# Kepler, Newton and Gravitation

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1 Kepler, Newton and Gravitation Kepler, Newton and Gravity 1

2 Using the unit of distance 1 AU = Earth-Sun distance PLANETS COPERNICUS MODERN Mercury Venus Earth Mars Jupiter Saturn Kepler, Newton and Gravity 2

3 Johannes Kepler A.D. foundation for modern cosmological concepts built on successes and failures demanded that his model s predictions be at least as accurate as the observations (1 arcmin error) I have discovered among the celestial movements the full nature of harmony. Kepler, Newton and Gravity 3

4 Kepler - the Ellipse Kepler, Newton and Gravity 4

5 Kepler - the Ellipse two focal points - foci focal distance Kepler, Newton and Gravity 5

6 Kepler - the Ellipse minor axis semi-minor axis - half the minor axis Kepler, Newton and Gravity 6

7 Kepler - the Ellipse major axis minor axis semi-major axis - half the major axis (a) semi-minor axis - half the minor axis Kepler, Newton and Gravity 7

8 Kepler - the Ellipse major axis two focal points - foci minor axis focal distance semi-major axis - half the major axis (a) semi-minor axis - half the minor axis eccentricity = focal distance major axis Kepler, Newton and Gravity 8

9 Kepler - the Ellipse major axis focal distance eccentricity = focal distance major axis e = 0 means???? e = 1 means???? Kepler, Newton and Gravity 9

10 Kepler s Three Laws I. Law of Ellipses Each planet s orbit is an ellipse with the Sun at one of the foci. implication: distance of the planet to the Sun varies Kepler, Newton and Gravity 10

11 Kepler s Three Laws II. Law of Equal Areas A line drawn from a planet to the Sun sweeps out equal areas in equal times. implication: orbital speeds are non-uniform yet vary in a regular way Closer a planet is the the Sun, the faster it moves in its orbit (force????) Kepler, Newton and Gravity 11

12 Kepler s Three Laws III. Harmonic Law Planet p (Earth years) a (AU) ============================ Mercury Venus Earth Mars Jupiter Saturn Kepler, Newton and Gravity 12

13 Kepler s Three Laws III. Harmonic Law Square of the orbital period is proportional to the cube of the average distance. p 2 = k a 3 implication: planets with large orbits move slowly Proportion holds for all planets => A PHYSICAL CAUSE!! Kepler, Newton and Gravity 13

14 Kepler s Third Law p 2 = k a 3 k = proportionality constant p (years) a (AU s) => k = 1 True for any body orbiting the Sun, even spacecraft! I contemplate its beauty with incredible and ravishing delight. Kepler, Newton and Gravity 14

15 Kepler made perhaps the greatest leap in scientific thinking predictions were 10 times more accurate than either Ptolemaic model (geocentric) Copernican model (heliocentric) gave birth to astronomy as a physical science Kepler, Newton and Gravity 15

16 Orbital Eccentricities Planet Orbital Eccentricity Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Kepler, Newton and Gravity 16

17 GALILEO, NEWTON, and GRAVITY Kepler, Newton and Gravity 17

18 Mechanics: the study of falling bodies speed: velocity: how fast you are going how fast you are going in a specific direction Kepler, Newton and Gravity 18

19 Albany to Portland: speed = 100 km/hr velocity = 100?????? km/hr north Portland to Albany: speed = 80 km/hr velocity =?????? 80 km/hr south Kepler, Newton and Gravity 19

20 velocity: speed and direction acceleration: CHANGE in velocity change in speed change in direction change in both speed and direction Kepler, Newton and Gravity 20

21 speed: measured in km/hr velocity: measured in km/hr in a direction acceleration: measured in km/hr unit of time Three Cases You are stopped at a red light. You step on the gas. Your speed changes from 0 to 15 km/hr. Are you accelerating? Yes, it is your speed that is changing. Kepler, Newton and Gravity 21

22 You are driving in a circle around a racetrack at a constant 20 km/hr. Are you accelerating? Yes. Your direction is changing. You are braking from 25 km/hr to 15 km/hr. Are you accelerating? Yes. Your speed is changing. Kepler, Newton and Gravity 22

23 Galileo and Motion downward motion: gravity pulling downward forced motion attractive force horizontal motion: would continue forever if no forces acted natural motion Kepler, Newton and Gravity 23

24 Galileo s Experiment inclined plane, perfectly smooth perfectly smooth ball Kepler, Newton and Gravity 24

25 Galileo s Experiment Kepler, Newton and Gravity 25

26 Galileo s Experiment Kepler, Newton and Gravity 26

27 Galileo s Experiment Kepler, Newton and Gravity 27

28 Galileo inertia: natural tendency for a body in motion to remain in motion or natural tendency for a body at rest to remain at rest Kepler, Newton and Gravity 28

29 Galileo also experimented with falling bodies falling is NOT a natural motion motion due to the force of gravity acceleration due to gravity constant velocity changes at a constant rate at earth s surface g = 9.8 m/s s We can use 10 m/s/s Kepler, Newton and Gravity 29

30 DROPPED ROCK How fast is it going after 1 second of falling? How fast is it going after 3 seconds of falling? Starts at 0 m/s 10 m/s 30 m/s How fast is it going after 5 seconds of falling? 50 m/s Kepler, Newton and Gravity 30

31 PARACHUTIST Ignore air resistance How fast is she going after 1 second of falling? How fast is she going after 3 seconds of falling? Starts at 0 m/s 10 m/s 30 m/s How fast is she going after 5 seconds of falling? 50 m/s Kepler, Newton and Gravity 31

32 ALL falling bodies fall with the SAME acceleration!! Kepler, Newton and Gravity 32

33 Newton 1600 A.D. Principia - Physics of motion and Concept of Gravitation Concept of Force force produces an acceleration acceleration is in the same direction as the force Kepler, Newton and Gravity 33

34 Increase the force => greater acceleration => object reaches a greater velocity What kind of relationship is this? ACCELERATION is directly proportional to FORCE Kepler, Newton and Gravity 34

35 Increase the MASS apply the original force => less acceleration What kind of relationship is this? ACCELERATION is inversely proportional to MASS Kepler, Newton and Gravity 35

36 Can you come up with a Pot O Gold type relationship for acceleration, force & mass? a = F m Forces have direction (same direction as the acceleration) Kepler, Newton and Gravity 36

37 Newton clarified these definitions: mass : the measure of an object s resistance to a change in motion velocity : how fast an object moves in a particular direction acceleration : how much the velocity (or direction) changes with time Kepler, Newton and Gravity 37

38 Newton s Three Laws of Motion INERTIAL LAW A body at rest remains at rest unless acted on by an outside force. A body in motion at a constant velocity along a straight line remains in motion unless acted on by an outside force. Kepler, Newton and Gravity 38

39 Newton s Three Laws of Motion INERTIAL LAW Implication: If we see an acceleration, we know there s a net force acting on the body in question (that is, change in speed direction, or both) Kepler, Newton and Gravity 39

40 Newton s Three Laws of Motion FORCE LAW rate of change in a body s velocity, due to an applied force (in other words, a body s acceleration) is in the same direction as the force proportional to the force inversely proportional to its mass Kepler, Newton and Gravity 40

41 Newton s Three Laws of Motion FORCE LAW Implication: can apply a force, measure acceleration and infer the mass of an object Kepler, Newton and Gravity 41

42 Newton s Three Laws of Motion REACTION LAW For every applied force, a force of equal size by opposite direction arises. Implication: forces act in pairs Kepler, Newton and Gravity 42

43 AN EXAMPLE You are an astronaut out in space. You push on your space capsule. What is the equal and opposite force? Spaceship pushes back on you. Since the forces are the SAME, exactly what is different? Kepler, Newton and Gravity 43

44 You Spaceship F F Kepler, Newton and Gravity 44

45 You Spaceship F m F M Kepler, Newton and Gravity 45

46 You Spaceship F m A F M a Kepler, Newton and Gravity 46

47 You Spaceship F m A F M a Which has the greater velocity? Kepler, Newton and Gravity 47

48 NEWTON S GREAT INSIGHT Gravitation is an interaction between two (or more) bodies, such as the Sun and the planets. Kepler, Newton and Gravity 48

49 Newton s Law of Gravitation What direction does the force of gravity act? (nature of the force) What is the amount of force? (physical properties that determine the strength of the force) Kepler, Newton and Gravity 49

50 LAWS OF MOTION + Kepler s Planetary Laws Law of Universal Gravitation Kepler, Newton and Gravity 50

51 GRAVITATION central force : type of force that causes elliptical orbits; force directed towards the center of motion planets moving in orbits under the influence of a central force followed Kepler s Second Law (Law of Areas) from the geometric properties of ellipses, force is described by a specific type of force law => re-derived Kepler s Third Law using this force law Newton s Laws and Kepler s Laws were in total agreement Kepler, Newton and Gravity 51

52 Consider the moon in orbit: Direction - changing? or not? curved path => changing Newton s First Law: direction is changing there is an acceleration Therefore, there is a force acting on the moon Kepler, Newton and Gravity 52

53 Force is directed towards the center of the Earth. We call a center-directed force a centripetal force. At every point in its orbit, a centripetal force acts on the moon to keep it bound to Earth. Kepler, Newton and Gravity 53

54 Billiard Ball Analogy Ball makes a collision with side Ball changes direction Ball accelerates from applied force of the wall Force is directed towards the center of the table Kepler, Newton and Gravity 54

55 Pentagonal Table Force still points to the center Angle of strike and rebound is smaller Kepler, Newton and Gravity 55

56 Hexagonal Table Force still points to the center Angle of strike and rebound is even smaller Kepler, Newton and Gravity 56

57 Circular Table Force points to center at every point Infinite number of sides, angle of strike and rebound is zero Kepler, Newton and Gravity 57

58 Moon in its circular orbit: at every point in its orbit, a centripetal force acts on the moon to keep it bound to Earth. GRAVITY!! Kepler, Newton and Gravity 58

59 Law of Gravitation Every body in the Universe attracts every other body with a gravitational force. Kepler, Newton and Gravity 59

60 Consider two objects only amount of gravitational force depends DIRECTLY on the amount of material each object has (mass) What would happen if you doubled the mass of one of the objects? Kepler, Newton and Gravity 60

61 The force between them was F. After the pink object doubles, the gravitational force is 2 times F or 2F TWICE AS MUCH Kepler, Newton and Gravity 61

62 Now what happens if we double the mass of the other object? The force between them was 2F After the purple object doubles, the gravitational force is 2 times 2F or 4F 4 TIMES the original amount of force Kepler, Newton and Gravity 62

63 To express that in algebra: F g is directly proportional to m purple x mass pink or F g m 1 x m 2 Kepler, Newton and Gravity 63

64 objects at greater separations have less gravitational force between them than those closer together 1 m When the distance is 1 meter, the force between them is F. Decrease of force with distance happens in a special way Force is inversely proportional to the square of the distance. Kepler, Newton and Gravity 64

65 Start with the objects 1 meter apart. What happens if we move them to 2 meters apart? (that is, we double the distance) 1 m 2 m Kepler, Newton and Gravity 65

66 2 m The force, when they were 1 meter apart, was F - now at 2 meters, the force is Less is it 1/2 as much? is it 1/4 as much? Force is proportional to 1 Force is 1 4 as much as it was. (distance) 2 Kepler, Newton and Gravity 66

67 1 m 3 m What happens to the force if we move them 3 meters apart? Force is then 1/9 as much as it was Kepler, Newton and Gravity 67

68 F is proportional to m 1 F is proportional to m 2 Force is proportional to 1 (distance) 2 Force m 1 m 2 (distance) 2 or m 1 m 2 R 2 Kepler, Newton and Gravity 68

69 Constant of proportionality is G - Universal Gravitational Constant G = 6.67 x Force = G m 1 m 2 R 2 Force (newtons) masses (kgs) R (meters) Kepler, Newton and Gravity 69

70 Law of Gravitation Fgravity = G m 1 m 2 R 2 Interpretation: Force due to gravity is directly proportional to the masses of the objects involved. Kepler, Newton and Gravity 70

71 Law of Gravitation Fgravity = G m 1 m 2 R 2 1 over R 2 Law Interpretation: Force due to gravity is inversely proportional to the distance between them squared. Kepler, Newton and Gravity 71

72 Newton worked this out mathematically (it s a model) - how did he test this? 1 R e 60 R e Moon is 60 times farther from the center of the Earth than the apple is. How much less gravitational force is felt at the location of the Moon? Kepler, Newton and Gravity 72

73 Force at Moon is 1 (60) 2 = Newton s mathematical model predicted the force (acceleration) should be 3600 times less. Comparing to measured observations of the Moon in its orbit, this was pretty nearly the same. Kepler, Newton and Gravity 73

74 Newton s Concept Extension Earth s gravity keeps the moon swinging around the Earth Sun s gravity keeps the planets swinging around the Sun Kepler, Newton and Gravity 74

75 Newton Revised Kepler s Third Law p 2 = k a 3 Kepler, Newton and Gravity 75

76 Newton Revised Kepler s Third Law p 2 = k a 3 p 2 = 4 2 a 3 G(M sun + m planet ) Kepler, Newton and Gravity 76

77 Kepler s Third Law Revised TREMENDOUSLY IMPORTANT!! p 2 = 4 2 a 3 G(M sun + m planet ) Can use this to find the mass of the Sun!! Kepler, Newton and Gravity 77

78 p 2 = 4 2 a 3 G(M sun + m planet ) measure the period of the Earth s orbit measure the average distance from the Sun approximate M sun + m earth» M sun look up the value of G calculate the mass of the Sun!! Kepler, Newton and Gravity 78

79 We can find the mass of the Sun using any body orbiting the Sun. We can find the mass of the Earth using any body orbiting the Earth. Kepler, Newton and Gravity 79

80 What would you do to find the mass of a planet that has no orbiting body? Kepler, Newton and Gravity 80

81 Center of Mass center of mass : balance point of a group of objects Where s the balance point? Kepler, Newton and Gravity 81

82 1 AU (150 million km) Earth A mere 500 km from the center of the Sun!!! What s your best guess as to the location of the center of mass between the Earth and the Sun? Kepler, Newton and Gravity 82

83 5 AU (750 million km) Jupiter Just barely outside the surface of the Sun What s your best guess as to the location of the center of mass between Jupiter and the Sun? Kepler, Newton and Gravity 83

84 Newton s Successful Predictions return of Halley s comet discovery of planet Neptune binary star systems follow Kepler s Laws Kepler, Newton and Gravity 84

85 Newton s Accomplishments found the physical interaction between the Sun and the planets revised Kepler s Third Law so that it became a tool for calculating masses of distant objects answered the question of how planets move made predictions far more accurate than any before physical support for the heliocentric model Kepler, Newton and Gravity 85

86 Newton s Law of Gravitation Finally, a universal law, that is, one that is the same for the heavens and the Earth. Kepler, Newton and Gravity 86

87 Escape Speed: minimum speed an object must have to break free of gravity 11 km/sec escape speed from Earth Kepler, Newton and Gravity 87

88 Mass and radius of planet determines the escape speed. v escape = 2 Gm R Kepler, Newton and Gravity 88

89 Newton s Cosmology cosmology : study of the origin, the nature and the evolution of the Universe Universe is infinite in extent If it were not infinite, but finite, gravitation would eventually pull all the matter in the Universe back together => one large mass! Kepler, Newton and Gravity 89

90 In an infinite universe, there would be an infinite number of small blobs of matter. => exactly the universe we see! Kepler, Newton and Gravity 90

91 One Small Teeny, Tiny Problem Orbit of Mercury had an unexplained wobble 41 arcsec per century Could not be explained with Newton s physics Hypothesis of a planet, hidden behind the Sun, never proved to be true (Vulcan) Kepler, Newton and Gravity 91

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