Chapter 13 Newton s Theory of Gravity

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1 Chapter 13 Newton s Theory of Gravity Chapter Goal: To use Newton s theory of gravity to understand the motion of satellites and planets. Slide 13-2

2 Chapter 13 Preview Slide 13-3

3 Chapter 13 Preview Slide 13-4

4 Chapter 13 Preview Slide 13-5

5 Chapter 13 Preview Slide 13-6

6 Chapter 13 Preview Slide 13-7

7 Chapter 13 Preview Slide 13-8

8 A Little History: Kepler s Laws From 1570 to 1600, Tycho Brahe compiled observations of stars and planets, and noted how the planets Mercury, Venus, Mars, Jupiter, and Saturn move through the sky. His mathematical assistant, Johannes Kepler, analyzed Brahe s data and discovered three laws that the planets obey as they orbit the sun. Slide 13-19

9 Kepler s First Law of Planetary Motion 1. Planets move in elliptical orbits, with the sun at one focus of the ellipse. Slide 13-20

10 Kepler s Second Law of Planetary Motion 2. A line drawn between the sun and a planet sweeps out equal areas during equal intervals of time. Slide 13-21

11 Kepler s Third Law of Planetary Motion 3. The square of a planet s orbital period is proportional to the cube of the semimajor-axis length. Slide 13-22

12 A Kepler s circular orbit Laws is a of special Planetary case of Motion an elliptical orbit. Slide 13-23

13 Isaac Newton Isaac Newton, Legend has it that Newton saw an apple fall from a tree, and it occurred to him that the apple was attracted to the center of the earth. If the apple was so attracted, why not the moon? Newton posited that gravity is a universal attractive force between all objects in the universe. Slide 13-24

14 Newton s Law of Gravity The moon is in free fall around the earth. Slide 13-25

15 Newton s Law of Gravity The gravitational constant is a universal constant with the value G = N m 2 /kg 2. Slide 13-26

16 Newton s Law of Gravity The gravitational force is an inverse-square force. Slide 13-27

17 Newton s Law of Gravity Since G is so small, it means that the attractive force between two 1.0 kg masses, whose centers are 1.0 m apart, is N. This is 100 billion times weaker than the force of gravity from the earth on either of the masses! Although weak, gravity is a long-range force. Gravity keeps the earth orbiting the sun and the solar system orbiting the center of the Milky Way galaxy. A galaxy of stars spanning a distance greater than 100,000 light years. Slide 13-28

18 The Principle of Equivalence The inertial mass of an object relates the net force acting on it to its acceleration: The gravitational mass of an object appears in Newton s law of gravity and determines the strength of the gravitational attraction: The assertion that m grav = m inert is called the principle of equivalence. Slide 13-35

19 Little g and Big G An object of mass m sits on the surface of Planet X. According to an observer on the planet, the gravitational force on m should be F G = mg surface. According to Newton s law of gravity, the gravitational force on m should be. These are the same if: Slide 13-36

20 Decrease of g with Distance A satellite orbits the earth at height h above the earth s surface. The value of g at height h above sea level is: where g earth = 9.83 m/s 2 and R e = m. Slide 13-39

21 Decrease of g with Distance Slide 13-40

22 Gravitational Potential Energy When two isolated masses m 1 and m 2 interact over large distances, they have a gravitational potential energy of: where we have chosen the zero point of potential energy at r =, where the masses will have no tendency, or potential, to move together. Note that this equation gives the potential energy of masses m 1 and m 2 when their centers are separated by a distance r. Slide 13-43

23 Gravitational Potential Energy Slide 13-44

24 Gravitational Potential Energy Suppose two masses a distance r 1 apart are released from rest. How will the small mass move as r decreases from r 1 to r 2? At r 1 U is negative. At r 2 U is larger and U is still negative, meaning that U has decreased. As the system loses potential energy, it gains kinetic energy while conserving E mech. The smaller mass speeds up as it falls. Slide 13-45

25 Example 13.2 Escape Speed Slide 13-48

26 Example 13.2 Escape Speed VISUALIZE Slide 13-49

27 Example 13.2 Escape Speed Slide 13-50

28 Example 13.2 Escape Speed Slide 13-51

29 Gravitational Potential Energy Consider a mass a distance y above the surface of the earth. If y is very small compared to the radius of the earth, we can use the binomial approximation to show: Set U 0 = 0 and this is our familiar formula for U g. Slide 13-52

30 Example 13.3 The Speed of a Satellite Slide 13-53

31 Example 13.3 The Speed of a Satellite VISUALIZE Slide 13-54

32 Example 13.3 The Speed of a Satellite Slide 13-55

33 Example 13.3 The Speed of a Satellite Slide 13-56

34 Satellite Orbits A circle is a special case of an ellipse. If a small mass m orbits a much larger mass M, the small mass is called a satellite. The speed of a satellite in a circular orbit is: Slide 13-57

35 Example 13.4 The Speed of the Space Shuttle Slide 13-62

36 Recall: Kepler s Third Law of Planetary Motion 3. The square of a planet s orbital period is proportional to the cube of the semimajor-axis length. The speed of a satellite in a circular orbit is: Squaring both sides and solving for T 2 gives: Planets are satellites of the sun, in orbits that are almost circular. The orbital radius = the semimajor-axis length for a circle. Slide 13-63

37 Useful Astronomical Data Slide 13-64

38 Example 13.5 Extrasolar Planets Slide 13-65

39 Recall: Kepler s Second Law of Planetary Motion 2. A line drawn between the sun and a planet sweeps out equal areas during equal intervals of time. The gravitational force exerts no torque, so the satellite s angular momentum, L, is conserved as it orbits. It can be shown that the area swept out per unit time is L/2m, which is constant for any satellite. Slide 13-66

40 Orbital Energetics We know that for a satellite in a circular orbit, its speed is related to the size of its orbit by v 2 = GM/r. The satellite s kinetic energy is thus: But GMm/r is the potential energy, U g, so: If K and U do not have this relationship, then the trajectory will be elliptical rather than circular. So, the mechanical energy of a satellite in a circular orbit is always: Slide 13-69

41 Orbital Energetics The figure shows the kinetic, potential, and total energy of a satellite in a circular orbit. Notice how, for a circular orbit, E mech = U g /2. It requires positive energy in order to lift a satellite into a higher orbit. Slide 13-70

42 Example 13.6 Raising a Satellite Slide 13-71

43 Orbital Energetics The figure shows the steps involved to lift a satellite to a higher circular orbit. The first kick increases K without increasing U g, so K is not U g /2, and the orbit is elliptical. The satellite then slows down as r increases. The second kick increases K again so that K = U g /2, and the orbit is circular. Slide 13-72

44 Chapter 13 Summary Slides Slide 13-73

45 General Principles Slide 13-74

46 Important Concepts Slide 13-75

47 Important Concepts Slide 13-76

48 Important Concepts Slide 13-77

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