# Chapter 13: Universal Gravitation

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2 Chapter 13: Universal Gravitation I. The Falling Apple (13.1) A. Isaac Newton ( ) 1. Formulated ideas based on earlier work by Galileo (concept of inertia) 2. Concept if object undergoes change in speed or direction, then a force is responsible

3 B. Related falling apple, to the motion of the moon (falling around the earth)

4 II. The Falling Moon (13.2) A. Newton hypothesized that moon was simply a projectile circling Earth under the attraction of gravity 1. Newton had to test hypothesis 2. Compared fall of apple to fall of moon

5 3. Newton reasoned that gravitational attraction was diluted by distance a. Moon is 60 times farther from the center of the Earth than the apple b. Calculated difference to be 1/(60) 2

6 B. Newton waited 20 years to prove hypothesis 1. Invented a new math (Calculus) to explain theory 2. Published his findings The Law of Universal Gravitation. (applied to all objects in the universe)

7

8 III. The Falling Earth (13.3) A. Newton s theory confirmed Coperincan theory of the solar system. 1. Earth no longer considered center of universe 2. Earth not even center of solar system

9 B. Planets tangential velocities enough to keep in orbit

10 IV. Newton s Law of Universal Gravitation (13.4) A. Law states: every object attracts every other object with a force that for any two objects is directly proportional to the mass of each object 1. The greater the mass the greater the attraction 2. The farther away the objects are from each other, the less the force of attraction between them

11 B. Law expressed as: F m m 1 d 2 2 m 1 is mass of one object m 2 is mass of other d is distance between their centers

12 C. The Universal Gravitational Constant (G) 1. The above equation is a proportional form of law. 2. Can be expressed exactly when universal gravitational constant (G) is introduced a. Value of G first measured 150 years later by Henry Cavendish b. G X10 N m/ kg F G m m 1 d 2 2

13 3. Value of G tells us that force of gravity is a very weak force (weakest of known four fundamental forces (electromagnetic and two known nuclear forces

14 D. Cavendish used value of G to calculate the mass of the Earth (mass of Earth = 6 X kilograms)

15 V. Gravity and Distance: The Inverse Square Law (13.5) A.Inverse Square Law when quantity varies as the inverse square of its distance from its source B. Also applies to light, radiation, and sound

16

17 I. Gravitational Fields (13.6) A. Gravitational Field force field that surrounds massive objects B. Can be represented by imaginary field lines.

18 C. The strength of Earth s gravitational field is the force per unit mass exerted by Earth on any object. g F m 9.8 N kg 9.8 m s 2

19 D. The strength of its force on objects follows the inverse-square law (so g weakens with increasing distance from Earth

20 VII. Weight and Weightlessness (13.7) A. The force of gravity causes acceleration 1. Since we are almost always in contact with the Earth, we sense gravity as something that presses us against Earth rather than something that accelerates us 2. This pressing against Earth is what we interpret as weight

21 B. More practical to define weight as the force you exert against a supporting floor (or weighing scales) you are as heavy as you feel.

22 C. Weightlessness is not absence of gravity; rather, it is absence of a support force.

23

24 IX. Ocean Tides (13.9) A. Newton showed that the ocean tides are caused by differences in the gravitational pull of the moon on opposite sides of earth 1. Oceans bulge about 1 meter on opposite sides of Earth 2. Because Earth spins, the tides change as Earth rotates

25 B. Sun also contributes to tides 1. Sun s pull 180 times greater than moon, but contributes only half as much as the moon 2. Because difference in gravitational pullby sun on opposite sides of Earth very small (0.017% compared to 6.7% for moon's gravitation)

26 X. Black Holes (13.10) A. There are two main processes going on continuously in stars like our sun. 1. Process of Gravitation tends to crunch all solar material toward the center 2. Process of thermonuclear fusion-consisting of reactions similar to those in a hydrogen bomb. Tend to blow solar material outward

27 B. Two processes balance each other, resulting in given size for sun. 1. If fusion rate increases, the sun will get bigger and hotter. ( red giant ) 2. If fusion decreases, sun will get cooler and smaller. ( black dwarf )

28

29 C. Black Holes For very heavy stars (at least 2 or 3 times more massive than our sun), when flame of thermonuclear fusion is extinguished, gravitational collapse occurs

30 1.Star caves in on itself, with atoms also compress so that there is no empty spaces- density becomes literally infinite 2. Gravitation near a black holes so great that nothing can get back out including light. 3. Although black holes can not be seen, their effects can be.

31 XI. Universal Gravitation (13.11) A. Law was used to explain perturbation of planets in our solar system 1. perturbation-deviations from normal orbits 2. Used law to explain Uranus s perturbation (caused by presence of another unknown planet-neptune)

32 B. Theory dramatically affected science and society 1. Ushered in the Age of Reason or Century of Enlightenment. 2. Nurtured the thinking of scientists, artists, writers, and philosophers of the 1700 s

33 a. John Locke argued that observation and reason, as demonstrated by Newton, should be our best judge and guide in all things b. Locke and followers modeled system of government that culminated in the Declaration of Independence and Constitution of the United States of America

34 Einstein s Theory of Gravity Newton s theories did not explain the origin of gravity Einstein proposed gravity is not a force, but an effect of space itself. Einstein said a mass changes the space around it, causing space to be curved.

35 Called the General Theory of Relativity This theory provides accurate predictions of gravity s effects, but is still not complete. It does not explain how masses curve space

36 Assessment Questions 1. Newton determined that the pull of Earth s gravity caused both apples and a. the moon to fall toward Earth. b. the moon to move away from Earth. c. the sun to move away from Earth. d. stars to fall toward Earth.

37 Assessment Questions 1. Newton determined that the pull of Earth s gravity caused both apples and a. the moon to fall toward Earth. b. the moon to move away from Earth. c. the sun to move away from Earth. d. stars to fall toward Earth. Answer: A

38 Assessment Questions 2. The moon falls toward Earth in the sense that it falls a. with an acceleration of 10 m/s 2, as apples fall on Earth. b. with an acceleration greater than 10 m/s 2. c. beneath the straight-line path it would take without gravity. d. above the straight-line path it would take without gravity.

39 Assessment Questions 2. The moon falls toward Earth in the sense that it falls a. with an acceleration of 10 m/s 2, as apples fall on Earth. b. with an acceleration greater than 10 m/s 2. c. beneath the straight-line path it would take without gravity. d. above the straight-line path it would take without gravity. Answer: C

40 Assessment Questions 3. Planets remain in orbit while falling around the sun due to their a. tangential velocities. b. zero tangential velocities. c. accelerations of about 10 m/s 2. d. centrifugal forces that keep them up.

41 Assessment Questions 3. Planets remain in orbit while falling around the sun due to their a. tangential velocities. b. zero tangential velocities. c. accelerations of about 10 m/s 2. d. centrifugal forces that keep them up. Answer: A

42 Assessment Questions 4. Newton did not discover gravity, for early humans discovered it whenever they fell. What Newton did discover is that gravity a. tells us about why the universe expands. b. tells us how to discover new planets. c. accounts for the existence of black holes. d. extends throughout the universe.

43 Assessment Questions 4. Newton did not discover gravity, for early humans discovered it whenever they fell. What Newton did discover is that gravity a. tells us about why the universe expands. b. tells us how to discover new planets. c. accounts for the existence of black holes. d. extends throughout the universe. Answer: D

44 Assessment Questions 5. Consider a space probe three times as far from Earth s center. Compared at Earth s surface, its gravitational attraction to Earth at this distance is about a. one third as much. b. one half as much. c. one ninth as much. d. zero.

45 Assessment Questions 5. Consider a space probe three times as far from Earth s center. Compared at Earth s surface, its gravitational attraction to Earth at this distance is about a. one third as much. b. one half as much. c. one ninth as much. d. zero. Answer: C

46 Assessment Questions 6. Compared to the gravitational field of Earth at its surface, Earth s gravitational field at a distance three times as far from Earth s center is about a. one third as much. b. one half as much. c. one ninth as much. d. zero.

47 Assessment Questions 6. Compared to the gravitational field of Earth at its surface, Earth s gravitational field at a distance three times as far from Earth s center is about a. one third as much. b. one half as much. c. one ninth as much. d. zero. Answer: C

48 Assessment Questions 7. Compared to the gravitational field of Earth at its surface, Earth s gravitational field at Earth s center is a. zero. b. half as much. c. twice as much. d. three times as much.

49 Assessment Questions 7. Compared to the gravitational field of Earth at its surface, Earth s gravitational field at Earth s center is a. zero. b. half as much. c. twice as much. d. three times as much. Answer: A

50 Assessment Questions 8. When an astronaut in orbit is weightless, he or she is a. beyond the pull of Earth s gravity. b. still in the pull of Earth s gravity. c. in the pull of interstellar gravity. d. beyond the pull of the sun s gravity.

51 Assessment Questions 8. When an astronaut in orbit is weightless, he or she is a. beyond the pull of Earth s gravity. b. still in the pull of Earth s gravity. c. in the pull of interstellar gravity. d. beyond the pull of the sun s gravity. Answer: B

52 Assessment Questions 9. The highest ocean tides occur when the Earth and moon are a. lined up with the sun. b. at right angles to the sun. c. at any angle to the sun. d. lined up during spring.

53 Assessment Questions 9. The highest ocean tides occur when the Earth and moon are a. lined up with the sun. b. at right angles to the sun. c. at any angle to the sun. d. lined up during spring. Answer: A

54 Assessment Questions 10. A black hole is a. simply a collapsed star. b. a two-dimensional surface in space. c. barely visible with high-powered telescopes. d. a new form of gravity.

55 Assessment Questions 10. A black hole is a. simply a collapsed star. b. a two-dimensional surface in space. c. barely visible with high-powered telescopes. d. a new form of gravity. Answer: A

56 Assessment Questions 11. Newton s law of universal gravitation had a great impact on society as many scientists, artists, writers, and philosophers hoped that a. more complex and universal laws would explain other phenomena of the world. b. greater observations would require fewer experimentations. c. no further explanation of other phenomena of the world would be required. d. studying other phenomena of the world would lead to just as simple and universal laws.

57 Assessment Questions 11. Newton s law of universal gravitation had a great impact on society as many scientists, artists, writers, and philosophers hoped that a. more complex and universal laws would explain other phenomena of the world. b. greater observations would require fewer experimentations. c. no further explanation of other phenomena of the world would be required. d. studying other phenomena of the world would lead to just as simple and universal laws. Answer: D

58 Gravitation

59 1. Kepler s Laws a. Prior to Kepler - Aristotle s concept of an Earth centered system dominant thought until Copernicus develops heliocentric model b. Based on data gathered by Tycho Brahe - Danish astronomy somtimes referred to as the great observer. c. Based on the Conservation of Energy and Angular Momentum

60 d. Three Laws (1) Paths of planets are ellipses (nearly circular) with the sun at one focus Exaggerated the ellipse Sun

61 (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? (b) Does this provide you with conservation of energy? Area 1 Area 2

62 (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 Why Average Radius? sun

63 2. 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 N-m 2 /kg 2 or m 3 /kgsec 2

64 c. 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 / T 2 F c = 4 2 r m / r 3 k 2 = k 3 m / r 2

65 d. Newton developed the concept but was not able to determine the value for G e. Value for G was found experimentally by Cavendish in 1798 (1) Led to the determination of the mass of the Earth (2) M E = 5.98 x kg

66 f. Gravitation - Distance and M 1 Mass Effects M 1 M 2 d M 2 d 2 M 1 M 2 M 1 M 1 2 d M 2 2 M 2 2 M 1 2 M 2 M 1 1/2 d M 2

67 g. Do Now 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

68 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.

69 h. If the sun suddenly collapsed to become a black hole, then the Earth would a. Leave the solar system in a straightline 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

70 We can see from Newton s Universal Law of Gravity equation, that 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 freezew and undergo enormous surface changes, its yearly path would continue

71 j. 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

72 m = 50 kg r = 35 m M = 150,000 kg v 0 = 0 F = G (M 1 M 2) / r 2 = 4.08 x 10-7 N F = m a so a = F / m = 8.16 x 10-9 m/s 2 s = v 0 t + 1/2 a t 2 of oxygen t = 92, 600 sec runs out

73 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.

74 3. Orbital Motion v - tangential velocity F - centripetal force v r - distance to center of mass of the Earth E r F c

75 a. Assume a circular orbit with gravity providing the centripetal Then force. F c mv 2 r F G GmM e r 2 which gives v GM e r b. Mass of the satellite is unimportant in describing its motion, only mass of planet.

76 c. We know that T 2 r v from v 2 r T so T 2 r 3 GM e d. Can be used for any body in orbit around another body.

77 e. Example Problems (1) A synchronous satellite will orbit at 3.6 x 10 7 m above the surface of the Earth. What is its speed? M e = 5.98x10 24 kg and r e = 6370 km. Givens: M e = v = 3068 m/s 5.98x10 24 kg r o = 36 x x 10 6 m G = 6.67 x N-m 2 /kg 2

78 (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. v 2 r T == v GM J r so that M J v 2 r G M J = 1.88 x kg

79 4. Law of Universal Gravitation and Weight a. Weight is due to gravity so w = G (M o M E) / r E 2 and since w = m g can determine g (acceleration due to gravity) g = G M E / r E 2 b. Weight changes with distance from the center of the Earth

80 c. Weight and Weightlessness (1). a GM e d 2 g changes with height and distance from center of the Earth (d) Therefore a g r e d 2 which allows you to calculate values for acceleration due to gravity for whatever distance you are above the Earth

81 (2) Weightlessness in space is not zero gravity, it is freefall. (a) Objects are still attracted by planet (b) Objects are falling toward the planet at the same rate the planet is falling away from them due to curvature of the planet s surface (c) Object still has weight in space

82 5. 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.

83 d. Described using vectors and concentrations (1) The closer the lines are together the more powerfull is the field. (2) Direction of arrows shows the direction of attraction.

84 6. Two Kinds of Mass a. Inertial mass equal to the net force exerted on an object divided by the object s acceleration 2 nd Law b. Gravitational Mass from Newton s Law of Universal Gravitation attraction of one mass for another c. Principal of Equivalence proposes they are equal in magnitude.

85 7. Einstein a. Stated gravity is not a force, but an effect of space. b. Mass causes space to be curved, and this curving accounts for acceleration. c. Even light bends with gravity. d. Concept behind warp drive on Star Trek.

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