relates the intensity of an effect to the inverse
-

square of the distance from the cause

â¢
in equation form:
intensity

= 1/
distance
2

â¢
for increases in distance, there are decreases in
force

â¢
even at great distances, force approaches but
never reaches zero

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Inverse
-
Square Law
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Inverse
-
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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 aboveGravity and Distance: The Inverse
-
Square Law

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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
Explanation
:The equation for gravitational force, cites only masses and distances asvariables. Rotation and atmospheres are irrelevant.

Gravity and Distance: The Inverse
-
Square Law

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If the masses of two planets are each somehow doubled,
the force of gravity between them

A.

doubles.

B.
reduces by half.

D.
reduces by one
-
quarter.Gravity and Distance: The Inverse
-
Square Law

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If the masses of two planets are each somehow doubled,
the force of gravity between them

A.

doubles.

B.
reduces by half.

D.
reduces by one
-
quarter.
Explanation
:Note that both masses double. Then double

double =

Gravity and Distance: The Inverse
-
Square Law

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If the mass of one planet is somehow doubled, the force of
gravity between it and a neighboring planet

A.

doubles.

B.
reduces by half.

D.
reduces by one
-
quarter.Gravity and Distance: The Inverse
-
Square Law

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If the mass of one planet is somehow doubled, the force of
gravity between it and a neighboring planet

A.

doubles.

B.
reduces by half.

D.
reduces by one
-
quarter.
Explanation
:Let the equation guide your thinking:Note that if one mass doubles, thenthe force between them doubles.

Gravity and Distance: The Inverse
-
Square Law

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Weight and Weightlessness

Weight

â¢
force an object exerts against a supporting
surface

examples:

â¢
standing on a scale in an elevator accelerating
downward, less compression in scale springs; weight
is less

â¢
standing on a scale in an elevator accelerating
upward, more compression in scale springs; weight is
greaterâ¢
at constant speed in an elevator, no change in weight
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Weight and Weightlessness

Weightlessness

â¢
no support force, as in free
-
fall

example:

astronauts in orbit are without
support forces and are in a

continual state of

weightlessness

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Weight and Weightlessness

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on a scale is

A.

greater.

B.
less.C.
zero.

D.
the normal weight.Weight and Weightlessness

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on a scale is

A.

greater.

B.
less.C.
zero.

D.
the normal weight.
Explanation
:The support force pressing on you is greater, so you weigh more.

Weight and Weightlessness

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When an elevator accelerates downward, your weight

A.

greater.

B.
less.C.
zero.

D.
the normal weight.
Weight and Weightlessness

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When an elevator accelerates downward, your weight

A.

greater.

B.
less.C.
zero.

D.
the normal weight.
Explanation
:The support force pressing on you is less, so you weigh less.
Question: Would you weigh less in an elevator that moves
downward at constant velocity?

Weight and Weightlessness

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When the elevator cable breaks, the elevator falls freely, so

A.

greater.

B.
less.

C.
zero.

D.
the normal weight
Weight and Weightlessness

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When the elevator cable breaks, the elevator falls freely, so

A.

greater.

B.
less.

C.
zero.

D.
the normal weight.
Explanation
:There is still a downward gravitational force acting on you, but
gravity is not felt as weight because there is no support force, so

Weight and Weightlessness

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If you weigh yourself in an elevator, youâll weigh more when
the elevator

A.

moves upward.

B.
moves downward.C.
accelerates upward.

D.
all of the aboveWeight and Weightlessness

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If you weigh yourself in an elevator, youâll weigh more when
the elevator

A.

moves upward.

B.
moves downward.C.
accelerates upward.

D.
all of the above
Explanation
:The support provided by the floor of an elevator is the same
whether the elevator is at rest or moving at constant velocity. Only
accelerated motion affects weight.

Weight and Weightlessness

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Universal Gravitation

Universal gravitation

â¢
everything attracts everything elseexample:
Earth is round because of gravitation
â
all
parts of Earth have been pulled in, making
the surface equidistant from the center

â¢
The universe is expanding and accelerating
outward.

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Projectile Motion

â¢
Without gravity, a tossed object follows a
straight
-
line path.

â¢
With gravity, the same object tossed at an angle
follows a curved path.Projectile

â¢
any object that moves through the air or space
under the influence of gravity, continuing in
motion by its own inertia

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Projectile Motion

Projectile motion is a combination of

â¢
a horizontal component
â¢
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Projectile Motion

Projectiles launched horizontally

Important points:

â¢
horizontal component of velocity doesnât change
(when air drag is negligible)

â
ball travels the same horizontal

distance in equal times (no

component of gravitational

force acting horizontally)

â
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Projectile Motion

â¢
vertical positions become farther apart with time

â
gravity acts downward, so ball accelerates downward

â¢
curvature of path is the combination of horizontal
and vertical components of motion
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Projectile Motion

Parabola

â¢
curved path of a projectile that undergoes
acceleration only in the vertical direction, while
moving horizontally at a constant speed
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Projectile Motion

Projectiles launched at an angle

â¢
paths of stone thrown at an angle upward and
downward

â
vertical and horizontal components are independent
of each other

Smashing Pumpkins

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Projectile Motion

â¢
paths of a cannonball shot at an upward angle

â
vertical distance that a stone falls is the same vertical
distance it would have fallen if it had been dropped
from rest and been falling for the same amount of
time (5
t
2
)
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Projectile Motion

â¢
paths of projectile following
a parabolic trajectory

â
horizontal component along
trajectory remains unchanged

â
only vertical component
changes

â
velocity at any point is
computed with the
Pythagorean theorem
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Projectile Motion

â¢
different horizontal distances

â
same range is obtained from two different launching
angles when the angles add up to 90

â¢
object thrown at an angle of 60

has the same
range as if it were thrown at an angle of 30 Projectile trajectories

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Projectile Motion

â¢
different horizontal distances (continued)

â
maximum range occurs for ideal launch at 45

â
with air resistance, the maximum range occurs for

a baseball

batted at less than 45

above thehorizontal

â
with air resistance the maximum range occurswhen a golf ball that is hit at an angle less than 38â¢
Without air resistance, the timefor a projectile to reach maximumheight is the same as the time forit to return to its initial level.

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The velocity of a typical projectile can be represented by
horizontal and vertical components. Assuming negligible air
resistance, the horizontal component along the path of the
projectile

A.

increases.

B.
decreases.C.
remains the same.

D.
not enough information
Projectile Motion

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The velocity of a typical projectile can be represented by
horizontal and vertical components. Assuming negligible air
resistance, the horizontal component along the path of the
projectile

A.

increases.

B.
decreases.C.
remains the same.

D.
not enough information
Explanation
:Since there is no force horizontally, no horizontal acceleration
occurs.

Projectile Motion

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When no air resistance acts on a fast
-
moving baseball, its
acceleration is

A.

downward,
g
.

B.
because of a combination of constant horizontal motion and
accelerated downward motion.C.
opposite to the force of gravity.

D.
centripetal.
Projectile Motion

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When no air resistance acts on a fast
-
moving baseball, its
acceleration is

A.

downward,
g
.

B.
because of a combination of constant horizontal motion and
accelerated downward motion.C.
opposite to the force of gravity.

D.
centripetal.Projectile Motion

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A ball tossed at an angle of 30

with the horizontal will go
as far downrange as one that is tossed at the same speed
at an angle of

A.

45
.

B.
60
.C.
75
.

D.
none of the aboveProjectile Motion

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A ball tossed at an angle of 30

with the horizontal will go
as far downrange as one that is tossed at the same speed
at an angle of

A.

45
.

B.
60
.C.
75
.

D.
none of the above
Explanation
:Same initial
-
speed projectiles have the same range when their launching angles
. Why this is true involves a bit of trigonometry
â
which, in the interest
of time, weâll not pursue here.

Projectile Motion

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Fast
-
Moving Projectiles
â
Satellites

â¢
satellite motion is an example of a high
-
speed
projectile

â¢
a satellite is simply a projectile that falls around
Earth rather than into it

â
sufficient tangential velocity needed for orbit

â
with no resistance to reduce speed, a satellite goes
around Earth indefinitely.

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As the ball leaves the girlâs hand, one second later it will
have fallen

A.

10 meters.

B.
5 meters below the dashed line.

C.
less than 5 meters below the straight
-
line path.

D.
none of the above

Fast
-
Moving Projectiles
â
Satellites

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As the ball leaves the girlâs hand, one second later it will
have fallen

A.

10 meters.

B.
5 meters below the dashed line.

C.
less than 5 meters below the straight
-
line path.

D.
none of the above
Comment
:Whatever the speed, the ball will fall a vertical distance of 5
meters below the dashed line.

Fast
-
Moving Projectiles
â
Satellites

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Circular Satellite Orbits

Satellite in circular orbit

â¢
speed

â
must be great enough to ensure
that its falling distance matches
Earthâs curvature

â
is constant
â
only direction
changes

â
unchanged by gravity

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Circular Satellite Orbits

â¢
positioningbeyond Earthâs atmosphere, where air
resistance is almost totally absent
example:
space shuttles are

launched to altitudes of 150

kilometers or more, to be above

air drag

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Circular Satellite Orbits

â¢
motionmoves in a direction perpendicular to the
force of gravity acting on it

â¢
period for complete orbitâ

â¢
for satellites close to Earth
â

â¢
for satellites at higher altitudes
â
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Circular Satellite Orbits

Curvature of the Earth

â¢
Earth surface drops a vertical distance of 5
meters for every 8000 meters tangent to the
surface
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Camera on balloon at 100,000 feet

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Circular Satellite Orbits What speed will allow the ball to clear the gap?

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When you toss a projectile sideways, it curves as it falls. It
will be an Earth satellite if the curve it makes

A.

matches the curved surface of Earth.

B.
results in a straight line.C.
spirals out indefinitely.

D.
none of the aboveCircular Satellite Orbits

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When you toss a projectile sideways, it curves as it falls. It
will be an Earth satellite if the curve it makes

A.

matches the curved surface of Earth.

B.
results in a straight line.C.
spirals out indefinitely.

D.
none of the above
Explanation
:For an 8
-
km tangent, Earth curves downward 5 m. Therefore, a
projectile traveling horizontally at 8 km/s will fall 5 m in that time,
and follow the curve of Earth.

Circular Satellite Orbits

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When a satellite travels at a constant speed, the shape of
its path is

A.

a circle.

B.
an ellipse.C.
an oval that is almost elliptical.

D.
a circle with a square corner, as seen throughout your book.
Circular Satellite Orbits

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When a satellite travels at a constant speed, the shape of
its path is

A.

a circle.

B.
an ellipse.C.
an oval that is almost elliptical.

D.
a circle with a square corner, as seen throughout your book.
Circular Satellite Orbits

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Circular Satellite Orbits

A payload into orbit requires control over

â¢
direction of rocket

â
initially, rocket is fired vertically, then tipped

â
once above the atmosphere, the rocket is aimed
horizontally

â¢
speed of rocket

â
payload is given a final thrust to orbital speed of 8
km/s to fall around Earth and become an Earth
satellite

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Elliptical Orbits

â¢
A projectile just above the atmosphere will
follow an elliptical path if given a horizontal speed
greater than 8 km/s.

Ellipse

â¢
specific curve, an oval path

example:
circle is a special case of an ellipse when its two

foci coincide
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Elliptical Orbits

Elliptical orbit

â¢
speed of satellite varies

â
initially, if speed is greater than needed for circular orbit,
satellite overshoots a circular path and moves away from
Earth

â
satellite loses speed and then regains it as it falls back
toward Earth

â
it rejoins its original path with the same speed it had
initially

â
procedure is repeated

Physics Classroom

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The speed of a satellite in an elliptical orbit

A.

varies.

B.
remains constant.C.
acts at right angles to its motion.

D.
all of the aboveElliptical Orbits

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The speed of a satellite in an elliptical orbit

A.

varies.

B.
remains constant.C.
acts at right angles to its motion.

D.
all of the above
Elliptical Orbits

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Energy Conservation and
Satellite Motion

Recall the following

â¢
object in motion possesses
KE

due to its motion

â¢
object above Earthâs surface possesses
PE

by
virtue of its position

â¢
satellite in orbit possesses
KE

and
PE

â
sum of
KE

and
PE

is constant at all points in the orbitCopyright © 2008 Pearson Education, Inc., publishing as Pearson Addison
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Energy Conservation and
Satellite Motion

PE
,
KE
, and speed in

â¢
circular orbit

â
unchanged

â
distance between the satellite
and center of the attracting
body does not change
â
PE

is
the same everywhere

â
no component of force acts
along the direction of motion
â
no change in speed and
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Energy Conservation and
Satellite Motion

â¢
elliptical orbit

â
varies

â¢
PE

is greatest when the satellite is farthest away
(apogee)

â¢
PE

is least when the satellite is closest (perigee)

â¢
KE

is least when
PE

is the most and vice versa

â¢
at every point in the orbit, sum of
KE

and
PE

is the
same

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Energy Conservation and
Satellite MotionWhen satellite gains altitude

and moves against gravitational

force, its speed and
KE

decrease

and continues to the apogee.

Past the apogee, satellite moves

in the same direction as the force

component and speed and
KEincreases. Increase continues

until past the perigee and cycle

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Escape Speed

Escape speed

â¢
the initial speed that an object must reach to escape
gravitational influence of Earth

â¢
11.2 kilometers per second from Earthâs surface
Escape velocity

â¢
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Escape Speed

First probe to escape the solar system is
Pioneer 10
, launched from Earth in 1972.

â¢
accomplished by directing the probe into the path of
oncoming Jupiter
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When a projectile achieves escape speed from Earth, it

A.

forever leaves Earthâs gravitational field.

B.
outruns the influence of Earthâs gravity, but is never beyond it.C.
comes to an eventual stop, returning to Earth at some future time.

D.
all of the above

Escape Speed

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When a projectile achieves escape speed from Earth, it

A.

forever leaves Earthâs gravitational field.

B.
outruns the influence of Earthâs gravity, but is never beyond
it.C.
comes to an eventual stop, returning to Earth at some future time.

D.
all of the aboveEscape Speed