Kepler's Laws and Universal Gravitation

1. Chapter 8 Universal Gravitation

2. Section 8.1 Objectives • List Kepler’s Laws and understand them • Calculate periods and velocities of orbiting objects • Understand that gravitational force is proportional to the masses and the inverse square of the distances between objects

3. 1. Kepler’s Laws • a. Prior to Kepler - Aristotle’s concept of an Earth centered system dominated thought until Copernicus develops heliocentric model • b. Based on data gathered by Tycho Brahe - • Danish astronomy sometimes referred to as the great observer.

4. Sun • Kepler’s Three Laws:(1) Paths of planets are ellipses (nearly circular) with the sun at one focus • Exaggerated the ellipse

5. (2) Line from the sun to a planet sweeps out equal areas in equal amounts of time • (a) At which point would have planet be moving faster? Area 2 Area 1

6. (3) Ratio of the average radius (r) cubed to the period (T) squared is constant for all planets • r 3 / T 2 = k • r = average radius from planet to the sun • T = period of revolution around the sun • Why Average Radius?

7. Example Problem • The moon Io is 4.2 units and Ganymede is 10.7 units from the center of Jupiter. Io has a period of 1.8 days what is the period of Ganymede? • Use (Ta/Tb)2 = (ra/rb)3 • Ta2 = Tb2(ra/rb)3 • Ta2 = (1.8days)2(10.7/4.2)3 • Ta = (52.8 days2)1/2 = 7.3 days

8. Newton and Gravity The Falling Apple According to legend, Newton discovered gravity while sitting under an apple tree.

9. The Falling Apple Newton saw the apple fall, or maybe even felt it fall on his head. Perhaps he looked up through the apple tree branches and noticed the moon. • He may have been puzzled by the fact that the moon does not follow a straight-line path, but instead circles about Earth. • He knew that circular motion is accelerated motion, which requires a force. • Newton had the insight to see that the moon is falling toward Earth, just as the apple is.

10. 13.2The Falling Moon The moon is actually falling toward Earth but has great enough tangential velocity to avoid hitting Earth.

11. The Falling Moon • Newton realized that if the moon did not fall, it would move off in a straight line and leave its orbit. • His idea was that the moon must be falling around Earth. • Thus the moon falls in the sense that it falls beneath the straight line it would follow if no force acted on it. • He hypothesized that the moon was simply a projectile circling Earth under the attraction of gravity.

12. The Falling Moon If the moon did not fall, it would follow a straight-line path.

13. 13.2The Falling Moon Newton’s Hypothesis Newton compared motion of the moon to a cannonball fired from the top of a high mountain. • If a cannonball were fired with a small horizontal speed, it would follow a parabolic path and soon hit Earth below. • Fired faster, its path would be less curved and it would hit Earth farther away. • If the cannonball were fired fast enough, its path would become a circle and the cannonball would circle indefinitely.

14. The Falling Moon This original drawing by Isaac Newton shows how a projectile fired fast enough would fall around Earth and become an Earth satellite.

15. The Falling Moon Both the orbiting cannonball and the moon have a component of velocity parallel to Earth’s surface. This sideways or tangential velocity is sufficient to ensure nearly circular motion around Earth rather than into it. With no resistance to reduce its speed, the moon will continue “falling” around and around Earth indefinitely.

16. 13.2The Falling Moon Tangential velocity is the “sideways” velocity—the component of velocity perpendicular to the pull of gravity.

17. Newton’s Law of Universal Gravitation Newton discovered that gravity is universal. Everything pulls on everything else in a way that involves only mass and distance. The force of gravity between objects depends on the distance between their centers of mass.

18. Developed by using Kepler’s Third Law and equating force to centripetal force. • r 3 / T 2 = k 1 implies T 2 = r 3 k 2 so • if F c = m v 2 / r and v = 2  r / T then F c =[ m (2  r / T) 2 ] /r = m 4  2 r / T2 F c = 4  2 r m / r 3 k2 = k 3m / r 2 or……… Fg = GMm/d2 where G is the gravitational constant and M is the mass of the planet

19. Newton developed the concept but was not able to determine the value for G • Value for G was found experimentally by Cavendish in 1798 • (pg 162 in text) • (1) Led to the determination of the mass of the Earth • (2) M E = 5.98 x 10 24 kg

20. Newton’s Law of Universal Gravitation Philipp von Jolly developed a simpler method of measuring the attraction between two masses.

21. Newton’s Law of Universal Gravitation • a. Every body attracts every other body with a force that varies based on the distance separating the bodies and their masses. F = G (M 1 M 2) / r 2 • b. G is the universal gravitational constant - similar to a constant such as the speed of light, Avogadro’s Number, etc…. G = 6.67 x 10 -11 N-m 2/kg 2 or m3/kg-sec 2

22. Newton’s Law of Universal Gravitation The value of G tells us that gravity is a very weak force. It is the weakest of the presently known four fundamental forces. We sense gravitation only when masses like that of Earth are involved.

23. Gravity and Distance: The Inverse-Square Law Gravitational force is plotted versus distance from Earth’s center.

24. Gravitational Forces F = G(m1m2)/d2 M2 M1 d M2 F • M1 F d M2 2 M1 2F M1 2 d M2 M1 ¼ F 2 M2 2F 2 M1 M2 M1 4F 2 M2 1/2 d 4F

25. Practice Exercise • Consider two satellites in orbit around a star (like our sun). If one satellite is twice as far from the star as the other, but both satellites are attracted to the star with the same gravitational force, how do the masses of the satellites compare? Sun

26. Answer • If both satellites had the same mass, then the one twice as far would be attracted to the star with only one-fourth the force (inverse square law). Since the force is the same for both, the mass of the farthermost satellite must be four times as great as the mass of the closer satellite.

27. If the sun suddenly collapsed to become a black hole, then the Earth would?? • a. Leave the solar system in a straight-line path. • b. Spiral into the black hole • c. Undergo a major increase in tidal forces • d. Continue to circle in its usual orbit. ?? Sun poof

28. From Newton’s Universal Law of Gravitation: The interaction F between the mass of the Earth and the Sun doesn’t change. This is because the mass of the Earth does not change, the mass of the sun does not change even though it is compressed, and the distance from the centers of the Earth and the sun, collapsed or not, does not change. • Although the Earth would very soon freeze and undergo enormous surface changes, its yearly path would continue as if the sun were its normal size.

29. Extra Credit Group Problem • A 50 kg astronaut is floating at rest in deep space 35 m from her stationary 150,000 kg spaceship. How long will it take her to float to the spaceship due to her attraction (gravity) with the ship? If she has a three hour supply of oxygen, will she make it to the ship in time? Help 35 meters

30. m1 = 50 kg d = 35 m • M2 = 150,000 kg v 0 = 0 • G = 6.67 x 10 -11 N-m 2/kg 2 • F = G(M 1 M 2)/d2= 4.08 x 10 -7 N • F = ma so a = F/m • a= 8.16 x 10-9 m/s2 • d = v0t + ½at2 or t= (2d/a)½ t = (2(35m)/(8.16x10-9m/s2))½ • t = 92,600 sec or ~26 hours • She’s toast ….. (runs out of oxygen)

31. k. Example Problem • Compare the gravitational pull on a spaceship at the surface of the Earth with the gravitational pull when the ship is orbiting 1000 km above the surface. (r E = 6370 km) • F = G (M 1 M 2) / r 2 • 25 % decrease. Note that the ship is still under effects of gravity and is NOT weightless.

32. Section 8.2 Using the Law of Universal Gravitation Objectives: • Solve problems using orbital velocity and period • Understand the term weightlessness of objects in free fall and orbit • Describe gravitational fields • Contrast Einstein's concept of gravity to that of Newton’s

33. v - tangential velocity F - centripetal force r - distance to center of mass of the Earth Orbital Motion v Fc E r

34. Assume a circular orbit with gravity providing the centripetal force. • Then which gives Mass of the satellite is unimportant in describing its motion, only mass of planet.

35. We know that And since… So the period of orbit: Can be used for any body in orbit around another body.

36. Example Problems (1) A synchronous satellite will orbit at 3.6 x 107 m above the surface of the Earth. What is its speed? Me = 5.98x1024 kg and re = 6370 km. • Givens: Me = 5.98x1024 kg ro = 36 x 106 + 6.37 x 10 6 m G = 6.67 x 10 -11 N-m 2/kg 2 Use the equation: • v = 3068 m/s

37. (2) A moon of Jupiter, called Calisto, circles Jupiter each 16.8 days. Its orbital radius is 1.88 x 10 6 km. Find the mass of Jupiter. *Convert radius from km to m and days to seconds = so that MJ = 1.88 x 10 27 kg

38. Law of Universal Gravitation and Weight • a. Weight is due to gravity so w = G(mME)/rE2 and since w = mg can determine g (acceleration due to gravity) since w=gm g = GM E/r E 2 • b. Weight changes with distance from the center of the Earth

39. c. Weight and Weightlessness • (1). g changes with height and distance from center of the Earth (d) Therefore taking the ratio of a/g gives: which allows you to calculate values for acceleration due to gravity for whatever distance you are above the Earth

40. “Weightlessness” in orbit is not zero gravity, it is freefall. • (a) Objects are still attracted by planet • (b) Objects are accelerating toward the planet at the same rate the planet is falling away from them due to curvature of the planet’s surface • You feel weightless because the space ship and you are in free fall around the planet (you orbit because your tangential velocity prevents you from hitting Earth)

41. Weight and Weightlessness The sensation of weight is equal to the force that you exert against the supporting floor.

42. Weight and Weightlessness When the elevator accelerates downward, the support force of the floor is less. The scale would show a decrease in your weight. If the elevator fell freely, the scale reading would register zero. According to the scale, you would be weightless. You would feel weightless, for your insides would no longer be supported by your legs and pelvic region. This is exactly what happens in orbit!

43. Gravitational Fields & Einstein Objectives: • Describe gravitational fields • Explain why the field concept does not tell us why gravity exists • Contrast gravity fields to Einstein's concept of gravity

44. 13.6Gravitational Field Earth can be thought of as being surrounded by a gravitational field that interacts with objects and causes them to experience gravitational forces.

45. Gravitational Field We can regard the moon as in contact with the gravitational field of Earth. A gravitational field occupies the space surrounding a massive body. A gravitational field is an example of a force field, for any mass in the field space experiences a force.

46. Gravitational Field A more familiar force field is the magnetic field of a magnet. • Iron filings sprinkled over a sheet of paper on top of a magnet reveal the shape of the magnet’s magnetic field. • Where the filings are close together, the field is strong. • The direction of the filings shows the direction of the field at each point. • Planet Earth is a giant magnet, and like all magnets, is surrounded in a magnetic field.

47. Gravitational Field Field lines can also represent the pattern of Earth’s gravitational field. • The field lines are closer together where the gravitational field is stronger. • Any mass in the vicinity of Earth will be accelerated in the direction of the field lines at that location. • Earth’s gravitational field follows the inverse-square law. • Earth’s gravitational field is strongest near Earth’s surface and weaker at greater distances from Earth.

48. Gravitational Field Field lines represent the gravitational field about Earth.

49. Gravitational Field • a. First type of field force we have encountered. • b. Use the concept of fields to explain how forces act through a distance, NOT WHY. • c. Fields describe how forces act on an object due to its location.

50. Described using vectors and concentrations • The closer the lines are together the more powerful the field. • Direction of arrows shows the direction of attraction. • Field Strength: g = F/m measured in Newtons/kg