This document provides a detailed overview of universal gravitation and its discovery. It discusses how Kepler summarized astronomical data and formulated his three laws of planetary motion in the early 1600s. Newton then used thought experiments involving cannonballs to deduce that gravity causes objects to fall and that the force of gravity follows an inverse-square law, decreasing with the square of the distance between objects. The document also explains how Cavendish experimentally determined the gravitational constant G.

1. Universal Gravitation Q&A using animations and questions from The Physics Classroom at http://www.glenbrook.k12.il.us/gbssci/phys/Class/circles/u6l3a.html

2. Gravity is More Than a Name Nearly every child knows of the word gravity . Gravity is the name associated with the reason for "what goes up, must come down." We all know of the word gravity - it is the thing which causes objects to fall to Earth. The role of physics is to explain phenomenon in terms of underlying principles; in terms of principles which are so universal that they are capable of explaining more than a single phenomenon but a wealth of phenomenon in a consistent manner.

3. The Force of Gravity (F grav ) Certainly gravity is a force which exists between the Earth and the objects which are near it. As you stand upon the Earth, you experience this force. When the only force acting upon an object is the force of gravity, the acceleration due to gravity ( g ) is the acceleration experienced by an object. On and near Earth's surface, the value for the acceleration of gravity is approximately 9.8 m/s 2 .

4. Objectives How and by whom was gravity discovered? What is the cause of this force which we refer to with the name of gravity? What variables affect the actual value of the force of gravity? Why does the force of gravity acting upon an object depend upon the location of the object relative to the Earth? How does gravity affect objects which are far beyond the surface of the Earth? How far-reaching is gravity's influence? And is the force of gravity which attracts my body to the Earth related to the force of gravity between the planets and the Sun?

5. The Apple, the Moon, and the Inverse Square Law In the early 1600's, German mathematician and astronomer Johannes Kepler mathematically analyzed known astronomical data in order to develop three laws to describe the motion of planets about the sun. Kepler's three laws emerged from the analysis of data carefully collected over a span of several years by his Danish predecessor and teacher, Tycho Brahe.

6. Kepler's three laws of planetary motion The path of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses) An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas) The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)

7. Kepler's laws provided a suitable framework for understanding the motion and paths of planets about the sun The cause for how the planets moved as they did was never stated. Kepler could only suggest that there was some sort of interaction between the sun and the planets which provided the driving force for the planet's motion. To Kepler, the planets were somehow "magnetically" driven by the sun to orbit in their elliptical trajectories. There was however no interaction between the planets themselves.

8. Newton knew that there must be some sort of force which governed the heavens Circular and elliptical motion were clearly departures from the inertial paths (straight-line) of objects. Circular motion (as well as elliptical motion) requires a centripetal force. The nature of such a force - its cause and its origin - bothered Newton for some time and was the fuel for much mental pondering.

9. Newton's Thought Experiment Suppose a cannonball is fired horizontally from a very high mountain in a region devoid of air resistance. In the presence of gravity, the cannonball would drop below this straight-line path and eventually fall to Earth. Now suppose that the cannonball is fired horizontally again, yet with a greater speed; in this case, the cannonball would eventually drop to earth. Only this time, the cannonball would travel further before striking the ground.

10. Now suppose that there is a speed at which the cannonball could be fired such that the trajectory of the falling cannonball matched the curvature of the earth. If such a speed could be obtained, then the cannonball would fall around the earth instead of into it; the cannonball would fall towards the Earth without ever colliding into it and subsequently become a satellite orbiting in circular motion.

11. And then at even greater launch speeds, a cannonball would once more orbit the earth in an elliptical path. The motion of the cannonball falling to the earth is analogous to the motion of the moon orbiting the Earth.

12. Newton's dilemma To provide reasonable evidence for the extension of the force of gravity from earth to the heavens. He needed to be able to show how the effect of gravity is diluted with distance.

13. Why is the acceleration of the moon so much smaller than the acceleration of the apple? GIVEN: The force of gravity causes earthbound objects to accelerate towards the earth at a rate of 9.8 m/s 2 . The moon accelerated towards the earth at a rate of 0.00272 m/s 2 . The more distant moon accelerates at a rate of acceleration which is approximately 1/3600-th the acceleration of the apple.

14. Compare the distance from the apple to the center of the earth with the distance from the moon to the center of the earth. The moon in its orbit about the earth is approximately 60 times further from the earth's center than the apple is. The force of gravity between the earth and any object is inversely proportional to the square of the distance which separates that object from the earth's center. The moon, being 60 times further away than the apple, experiences a force of gravity which is 1/(60) 2 times that of the apple.

15. The force of gravity follows an inverse square law . The relationship between the force of gravity ( F grav ) between the earth and any other object and the distance which separates their centers ( d ) can be expressed by the following relationship Since the distance d is in the denominator of this relationship, it can be said that the force of gravity is inversely related to the distance. And since the distance is raised to the second power, it can be said that the force of gravity is inversely related to the square of the distance.

16. 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 (d) 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.

17. Increasing the separation distance decreases 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. Example: 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). Example: 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).

18. Check Your Understanding 1 . Suppose that two objects attract each other with a force of 16 units. If the distance between the two objects is doubled, what is the new force of attraction between the two objects? 2. Suppose that two objects attract each other with a force of 16 units. If the distance between the two objects is tripled, then what is the new force of attraction between the two objects?

19. Check Your Understanding 3. Suppose that two objects attract each other with a force of 16 units. If the distance between the two objects is reduced in half, then what is the new force of attraction between the two objects? 4. Suppose that two objects attract each other with a force of 16 units. If the distance between the two objects is reduced by a factor of 5, then what is the new force of attraction between the two objects?

20. Check Your Understanding 5. If you wanted to make a profit in buying gold by weight at one altitude and selling it at another altitude for the same price per weight, should you buy or sell at the higher altitude location? 6. What kind of scale must you use for this work?

21. Newton's Law of Universal Gravitation Newton knew that the force which caused the apple's acceleration (gravity) must be dependent upon the mass of the apple. And since the force acting to cause the apple's downward acceleration also causes the earth's upward acceleration (Newton's third law), that force must also depend upon the mass of the earth.

22. ALL objects attract each other with a force of gravitational attraction. This force of gravitational attraction is directly dependent upon the masses of both objects and inversely proportional to the square of the distance which separates their centers.

23. TRENDS with MASS More massive objects will attract each other with a greater gravitational force. As the mass of either object increases, the force of gravitational attraction between them also increases. If the mass of one of the objects is doubled, then the force of gravity between them is doubled; if the mass of one of the objects is tripled, then the force of gravity between them is tripled; if the mass of both of the objects is doubled, then the force of gravity between them is quadrupled; and so on.

24. TRENDS with DISTANCE More separation distance will result in weaker gravitational forces. As two objects are separated from each other, the force of gravitational attraction between them also decreases. If the separation distance between two objects is doubled (increased by a factor of 2), then the force of gravitational attraction is decreased by a factor of 4 (2 raised to the second power).

25. Illustrating the proportionalities of the universal law of gravitation

26. The Constant of Proportionality The universal gravitation constant symbolized ‘G’. The precise value of G was determined experimentally by Henry Cavendish in the century after Newton's death. The value of G is found to be G = 6.67 x 10 -11 N m 2 /kg 2 The units on G may seem rather odd, but when the units on G are substituted into the equation above and multiplied by m 1 *m 2 units and divided by d 2 units, the result will be Newtons - the unit of force.

27. Sample Problem #1 Determine the force of gravitational attraction between the earth (m = 5.98 x 10 24 kg) and a 70-kg physics student if the student is standing at sea level, a distance of 6.37 x 10 6 m from earth's center. The solution of the problem involves substituting known values of G ( 6.67 x 10 -11 N m 2 /kg 2 ), m 1 (5.98 x 10 24 kg ), m 2 (70 kg) and d (6.37 x 10 6 m) into the universal gravitation equation and solving for F grav .

28. Sample Problem #2 Determine the force of gravitational attraction between the earth (m = 5.98 x 10 24 kg) and a 70-kg physics student if the student is in an airplane at 40000 feet above earth's surface. This would place the student a distance of 6.38 x 10 6 m from earth's center. This yields the same result as when calculating it using the equation w = m*g = (70 kg)*(9.8 m/s 2 ) = 686 N

29. Gravitational interactions exist between all objects with an intensity which is directly proportional to the product of their masses So as you sit in your seat in the physics classroom, you are gravitationally attracted to your lab partner, to the desk you are working at, and even to your physics book. Of course, most gravitational forces are so minimal to be noticed. Gravitational forces only are recognizable as the masses of objects become large.

30. Check Your Understanding 2 Suppose that two objects attract each other with a force of 16 units. If the distance between the two objects is doubled, what is the new force of attraction between the two objects? Suppose that two objects attract each other with a force of 16 units. If the distance between the two objects is reduced in half, then what is the new force of attraction between the two objects?

31. Check Your Understanding 3. Suppose that two objects attract each other with a force of 16 units. If the mass of both objects was doubled, and if the distance between the objects remained the same, then what would be the new force of attraction between the two objects? 4. Suppose that two objects attract each other with a force of 16 units. If the mass of both objects was doubled, and if the distance between the objects was doubled, then what would be the new force of attraction between the two objects?

32. Check Your Understanding 5. Suppose that two objects attract each other with a force of 16 units. If the mass of both objects was tripled, and if the distance between the objects was doubled, then what would be the new force of attraction between the two objects? 6. Suppose that two objects attract each other with a force of 16 units. If the mass of object 1 was doubled, and if the distance between the objects was tripled, then what would be the new force of attraction between the two objects?

33. Check Your Understanding 7. If our sun shrank in size to a black hole, would the Earth's orbit be affected? Explain. 8. If you wanted to make a profit in buying gold by weight at one altitude and selling it at another altitude for the same price per weight, should you buy or sell at the higher altitude location? What kind of scale must you use for this work?

34. Check Your Understanding 9. What would happen to your weight if the mass of the Earth somehow increased by 10%? 10. 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.

35. Cavendish and the Value of G The value of G was not experimentally determined until nearly a century later by Lord Henry Cavendish using a special apparatus. Cavendish's measurements resulted in an experimentally determined value of 6.75 x 10 -11 N m 2 /kg 2 . Today, the currently accepted value is 6.67259 x 10 -11 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.

36. Check Your Understanding 3 Suppose that you have a mass of 70 kg (equivalent to a 154-pound person). How much mass would another object have to have in order for your body and the object to attract each other with a force of 1-Newton when separated by 10 meters?

37. The Value of g g is referred to as the acceleration of gravity. It's value is 9.8 m/s 2 on the surface of the Earth. The value of g is dependent upon location. There are slight variations in the value of g about earth's surface. These variations result from: the varying density of the geologic structures below each specific surface location the earth is not truly spherical a location of orbit about the earth changes the value of g

38. The value of g at various locations from Earth's center. 0.13 5.64 x 10 7 m 50000 km above surface 1.49 1.64 x 10 7 m 10000 km above surface 2.60 1.24 x 10 7 m 6000 km above surface 3.08 1.14 x 10 7 m 5000 km above surface 3.70 1.04 x 10 7 m 4000 km above surface 4.53 9.38 x 10 6 m 3000 km above surface 5.68 8.38 x 10 6 m 2000 km above surface 7.33 7.38 x 10 6 m 1000 km above surface 9.8 6.38 x 10 6 m Earth's surface Value of g Distance from Earth’s center Location

39. The equation takes the following form: g = G * M planet / (R planet ) 2 The value of g varies inversely with the distance from the center of the earth. The same equation used to determine the value of g on earth can also be used to determine the acceleration of gravity on the surface of other planets. The value of g on any other planet can be calculated from the mass of the planet and the radius of the planet.

40. Kepler's Three Laws early 1600s summarize data of his mentor - Tycho Brahe described the motion of planets in a sun-centered solar system Kepler's explanations of the underlying reasons for such motions are no longer accepted the actual laws are still an accurate description of the motion of any planet and any satellite

41. Kepler's Laws of Planetary Motion The path of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses) An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas) The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)

42. Kepler's first law Explains that planets are orbiting the sun in a path described as an ellipse. An ellipse is a special curve in which the sum of the distances from every point on the curve to two other points is a constant. (The two other points are known as the foci of the ellipse.) the sun being located at one of the foci of that ellipse.

43. Kepler's second law Describes the speed at which any given planet will move while orbiting the sun. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if a line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time.

44. Kepler's Third Law Compares the orbital period and radius of orbit of a planet to those of other planets. Ratio of T 2 /R 3 is the same for all planets and for any satellite (whether a moon or a man-made satellite) about any planet. (T is the period and R is the average distance from the sun)

45. Check Your Understanding 1 . Today it is widely believed that the planets travel in elliptical paths around the sun. a. Who gathered the data to support this fact? b. Who analyzed the data to support this fact? c. Who provided the accurate explanation for why this occurs?

46. Check Your Understanding 2. Galileo discovered four moons of Jupiter. One moon - Io - which he measured to be 4.2 units from the center of Jupiter and had an orbital period of 1.8 days. Galileo measured the radius of Ganymede to be 10.7 units from the center of Jupiter. Use Kepler's third law to predict the orbital period of Ganymede.

47. Check Your Understanding 3. If a small planet were located eight times as far from the sun as the Earth's distance from the sun (1.5 x 10 11 m), how many years would it take the planet to orbit the sun. GIVEN: T 2 /R 3 = 2.97 x 10 -19 s 2 /m 3 4. On average, the planet Mars is 1.52 times further from the sun as is Earth. Given that the Earth orbits the sun in approximately 365 earth days, predict the time required for Mars to orbit the sun.

48. Velocity of Planets and Satellites Use the equation from circular motion F= mv 2 /R and set it equal to the law of universal gravitation. This gives m 1 v 2 /R = m 1 *m 2 /R 2 When rearranging this, you find that v =  Gm 2 /R