Download Newton`s Universal Law of Gravitation “The Apple and the Moon

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Newton’s Universal Law of Gravitation
“The Apple and the Moon” Video
Objectives
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Recognize that a gravitational force exists between any two objects and that the force is
directly proportional to the product of the masses and inversely proportional to the square
of the distance between them.
Understand the functional dependence of the gravitational force on mass and distance.
Use the following expressions to solve problems: F = GMm/r2, F = ma, a = GM/r2.
Recognize that, for small enough velocities, the time for a projectile to fall to earth is
independent of its horizontal velocity, but for very large horizontal velocities, the effect of
the earth's curvature must be taken into consideration.
Describe orbital motion in terms of the law of universal gravitation and inertia.
Watch the Apple and the Moon
(http://video.google.com/videoplay?docid=3265790422523131277# )video and answer
the following questions.
1. According to the video, what was Newton’s great accomplishment?
2. What was Copernicus’ contribution to astronomy?
3. Who was the first person to discover that all objects fall at the same rate near the
surface of the earth?
4. Whose three laws explain how things work above and beyond the earth?
5. What are the three laws?
6. How long did it take Newton to develop his law of gravity and how long did Newton
wait to publish his theory of universal gravity?
7. Fill in the blanks. The force of attraction between two bits of matter is ___________
to the each of their masses. As the distance between the two bodies increases, the force
weakens as ___________________________.
8. What is equation for the Universal Law of Gravitation shown in the video? (Ignore the
r to the right of the equation. List what each individual variable is along with what it
represents.
9. The video shows the equation for universal gravitation with a negative sign. What does
the negative sign indicate?
10. Newton’s law says that there is a force between any two particles of mass anywhere
in the universe. What’s the net effect of all these forces?
11. As shown in the video, use the law of gravitation and Netwon’s second law to derive
an expression for the acceleration of gravity.
12. Acceleration of gravity on the moon is ______________ what it is on the earth.
13. What argument did Newton use to explain why the moon doesn’t fall to the earth?
14. Explain what “zero g” or “weightlessness” really means.
15. Why doesn’t the moon want to travel in a circle?
***End of Video***
Read the following and answer the questions. Whatever is not completed is to be done
for homework .
Using Equations as a Guide to Thinking
The inverse square law proposed by Newton suggests that the force of gravity acting
between any two objects is inversely proportional to the square of the separation distance
between the object's centers. Altering the separation distance (r) results in an alteration in
the force of gravity acting between the objects. Since the two quantities are inversely
proportional, an increase in one quantity results in a decrease in the value of the other
quantity. That is, an increase in the separation distance causes a decrease in the force of
gravity and a decrease in the separation distance causes an increase in the force of gravity.
Furthermore, the factor by which the force of gravity is changed is the square of the
factor by which the separation distance is changed. So if the separation distance is
doubled (increased by a factor of 2), then the force of gravity is decreased by a factor of
four (2 raised to the second power). And if the separation distance is tripled (increased by
a factor of 3), then the force of gravity is decreased by a factor of nine (3 raised to the
second power). Thinking of the force-distance relationship in this way involves using a
mathematical relationship as a guide to thinking about how an alteration in one variable
affects the other variable. Equations can be more than merely recipes for algebraic
problem-solving; they can be "guides to thinking." Check your understanding of the
inverse square law as a guide to thinking by answering the following questions below.
16. Suppose that two objects attract each other with a force of 16 N. If the distance
between the two objects is reduced in half, what is the new force of attraction between
the two objects?
17. Suppose that two objects attract each other with a force of 16 N. If the distance
between the two objects is reduced by a factor of 4, then what is the new force of
attraction between the two objects?
18. Suppose that two objects attract each other with a force of 16 N. If the distance
between the two objects is doubled and the mass of one of the object is doubled, then
what is the new force of attraction between the two objects?
19. Suppose that two objects attract each other with a force of 16 N. If the distance
between the two objects is doubled and the mass of both of the objects is doubled, then
what is the new force of attraction between the two objects?
20. The planet Jupiter is more than 300 times as massive as Earth, so it might seem that a
body on the surface of Jupiter would weigh 300 times as much as on Earth. But it so
happens a body would scarcely weigh three times as much on the surface of Jupiter as it
would on the surface of the Earth. Explain why this is so.
GM
) to determine the acceleration of gravity
r2
for the various planets. G = 6.67 x 10-11 Nm2/kg2, M is the mass of the planet and r is the
radius of the planet in this case.
21. Use the equation derived earlier ( g 
Planet
Radius (m)
Mass (kg)
Mercury
2.43 x 106
3.2 x 1023
Venus
6.073 x 106
4.88 x1024
Mars
3.38 x 106
6.42 x 1023
Jupiter
6.98 x 107
1.901 x 1027
Saturn
5.82 x 107
5.68 x 1026
Uranus
2.35 x 107
8.68 x 1025
Neptune
2.27 x 107
1.03 x 1026
Pluto
1.15 x 106
1.2 x 1022
g (m/s2)
Soda Can
Fg (N)
To get a sense of the acceleration of gravity on other planets, use the values above to
calculate the force of gravity acting on a can of soda from each of the planets. The mass
of a can of soda is 0.380 kg.
There are a set of soda cans in the classroom. Each represents the weight of the can on
different planets. You will get a chance to see how they feel compared to a can on earth.
Please read the following page on the determination of the Gravitational Constant, G.
Also, read Chapter 13 section 1-2, 6-9, 14-15 and 17-18 in your textbook.
IN SEARCH OF “G”
In 1665, 22 year-old Isaac Newton derived the law of universal gravitation. Although
Newton was able to verify the validity of the “inverse-square” nature of this formula
through calculations involving the orbit of the moon, he was never able to determine the
value of the constant G (the universal gravitation constant). The equation could only be
used as the proportion until the value of G was determined. The problem in determining
G was that in order to do so, you needed to know the gravitational force between two
known masses separated by a known distance. One could easily determine the attraction
between an object of known mass and the earth by weighing it, but the mass of the earth
was not known during Newton’s lifetime.
Before the end of the eighteenth century, however, the answer would be found!
THE CAVENDISH EXPERIMENT
In 1798, Henry Cavendish devised an experiment to determine the value of G. Instead of
involving the gravitational force exerted by the earth, he measured the gravitational
attraction between two known masses in his lab. Measuring this force was difficult
because it was exceedingly small (about 10–9 N or 0.000000001 N). Cavendish's
apparatus for experimentally determining the value of G involved a light rod which was
6-feet long. Two small metal spheres were attached to the ends of the rod and the rod was
suspended by a wire. When the long rod becomes twisted, the torsion of the wire begins
to exert a torsional force which is proportional to the angle of rotation of the rod.
Cavendish had calibrated his instrument to determine the relationship between the angle
of rotation and the amount of torsional force. A diagram of the apparatus is shown.
Cavendish then brought two large lead spheres near the
smaller spheres attached to the rod. Since all masses
attract, the large spheres exerted a gravitational force upon
the smaller spheres and twisted the rod a measurable
amount. Once the torsional force balanced the
gravitational force, the rod and spheres came to rest and
Cavendish was able to determine the gravitational force of
attraction between the masses. By measuring m1, m2, r and
Fgrav, the value of G could be determined. Cavendish's
measurements resulted in an experimentally determined
value of 6.75 x 10-11 N m2/kg2. Today, the currently
accepted value is 6.67259 x 10-11 N m2/kg2.
The value of G is an extremely small numerical value. Its smallness accounts for the fact
that the force of gravitational attraction is only appreciable for objects with large mass.
While two students will indeed exert gravitational forces upon each other, these forces
are too small to be noticeable. Yet if one of the students is replaced with a planet, then the
gravitational force between the other student and the planet becomes noticeable.