Download PHYSICS CHAPTER 8 : Universal Gravitation

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 contents seem meager by
today's standards:
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.
Thermometer
Battery
Radio transmitter
beeps
- changed the tone of its
to match temperature changes
Nitrogen gas - pressurized the interior of the satellite
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
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 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
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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 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 4megaton 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.
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
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 nineteenyear-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 x 2 to cube number then y2 and
number 3 to square root number then square of x to cube root number then 2 nd 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
Sun
6.960 e8
1.991e30
Mercury
2.430 e6
3.200e23
5.800e10
Venus
Earth
6.073 e6
6.3713 e6
4.880e24
5.979e24
1.081e11
1.4957e11
3
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 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, m 1 and m2. He proposed
__________
F = G (m1m2 ) / d2
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 onequarter 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
___________
Tp2 = rps2 x Mp (4π2rps) / 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.
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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 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
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
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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 the one below 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 curvature of
Earth will just match the curvature of the trajectory, and the ball will orbit Earth.
The drawing 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 the sketch
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.
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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
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
_____
G = 6.67e –11 Nxm2/kg2
v = √GME/r
Earth’s radius = 6.37e 10 m
_______________________________________________
v = √6.67e –11 Nxm2/kg2 x 5.98e24 kg / (6.37e 10 m + 225 km)
7.77 km/s x 3600s x .62 = 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.
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
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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 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,
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 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.
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One way to picture how space is affected by mass is to compare it to a large two-dimensional rubber sheet,
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,
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.
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