Download Use Example problem 8-2 to solve practice

PHYSICS CHAPTER 8 : Universal Gravitation
The Soviet Sputnik satellite was the first to orbit Earth, launched on October 4,
1957. Because of Soviet government secrecy at the time, no photographs were
taken of this famous launch. Sputnik was a 23-inch (58-cm), 184-pound (83-kg)
metal ball. Although it was a remarkable achievement, Sputnik's carried only a
thermometer, battery, and a radio transmitter, which seem meager by today's
standards. It changed the tone of its beeps to match temperature changes it was
experiencing. The interior of the satellite was pressurized with Nitrogen gas.
On the outside of Sputnik, four whip antennas transmitted on short-wave
frequencies above and below what is today's Citizens Band (27 MHz). According
to the Space Satellite Handbook, by Anthony R. Curtis:
After 92 days, gravity took over and Sputnik burned in Earth's atmosphere. Thirty
days after the Sputnik launch, the dog Laika orbited in a half-ton Sputnik satellite
with an air supply for the dog. It burned in the atmosphere in April 1958.
Sputnik is a good example of just how simple a satellite can be. As we will see
later, today's satellites are generally far more complicated, but the basic idea is a
straightforward one.
When astronauts, a satellite, and the space shuttle are seen in orbit about Earth,
they appear to be floating weightless in space. Does Earth's gravitational force
reach into space? Is this force the same strength as it is on Earth?
Why do objects fall toward Earth? To ancient Greek scientists, certain things, like
hot air or smoke, rose; others, like rocks and shoes, fell simply because they had
some built-in desire to rise or fall. The Greeks gave the names "levity" and
"gravity" to these properties. If you ask a child today why things fall, he or she will
probably say "because of gravity." But, giving something a name does not explain
it. Galileo and Newton gave the name gravity to the force that exists between
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Earth and objects. Newton showed that the same force exists between all bodies.
In this century, Einstein gave a much different and deeper description of the
gravitational attraction. But, we still know only how things fall, not why.
As Galileo wrote almost four hundred years ago in response to a statement that
"gravity" is why stones fall downward,
What I am asking you for is not the name of the thing, but its essence, of which
essence you know not a bit more than you know about the essence of whatever
moves the stars around. We do not really understand what principle or what force
it is that moves stones down ward.
Today the debate is still on and the search for what actually causes gravity and
how to explain gravity. One of the most prominent explanations is that energy in
linear motion causes the effect of gravity. (Honors report on latest on gravity)
8.1 MOTION IN THE HEAVENS AND ON EARTH
We know how objects move on Earth. We can describe and even calculate
projectile motion. Early humans could not do that, but they did notice that the
motion of stars and other bodies in the heavens was quite different. Stars moved
in regular paths. Planets, or wanderers, moved through the sky in much more
complicated paths. Astrologers claimed that the motions of these bodies were able
to control events in human lives. Comets (Aristotle named them Kome or stars
with hair) were even more erratic. These mysterious bodies appeared without
warning, spouting bright tails, and were considered bearers of evil omens.
Comets are actually small bodies of frozen rock ice or dust that have tails coming
off from being broken apart in orbit by the sun.
Halley's Comet (officially designated 1P/Halley) is the most famous of the
periodic comets and can currently be seen every 75–76 years. Many comets with
long orbital periods may appear brighter and more spectacular, but Halley is the
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only short-period comet that is clearly visible to the naked eye, and thus, the only
naked-eye comet certain to return within a human lifetime. During its returns to the
inner solar system, it has been observed by astronomers since at least 240 BC,
but it was not recognized as a periodic comet until the eighteenth century when its
orbit was computed by Edmond Halley, after whom the comet is now named.
Halley's Comet last appeared in the inner Solar System in 1986, and will next
appear in mid-2061. I will be 103 how old will you be?
Asteroids are made of rock and debris coming out of space. There is about a 1 in
10 million chance of them hitting the earth. If the asteroid is 1 km in size it is called
a planet killer. It would be comparable of being struck by a 4-megaton nuclear
warhead. We had 10 near misses in 2002 (about 1 - 2 times the distance from the
moon - 230 to 460 thousand miles.) In 2029 one should come as close as 3 earth
diameters away – 24 000 miles.
Small objects frequently collide with the Earth. There is an inverse relationship
between the size of the object and the frequency that such objects hit the earth.
Asteroids with a 1 km (0.62 mi) diameter strike the Earth every 500,000 years on
average.[2] Large collisions – with 5 km (3 mi) objects – happen approximately
once every ten million years. The last known impact of an object of 10 km (6 mi) or
more in diameter was at the Cretaceous-Tertiary extinction event 65 million years
ago.
Asteroids with diameters of 5 to 10 m (16 to 33 ft) enter the Earth's atmosphere
approximately once per year, with as much energy as Little Boy, the atomic bomb
dropped on Hiroshima, approximately 15 kilotonnes of TNT. These ordinarily
explode in the upper atmosphere, and most or all of the solids are vaporized.[3]
Objects with diameters over 50 m (164 ft) strike the Earth approximately once
every thousand years, producing explosions comparable to the one known to have
detonated above Tunguska in 1908.[4] At least one known asteroid with a diameter
of over 1 km (0.62 mi), (29075) 1950 DA, has a possibility of colliding with Earth
on March 16, 2880, but the torino scale only works for impact possibilities within
100 years, and thus cannot apply to this asteroid.
Objects with diameters smaller than 10 m (33 ft) are called meteoroids (or
meteorites if they strike the ground). An estimated 500 meteorites reach the
surface each year, but only 5 or 6 of these are typically recovered and made
known to scientists.
Because of the works of Galileo, Kepler, Newton, and others, we now know that
all of these objects follow the same laws that govern the motion of golf balls and
other objects on Earth.
Kepler's Laws of Planetary Motion
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As a boy of fourteen in Denmark, Tycho Brahe (1546-1601) observed an eclipse
of the sun on August 21, 1560, and vowed to become an astronomer. In 1563, he
observed two planets in conjunction, that is, located at the same point in the sky.
The date of the event, predicted by all the books of that period, was off by two
days, so Brahe decided to dedicate his life to making accurate predictions of
astronomical events.
Brahe studied astronomy as he traveled throughout Europe for five years. In
1576, he persuaded King Frederick II of Denmark to give him the island of Hven
as the site for the finest observatory of its time. He constructed huge instruments
like the astrolabe, the sextant, and the quadrant, and spent the next 20 years
using them to carefully recording the exact positions of the planets and stars.
Brahe was not known for his sunny disposition, so in 1597, out of favor with the
new Danish king, Brahe moved to Prague. He became the astronomer to the court
of Emperor Rudolph of Bohemia where, in 1600, a nineteen-year-old German
Johannes Kepler (1571-1630) became one of his assistants. Although Brahe still
believed strongly that Earth was the center of the universe, Kepler wanted to use
a sun-centered system to explain Brahe's precise data. He was convinced that
geometry and mathematics could be used to explain the number, distance, and
motion of the planets. By doing a careful mathematical analysis of Brahe's data,
Kepler discovered three laws that still describe the behavior of every planet and
satellite. The theories he developed to explain his laws, however, are no longer
considered correct. The three laws can be stated as follows.
1. The paths of the planets are ellipses with the center of the sun at one focus.
2. An imaginary line from the sun to a planet sweeps out equal areas in equal
time intervals. Thus, planets move fastest when closest to the sun, slowest
when farthest away, Figure 8-2.
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3. The ratio of the squares of the periods of any two planets revolving about
the sun is equal to the ratio of the cubes of their average distances from the
sun. Thus, if Ta and Tb are their periods and ra and rb their average
distances,
(Ta/Tb)2 = (ra/rb)3 Ta2rb3 = Tb2ra3
________
_________
_________
_________
Ta = √ Tb2ra3 / rb3 Tb = √ Ta2rb3 / ra3 rb = 3√ Tb2ra3/ Ta2
ra = 3√ Ta2rb3/ Tb2
Notice that the first two laws apply to each planet, moon, or satellite individually.
The third law relates the motion of several satellites about a single body. For
example, it can be used to compare the distances and periods of the planets
about the sun. It can also be used to compare distances and periods of the moon
and artificial satellites around Earth.
PROBLEM SOLVING STRATEGY
When working with equations that involve squares and square roots, or cubes
and cube roots, your solution is more precise if you keep at least one extra digit in
your calculations until you reach the end.
When you use Kepler's third law to find the radius of the orbit of a planet or
satellite, first solve for the cube of the radius, then take the cube root. This is
easier to do if your calculator has a cube-root key. On TI-83 push math:4, then put
in number, and press enter. Small calculator to square number then x2 to cube
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number then y2 and number 3 to square root number then square of x to cube root
number then 2nd then the square of y to the x power.
Table 8-1 Planetary Data
Name
Average radius (m)
Mass (kg) Mean distance from sun (m)
Orbit around itself
orbit around the sun –this is
Plus the distance
the radius from the sun
8
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Sun
6.960 e
1.991e
6
Mercury
2.430 e
3.200e23
5.800e10
Venus
6.073 e6
4.880e24
1.081e11
Earth
6.3713 e6
5.979e24
1.4957e11
Mars
3.380 e6
6.420e23
2.278e11
Jupiter
6.980 e7
1.901e27
7.781e11
Saturn
5.820 e7
5.680e26
1.427e12
Uranus
2.350 e7
8.680e25
2.870e12
Neptune 2.270 e7
1.030e26
4.500e12
Pluto
1.150 e6
1.200e22
5.900e12
Pluto was drummed out of planet status in 2006 by the IAU because it does not
meet all 3 requirements to be considered a planet – it does not clear the
neighborhood around its orbit
Use example problem 8-1a and 8-1b to help solve practice problems 8-1
Universal Gravitation
In 1666, some 45 years after Kepler's work, young Isaac Newton was living at
home in rural England because the plague had closed all schools. Newton had
used mathematical arguments to show that if the path of a planet were an ellipse,
in agreement with Kepler's first law, then the net force, F, on the planet must vary
inversely with the square of the distance between the planet and the sun. That is,
he could write an equation,
F α 1/d2
where the symbol α means "is proportional to,"and d is the average distance
between the centers of the two bodies. He also showed that the force acted in the
direction of a line connecting the centers. But, at this time, Newton could go no
further because he could not measure the magnitude of the force, F.
Newton later wrote that the sight of a falling apple made him think of the
problem of the motion of the planets. Newton recognized that the apple fell
straight down because Earth attracted it. Might not this force extend beyond the
trees, to the clouds, to the moon, and even beyond? Gravity could even be the
force that attracts the planets to the sun. Newton recognized that the force on the
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apple must be proportional to its mass. Further, according to his own third law of
motion, the apple would also attract Earth, so the force of attraction must be
proportional to the mass of Earth as well. He was so confident the laws that
governed motion on Earth would work anywhere that he assumed the same force
of attraction acted between any two masses, m1 and m2. He proposed
__________
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F = G (m1m2 ) / d
d = √G (m1m2 ) /F m2 = Fd2/Gm1 G = Fd2/m1m2
where d is the distance between the centers of the spherical masses, and G is a
universal constant, one that is the same everywhere. According to Newton's
equation, if the mass of a planet were doubled, the force of attraction would be
doubled. Similarly, if the planet were attracted toward a star with twice the mass of
the sun, the force would be twice as great. And, if the planet were twice the
distance from the sun, the force would be only one-quarter as strong. Figure 8-5
illustrates these relationships. Because the force depends on 1/d2, it is called an
inverse square law.
Newton's Use of His Law of Universal Gravitation
Newton applied his inverse square law to the motion of the planets about the
sun. He used the symbol Mp for the mass of the planet, Ms for the mass of the sun,
and rps for the radius of the planet's orbit. He then used his second law of motion,
F = ma, with F the gravitational force and a the centripetal acceleration. That is, F
= Mpa and a = 4 π2rps/Tp2. For the sake of simplicity, we assume circular orbits.
F = F so G (MsMp ) = Mp (4π2rps) so G (MsMp)xTp2= rps2 x Mp (4π2rps)
rps2
Tp2
___________
2
2
2
Tp = rps x Mp (4π rps) / G (MsMp) = T = √ 4π2 rps3/ GMs
This equation is Kepler's third law ---- the square of the period is proportional to
the cube of the distance. The proportionality constant, 4π2/GMs, depends only on
the mass of the sun and Newton's universal gravitational constant G. It does not
depend on any property of the planet. Thus, Newton's law of gravitation not only
leads to Kepler's third law, but it also predicts the value of the constant.
G = 4π2 rps3 / MsTp2
In our derivation of this equation, we have assumed the orbits of the planets are
circles. Newton found the same result for elliptical orbits.
Weighing Earth
As you know, the force of gravitational attraction between two objects on Earth is
very small. You cannot feel the slightest attraction even between two massive
bowling balls. In fact, it took 100 years after Newton's work to develop an
apparatus that was sensitive enough to measure the force. In 1798, the
Englishman Henry Cavendish (1731-1810) used equipment like that sketched in
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Figure 8-7. A rod about 20 cm long had two small lead balls attached. A thin wire
suspended the rod so it could rotate. Cavendish measured the force on the
spheres needed to rotate the rod through given angles. Then, he placed two large
lead balls close to the small ones. The force of attraction between the balls
caused the rod to rotate. By measuring the angle through which it turned,
Cavendish was able to calculate the attractive force between the masses. He
found that the force agreed with Newton's law of gravitation.
Cavendish measured the masses of the balls and the distance between their
centers. Substituting these values for force, mass, and distance into Newton's law,
he found the value of G.
Newton's law of universal gravitation says d=r
F = G (m1m2)
d2
When m1 and m2 are measured in kilograms, d in meters, and F in newtons, then
G = 6.67e-11 N-m2/kg2. For example, the attractive gravitational force between two
bowling balls, each of mass 7.26 kg, with their centers separated by 0.30 m is
Fg = (6.67e-11 N-m2/kg2)(7.26 kg)(7.26 kg)
(0.30 m)2
= 3.91 x 10-8 N.
Cavendish's experiment is often called "weighing the earth." You know that on
Earth's surface the weight of an object is a measure of Earth's gravitational
attraction: F = W = mg. According to Newton, however, F = F
F = GMem = mg
r2
so, g = GMe
r2
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Because Cavendish measured the constant G, we can rearrange this equation as
Me = gre2
G
Using modern values of the constants, we find
Me = (9.80 m/s2)(6.37 X 106 m)2
6.67 x 10-11 N-m2/kg2
= 5.98 X 1024 kg.
Comparing the mass of Earth to that of a bowling ball, you can see why the
gravitational attraction of everyday objects is not easily sensed.
Do concept review 8-1
8.2 USING THE LAW OF UNIVERSAL GRAVITATION
The planet Uranus was discovered in 1741. By 1830, it appeared that Newton's
law of gravitation didn't correctly predict its orbit. Some astronomers thought
gravitational attraction from an undiscovered planet might be changing its path. In
1845, the location of such a planet was calculated, and astronomers at the Berlin
Observatory searched for it. During the first evening, they found the giant planet
now called Neptune.
Motion of Planets and Satellites
Newton used a drawing similar to Figure 8-9 to illustrate a "thought
experiment."
Imagine a cannon, perched atop a high mountain, shooting a cannonball
horizontally. The cannonball is a projectile and its motion has vertical and
horizontal components. It follows a parabolic trajectory. During the first second the
ball is in flight, it falls 4.9 m. If its speed increases, it will travel farther across the
surface of Earth, but it will still fall 4.9 m in the first second of flight. Meanwhile, the
surface of Earth is curved. If the ball goes just fast enough, after one second, it will
reach a point where Earth has curved 4.9 m away from the horizontal. That is, the
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curvature of Earth will just match the curvature of the trajectory, and the ball will
orbit Earth.
Figure 8-9 shows that Earth curves away from a line tangent to its surface at a
rate of 4.9 m for every 8 km. That is, the altitude of the line tangent to Earth at A
will be 4.9 m above Earth at B. If the cannonball in Figure 8-9 were given just
enough horizontal velocity to travel from A to B in one second, it would also faIl
4.9 m and arrive at C. The altitude of the ball in relation to Earth's surface would
not have changed. The cannonball would fall toward Earth at the same rate that
Earth's surface curves away. An object with a horizontal speed of 8 km/s will keep
the same altitude and circle Earth as an artificial satellite. (17 895mph)
Newton's thought experiment ignored air resistance. The mountain would have
to be more than 150 km above Earth's surface (93miles) to be above most of the
atmosphere. A satellite at this altitude encounters little air resistance and can orbit
Earth for a long time.
A satellite in an orbit that is always the same height above Earth moves with
uniform circular motion. Its centripetal acceleration is ac = v2/r. Using Newton's
second law, F = ma, with the gravitational force between Earth and the satellite,
F=ma and F=GMEm/r2 and ac = v2/r so F=mv2/r this leads to
GMEm = mv2
r2
r
Solving this for the velocity, v, we find
_____
v = √GME/r
By using Newton's law of universal gravitation, we have shown that the time for
a satellite to circle Earth, its period, is given by
______
T = 2π √r3/GME
Note that the orbital velocity and period are independent of the mass of the
satellite. Satellites are accelerated to the speeds needed to achieve orbit by large
rockets, such as the shuttle booster rocket. The acceleration of any mass must
follow Newton's law, F = ma, so a more massive satellite requires more force to
put it into orbit. Thus, the mass of a satellite is limited by the capability of the
rocket used to launch it.
Note that these equations for the velocity and period of a satellite can be used
for any body in orbit about another. The mass of the central body, like the sun,
would replace ME in the equations, and r is the distance from the sun to the
orbiting body.
Example problem 8-2
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A satellite is orbiting the Earth 225 km from the Earth’s surface. What is the orbital
velocity of the satellite?
Given: 225 km above earth
Find: Orbital velocity of satellite
Me = 5.98e24 kg
_____
2
2
G = 6.67e –11 Nxm /kg
v = √GME/r
Earth’s radius = 6.37e 6 m
+ 2.25e5m = 6.60e6m
_______________________________________________
v = √6.67e –11 Nxm2/kg2 x 5.98e24 kg / (6.60e 6 m )
=7.78e3m/s 7.78 km/s x 3600s x .62 = 17364 – rounding 17380 mph
http://science.howstuffworks.com/satellite3.htm satellite in orbit
Use Example problem 8-2 to solve practice problems 8-2
Weight and Weightlessness
The acceleration of objects due to Earth's gravitation can be found by using the
inverse square law and Newton's second law. Since
F = GMEm = F = ma, so, a = GME so ad2 = GME
d2
d2
but on Earth’s surface, the equation can be written as
g = GME so gRE2 = GME so gRE2 = ad2
RE2
Thus, a = g(RE/d)2
As we move farther from Earth's center, the acceleration due to gravity is
reduced according to this inverse square relationship.
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You have probably seen astronauts on a space shuttle working and relaxing in
"zero-g", or "weightlessness." The shuttle orbits Earth about 400 km (250miles)
above its surface. At that distance, g = 8.70 m/s2, only slightly less than on Earth's
surface. Thus, Earth's gravitational force is certainly not zero in the shuttle. In fact,
gravity causes the shuttle to circle Earth. Why, then, do the astronauts appear to
have no weight? Just as with Newton's cannonball, the shuttle and everything in it
are falling freely toward Earth as they orbit around it.
How do you measure weight? You either stand on a spring scale or hang an
object from a scale. Weight is found by measuring the force the scale exerts in
opposing the force of gravity. As we saw in Chapter 5, if you stand on a scale in
an elevator that is accelerating downward, your weight is reduced. If the elevator
is in freefall, that is, accelerating downward at 9.80 m/s2, then the scale exerts no
force on you. With no force on your feet, you feel weightless. So it is in an orbiting
satellite. The satellite, the scale, you, and everything else in it are accelerating
toward the Earth.
The Gravitational Field
Many of the common forces are contact forces. Friction is exerted where two
objects touch; the floor and your chair or desk push on you. Gravity is different. It
acts on an apple falling from a tree and on the moon in orbit; it even acts on you in
midair. In other words, gravity acts over a distance, Newton himself was uneasy
with such an idea. How can the sun exert a force on Earth 150 million kilometers
away?
In the nineteenth century, Michael Faraday invented the concept of the field to
explain how a magnet attracts objects. Later, the field concept was applied to
gravity. Anything that has mass is surrounded by a gravitational field. It is the field
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that acts on a second body, resulting in a force of attraction. The field acts on the
second body at the location of that body. In general, the field concept makes the
idea of a force acting across great distances unnecessary.
To find the strength of the gravitational field, place a small body of mass m in
the field and measure the force. We define the field strength, g, to be the force
divided by a unit mass, FIm. It is measured in newtons per kilogram. The direction
of g is in the direction of the force. Thus,
g=F
m
Note that the field is numerically equal to the acceleration of gravity at the
location of the mass. On Earth's surface, the strength of the gravitational field is
9.8 N/kg. It is independent of the size of the test mass. The field can be
represented by a vector of length g and pointing toward the object producing the
field. We can picture the gravitational field of Earth as a collection of vectors
surrounding Earth and pointing toward it, Figure 8-12.
The strength of the field varies inversely with the square of the distance from the
center of Earth. To get a feeling for the field, hold a heavy book in your hands and
close your eyes. Imagine a spring pulling the book back toward the center of
Earth. As you lift the book, the spring stretches. Both the spring and gravitational
field are invisible.
Einstein's Theory of Gravity
Newton's law of universal gravitation allows us to calculate the force that exists
between two bodies because of their masses. The concept of a gravitational field
allows us to picture the way gravity acts on bodies far away. Neither explains the
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origin of gravity.
Albert Einstein (1879-1955) proposed that gravity is not a force, but an effect of
space itself. According to Einstein, a mass changes the space about it. Mass
causes space to be curved, and other bodies are accelerated because they move
in this curved space.
One way to picture how space is affected by mass is to compare it to a large
two-dimensional rubber sheet, Figure 8-13.
The large ball in the middle of the sheet represents a massive object. It forms
an indentation. A marble rolling across the sheet simulates the motion of an object
in space. If the marble moves near the sagging region of the sheet, its path will be
curved. In the same way, Earth orbits the sun because space is distorted by the
two bodies.
Einstein's theory, called the general theory of relativity, makes predictions that
differ slightly from the predictions of Newton's laws. In every test, Einstein's theory
has been shown to give the correct results.
Perhaps the most interesting prediction is the deflection of light by massive
objects. In 1919, during an eclipse of the sun, astronomers found that light from
distant stars that passed near the sun was deflected in agreement with Einstein's
predictions. Astronomers have seen light from a distant, bright galaxy bent as it
passed by a closer, dark galaxy.
The result is two or more images of the bright galaxy. Another result of general
relativity is the effect on light of very massive objects. If the object is massive
enough, light leaving it will be totally bent back to the object, Figure 8-14.
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No light ever escapes. Such an object, called a black hole, has been identified
as a result of its effect on nearby stars.
Einstein's theory is not yet complete. It does not explain how masses curve
space. Physicists are still working to understand the true nature of gravity.
Do Concept Review 8-2
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