Transcript Gravitational Field

Chapter 8
Universal Gravitation
Section 8.1 Objectives
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List Kepler’s Laws and understand
them
Calculate periods and velocities of
orbiting objects
Understand that gravitational force is
proportional to the masses and the
inverse square of the distances
between objects
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1. Kepler’s Laws
• a. Prior to Kepler - Aristotle’s concept of
an Earth centered system dominated
thought until Copernicus develops
heliocentric model
• b. Based on data gathered by Tycho
Brahe 
Danish astronomy sometimes referred to as
the great observer.
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Kepler’s Three Laws:
(1) Paths of planets are ellipses (nearly
circular) with the sun at one focus
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Exaggerated the ellipse
Sun
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(2) Line from the sun to a planet
sweeps out equal areas in equal
amounts of time
• (a) At which point would have planet
be moving faster?
Area 1
Area 2
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(3) Ratio of the average radius (r)
cubed to the period (T) squared is
constant for all planets
•r 3 / T 2 = k
• r = average radius from planet to the
sun
• T = period of revolution around the
sun
• Why Average Radius?
Example Problem
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The moon Io is 4.2 units and
Ganymede is 10.7 units from the
center of Jupiter. Io has a period of
1.8 days what is the period of
Ganymede?
Use (Ta/Tb)2 = (ra/rb)3
Ta2 = Tb2(ra/rb)3
Ta2 = (1.8days)2(10.7/4.2)3
Ta = (52.8 days2)1/2 = 7.3 days
Newton and Gravity
The Falling Apple
According to legend, Newton
discovered gravity while
sitting under an apple tree.
The Falling Apple
Newton saw the apple fall, or maybe even felt it fall
on his head. Perhaps he looked up through the apple
tree branches and noticed the moon.
• He may have been puzzled by the fact that the
moon does not follow a straight-line path, but
instead circles about Earth.
• He knew that circular motion is accelerated
motion, which requires a force.
• Newton had the insight to see that the moon is
falling toward Earth, just as the apple is.
13.2 The Falling Moon
The moon is actually falling toward
Earth but has great enough tangential
velocity to avoid hitting Earth.
The Falling Moon
Newton realized that if the moon did not fall, it
would move off in a straight line and leave its
orbit.
His idea was that the moon must be falling
around Earth.
Thus the moon falls in the sense that it falls
beneath the straight line it would follow if no
force acted on it.
He hypothesized that the moon was simply a
projectile circling Earth under the attraction of
gravity.
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The Falling Moon
If the moon did not fall, it would follow a straightline path.
13.2 The Falling Moon
Newton’s Hypothesis
Newton compared motion of the moon to a
cannonball fired from the top of a high mountain.
• If a cannonball were fired with a small
horizontal speed, it would follow a parabolic
path and soon hit Earth below.
• Fired faster, its path would be less curved and it
would hit Earth farther away.
• If the cannonball were fired fast enough, its
path would become a circle and the cannonball
would circle indefinitely.
The Falling Moon
This original drawing by
Isaac Newton shows how
a projectile fired fast
enough would fall around
Earth and become an
Earth satellite.
The Falling Moon
Both the orbiting cannonball and the moon have a
component of velocity parallel to Earth’s surface.
This sideways or tangential velocity is sufficient to
ensure nearly circular motion around Earth rather than
into it.
With no resistance to reduce its speed, the moon will
continue “falling” around and around Earth indefinitely.
13.2 The Falling Moon
Tangential velocity is the “sideways” velocity—the
component of velocity perpendicular to the pull of
gravity.
Newton’s Law of Universal Gravitation
Newton discovered that gravity is universal.
Everything pulls on everything else in a way
that involves only mass and distance.
The force of gravity between objects depends on the
distance between their centers of mass.
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Developed by using Kepler’s Third Law
and equating force to centripetal force.
•r
3
/T
2
=k
1
implies
T 2 = r 3 k 2 so
• if F c = m v 2 / r
and v = 2  r / T
then
F c =[ m (2  r / T) 2 ] /r
= m 4  2 r / T2
F c = 4  2 r m / r 3 k2 = k 3 m / r 2
or……… Fg = GMm/d2
where G is the
gravitational constant and M is the mass of the
planet
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Newton developed the concept but was
not able to determine the value for G
Value for G was found experimentally
by Cavendish in 1798
(pg 162 in text)
• (1) Led to the determination of the mass
of the Earth
• (2) M E = 5.98 x 10 24 kg
Newton’s Law of Universal Gravitation
Philipp von Jolly developed a simpler method of
measuring the attraction between two masses.
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Newton’s Law of Universal Gravitation
• a. Every body attracts every other body
with a force that varies based on the
distance separating the bodies and their
masses.
F = G (M 1 M 2) / r 2
• b. G is the universal gravitational
constant - similar to a constant such as
the speed of light, Avogadro’s Number,
etc….
G = 6.67 x 10
-11
N-m 2/kg 2 or m3/kg-sec 2
Newton’s Law of Universal Gravitation
The value of G tells us that gravity is a very weak
force.
It is the weakest of the presently known four
fundamental forces.
We sense gravitation only when masses like that of
Earth are involved.
Gravity and Distance: The Inverse-Square Law
Gravitational force is plotted versus distance
from Earth’s center.
Gravitational Forces F = G(m1m2)/d2
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M1
F
M1
M2
M2
d
F
d
2 M1
2F
M2
M1
2d
M1
2 M1
2F
4F
2 M2
2 M2
M2
¼F
M2
M1
1/2 d
4F
Practice Exercise
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Consider two satellites in orbit around a
star (like our sun). If one satellite is
twice as far from the star as the other,
but both satellites are attracted to the
star with the same gravitational force,
how do the masses of the satellites
compare?
Sun
Answer
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If both satellites had the same mass,
then the one twice as far would be
attracted to the star with only onefourth the force (inverse square law).
Since the force is the same for both,
the mass of the farthermost satellite
must be four times as great as the
mass of the closer satellite.
If the sun suddenly collapsed to become a
black hole, then the Earth would??
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a. Leave the solar system in a straight-line
path.
b. Spiral into the black hole
c. Undergo a major increase in tidal forces
d. Continue to circle in its usual orbit.
??
Sun
poof
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From Newton’s Universal Law of Gravitation:
The interaction F between the mass of the
Earth and the Sun doesn’t change. This is
because the mass of the Earth does not
change, the mass of the sun does not change
even though it is compressed, and the distance
from the centers of the Earth and the sun,
collapsed or not, does not change.
Although the Earth would very soon freeze and
undergo enormous surface changes, its yearly
path would continue as if the sun were its
normal size.
Extra Credit Group Problem
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A 50 kg astronaut is floating at rest in deep
space 35 m from her stationary 150,000 kg
spaceship. How long will it take her to float
to the spaceship due to her attraction
(gravity) with the ship? If she has a three
hour supply of oxygen, will she make it to the
ship in time?
Help
35 meters
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m1 = 50 kg d = 35 m
M2 = 150,000 kg
v0= 0
G = 6.67 x 10
-11
N-m 2/kg 2
F = G(M 1 M 2)/d2= 4.08 x 10 -7 N
F = ma
so a = F/m
a= 8.16 x 10-9 m/s2
d = v0t + ½at2 or t= (2d/a)½
t
= (2(35m)/(8.16x10-9m/s2))½
t = 92,600 sec or ~26 hours
She’s toast ….. (runs out of oxygen)
k. Example Problem
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Compare the gravitational pull on a
spaceship at the surface of the
Earth with the gravitational pull
when the ship is orbiting 1000 km
above the surface. (r E = 6370 km)
F = G (M 1 M 2) / r 2
25 % decrease. Note that the ship
is still under effects of gravity and
is NOT weightless.
Section 8.2 Using the Law of
Universal Gravitation
Objectives:
 Solve problems using orbital velocity
and period
 Understand the term weightlessness
of objects in free fall and orbit
 Describe gravitational fields
 Contrast Einstein's concept of gravity
to that of Newton’s
Orbital Motion
v - tangential velocity
F - centripetal force
v
E
Fc
r
r - distance to center of
mass of the Earth
Assume a circular orbit with gravity
providing the centripetal force.
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Then
2
GmM e
mv
Fc  FG 

2
r
r
which gives
v 
GM e
r
Mass of the satellite is unimportant
in describing its motion, only mass of planet.
We know that
And since…
2r
T
v
GMe
v
r
So the period of orbit:
3
r
r
T  2r
; T  2
GM e
GM e
Can be used for any body in orbit around
another body.
Example Problems
(1) A synchronous satellite will orbit at 3.6 x 107
m above the surface of the Earth. What is its
speed?
Me = 5.98x1024 kg and re = 6370 km.
24 kg
 Givens: Me = 5.98x10
ro = 36 x 106 + 6.37 x 10 6 m
G = 6.67 x 10 -11 N-m 2/kg 2
Use the equation:
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v = 3068 m/s
GMe
v
r
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(2) A moon of Jupiter, called Calisto,
circles Jupiter each 16.8 days. Its
orbital radius is 1.88 x 10 6 km. Find
the mass of Jupiter.
*Convert radius from km to m and days to seconds
2r
v
T
so that
v2r
MJ 
G
=
GM J
v
r
d
2r
v 

t
t
MJ = 1.88 x 10 27 kg
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Law of Universal Gravitation and
Weight
• a. Weight is due to gravity so
w = G(mME)/rE2
and since w = mg can determine g
(acceleration due to gravity)
since w=gm
g = GM E/r
E
2
• b. Weight changes with distance from
the center of the Earth
c. Weight and Weightlessness
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(1).
GMe
a 2
d
g changes with height and
distance from center of the
Earth (d)
Therefore taking the ratio of a/g gives:
GM e
2
2
a
 re 
d

 a  g 
GM e
g
d 
re2
which allows you to
calculate values for
acceleration due to gravity
for whatever distance you
are above the Earth
“Weightlessness” in orbit is not
zero gravity, it is freefall.
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(a) Objects are still attracted by planet
(b) Objects are accelerating toward the
planet at the same rate the planet is
falling away from them due to
curvature of the planet’s surface
You feel weightless because the space ship
and you are in free fall around the planet
(you orbit because your tangential velocity
prevents you from hitting Earth)
Weight and Weightlessness
The sensation of weight is equal to the force
that you exert against the supporting floor.
Weight and Weightlessness
When the elevator accelerates downward, the
support force of the floor is less.
The scale would show a decrease in your
weight.
If the elevator fell freely, the scale reading
would register zero. According to the scale,
you would be weightless.
You would feel weightless, for your insides
would no longer be supported by your legs and
pelvic region.
This is exactly what happens in orbit!
Gravitational Fields & Einstein
Objectives:
 Describe gravitational fields
 Explain why the field concept does
not tell us why gravity exists
 Contrast gravity fields to Einstein's
concept of gravity
13.6 Gravitational Field
Earth can be thought of as being
surrounded by a gravitational field
that interacts with objects and causes
them to experience gravitational
forces.
Gravitational Field
We can regard the moon as in contact with the
gravitational field of Earth.
A gravitational field occupies the space surrounding
a massive body.
A gravitational field is an example of a force field, for
any mass in the field space experiences a force.
Gravitational Field
A more familiar force field is the
magnetic field of a magnet.
• Iron filings sprinkled over a sheet
of paper on top of a magnet
reveal the shape of the magnet’s
magnetic field.
• Where the filings are close
together, the field is strong.
• The direction of the filings shows
the direction of the field at each
point.
• Planet Earth is a giant magnet,
and like all magnets, is
surrounded in a magnetic field.
Gravitational Field
Field lines can also represent the pattern of Earth’s
gravitational field.
• The field lines are closer together where the
gravitational field is stronger.
• Any mass in the vicinity of Earth will be
accelerated in the direction of the field lines at
that location.
• Earth’s gravitational field follows the inversesquare law.
• Earth’s gravitational field is strongest near Earth’s
surface and weaker at greater distances from
Earth.
Gravitational Field
Field lines represent the gravitational field about
Earth.
Gravitational Field
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a. First type of field force we have
encountered.
b. Use the concept of fields to
explain how forces act through a
distance, NOT WHY.
c. Fields describe how forces act on
an object due to its location.
Described using vectors and
concentrations
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The closer the lines are together the more
powerful the field.
Direction of arrows shows the direction of
attraction.
Field Strength: g = F/m measured in
Newtons/kg
Example Field Problem
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A 2 kg mass has a weight of 10N at a
specific location in space watch is the
strength of the gravitational field at
that location?
g = F/m = 10N/2kg = 5 N/kg
Einstein
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a. Stated gravity is not a force, but
an effect of space.
b. Mass causes space to be curved,
and this curving accounts for
acceleration.
c. Even light bends with gravity.
d. Concept behind ‘warp drive” on
Star Trek.
Einstein Gravity Demos
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http://www.youtube.com/watch?v=M
TY1Kje0yLg
http://www.youtube.com/watch?v=Y
ByqTYzeJww
http://www.youtube.com/watch?v=G
ShDvKmpkKw
http://www.youtube.com/watch?v=0
rocNtnD-yI
The concept behind ‘warp drive” on
Star Trek.
Make it so……..