Newton s Universal Law of Gravitation The Apple and the Moon Video

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1 Name Date Pd Newton s Universal Law of Gravitation The Apple and the Moon Video Objectives 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/r 2, F = ma, a = GM/r 2. 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= # )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?

2 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?

3 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.

4 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.

5 GM 21. Use the equation derived earlier ( g ) to determine the acceleration of gravity 2 r for the various planets. G = 6.67 x Nm 2 /kg 2, M is the mass of the planet and r is the radius of the planet in this case. Planet Radius (m) Mass (kg) Mercury 2.43 x x Venus x x10 24 Mars 3.38 x x Jupiter 6.98 x x Saturn 5.82 x x Uranus 2.35 x x Neptune 2.27 x x Pluto 1.15 x x g (m/s 2 ) Soda Can F g (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 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, and in your textbook.

6 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 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 m 1, m 2, r and F grav, the value of G could be determined. Cavendish's measurements resulted in an experimentally determined value of 6.75 x N m 2 /kg 2. Today, the currently accepted value is x N m 2 /kg 2. 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.

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