Chapter 12

1. Chapter 12 Gravitation

2. Theories of Gravity Newton’s Einstein’s

3. Newton’s Law of Gravitation • any two particles m1, m2 attract each other • Fg= magnitude of their mutual gravitational force

4. Newton’s Law of Gravitation • Fg = magnitude of the force that: • m1 exerts on m2 • m2 exerts on m1 • direction: along the line joining m1, m2

5. Newton’s Law of Gravitation • Also holds if m1, m2 are two bodies with spherically symmetric mass distributions • r = distance between their centers

6. Newton’s Law of Gravitation • ‘universal’ law: • G = fundamental constant of nature • careful measurements: G=6.67×10-11Nm2/kg2

7. How Cavendish Measured G(in 1798, without a laser)

8. 1798: small masses (blue) on a rod gravitate towards larger masses (red), so the fiber twists

9. 2002: we can measure the twist by reflecting laser light off a mirror attached to the fiber

10. If a scale calibrates twist with known forces, we can measure gravitational forces, hence G

11. What about g = 9.8 m/s2 ? • How is G related to g? • Answer: Show that g = GmE/RE2

12. Other planets, moons, etc? • gp=acceleration due to gravity at planet’s surface • density assumed spherically symmetric, but not necessarily uniform

13. Example:Earth’s density is not uniform

14. Yet to an observer (m) outside the Earth, its mass (mE) acts as if concentrated at the center

15. Newton’s Law of Gravitation • This is the magnitude of the force that: • m1 exerts on m2 • m2 exerts on m1 • What if other particles are present?

16. Superposition Principle • the gravitational force is a vector • so the gravitational force on a body m due to other bodies m1 , m2 , ... is the vector sum: Do Exercise 12-8 Do Exercise 12-6

17. Superposition Principle • Example 12-3 • The total gravitationalforce on the mass at Ois the vector sum: Do some of Example 12-3 and introduce Extra Credit Problem 12-42

18. Gravitational Potential Energy, U • This follows from: Derive U = - Gm1m2/r

19. Gravitational Potential Energy, U • Alternatively: a radial conservative force has a potential energy U given by F = – dU/dr

20. Gravitational Potential Energy, U • U is shared between both m1 and m2 • We can’t divide up U between them Example: Find U for the Earth-moon system

21. Superposition Principle for U • For many particles,U = total sharedpotential energy of the system • U = sum of potentialenergies of all pairs Write out U for this example

22. Total Energy, E • If gravity is force is the only force acting, the total energy E is conserved • For two particles,

23. Application: Escape Speed • projectile: m • Earth: mE • Find the speed that m needs to escape from the Earth’s surface Derive the escape speed: Example 12-5

24. Orbits of Satellites

25. Orbits of Satellites • We treat the Earth as a point mass mE • Launch satellite m at A with speed v toward B • Different initial speeds v give different orbits, for example (1) – (7)

26. Orbits of Satellites • Two of Newton’s Laws predict the shapes of orbits: • 2nd Law • Law of Gravitation

27. Orbits of Satellites • Actually: • Both the satellite and the point C orbit about their common CM • We neglect the motion of point C since it very nearly is their CM

28. Orbits of Satellites • If you solve the differential equations, you find the possible orbit shapes are: • (1) – (5): ellipses • (4): circle • (6): parabola • (7): hyperbola

29. Orbits of Satellites • (1) – (5): closed orbits • (6) , (7): open orbits • What determines whether an orbit is open or closed? • Answer: escape speed

30. Escape Speed • Last time we launched m from Earth’s surface (r = RE) • We set E = 0 to find

31. Escape Speed • We could also launch m from point A(any r > RE) • so use r instead of RE :

32. Orbits of Satellites • (1) – (5): ellipses • launch speed v < vesc • (6): parabola • launch speed v = vesc • (7): hyperbola • launch speed v > vesc

33. Orbits of Satellites • (1) – (5): ellipses • energy E < 0 • (6): parabola • energy E = 0 • (7): hyperbola • energy E > 0

34. Circular Orbits

35. Derive speed v Circular Orbit: Speed v • uniform motion • independent of m • determined by radius r • large r means slow v

36. Compare to Escape Speed • If you increase your speed by factor of 21/2 you can escape!

37. Derive period T Circular Orbit: Period T • independent of m • determined by radius r • large r means long T Do Problem 12-45

38. Derive energy E Circular Orbit: Energy E • depends on m • depends on radius r • large r means large E

39. Orbits of Planets

40. Same Math as for Satellites • Same possible orbits, we just replace the Earth mE with sun ms • (1) – (5): ellipses • (4): circle • (6): parabola • (7): hyperbola

41. Orbits of Planets • Two of Newton’s Laws predict the shapes of orbits: • 2nd Law • Law of Gravitation • This derives Kepler’s Three Empirical Laws

42. Kepler’s Three Laws • planet orbit = ellipse(with sun at one focus) • Each planet-sun line sweeps out ‘equal areas in equal times’ • For all planet orbits, a3/T2 = constant

43. Kepler’s First Law • planet orbit = ellipse • P = planet • S = focus (sun) • S’ = focus (math) • a = semi-major axis • e = eccentricity0 < e < 1e = 0 for a circle Do Problem 12-64

44. Kepler’s Second Law • Each planet-sun line sweeps out ‘equal areas in equal times’

45. Kepler’s Second Law Present some notes on Kepler’s Second Law

46. Kepler’s Third Law • We proved this for a circular orbit (e = 0) • T depends on a, not e

47. Kepler’s Third Law • Actually: • Since both the sun and the planet orbit about their common CM

48. Theories of Gravity Newton’s Einstein’s

49. Einstein’s Special Relativity • all inertial observers measure the same value c = 3.0×108 m/s2 for the speed of light • nothing can travel faster than light • ‘special’ means ‘not general’: • spacetime (= space + time) is flat

50. Einstein’s General Relativity • nothing can travel faster than light • but spacetime is curved, not flat • matter curves spacetime • if the matter is dense enough, then a ‘black hole’ forms