1. Universal Gravitation
Chapter 8
Universal Gravitation

2. Drill:
Calculate ac & Fc of a 25 kg ball rotation at the end of a 4.0 m rope at revolutions per second.

3. Planetary Motion
Galileo
Tycho Brahe
Johannes Kepler
Isaac Newton

4. Kepler’s Laws of Planetary Motion

5. 1) The paths of planets are ellipses, with the sun as one focus

6. 2) An imaginary line from the sun to any planet sweeps out equal areas in equal time intervals. Planets move faster when closer to the sun

7. 3) The square of the ratio of the periods of any two planets = the cube of the ratio of their orbits average radii

8. ( )2=( )3
TA rA
TB rB

9. Drill: Planet. Orbital. Period Name. Rad. (km). (s) Mo 2. 0 x 108. 4
Drill: Planet Orbital Period Name Rad. (km) (s) Mo x x 106 Smit x ? Solve for ?

10. Drill: Planet. Orbital. Period Name. Rad. (km). (s) Smo 2. 0 x 108. 4
Drill: Planet Orbital Period Name Rad. (km) (s) Smo x x 106 Mit ? x 109 Solve for ?

11. Universal Gravitation
Isaac Newton
Henry Cavendish
Michael Faraday
Albert Einstein

12. An attractive force that exist between all objects
Gravitational Force
An attractive force that exist between all objects

13. Found to directly proportioned to the masses of the two objects
Gravitational Force
Found to directly proportioned to the masses of the two objects

14. Gravitational Force
Fg  m

15. Found to inversely proportioned to the distance2 between two objects
Gravitational Force
Found to inversely proportioned to the distance2 between two objects

16. Gravitational Force 1 d2
Fg 

17. Universal Law
of Gravitation
mAmB d2
Fg = G

18. Gravitational Constant
6.67 x Nm2/kg2

19. Calculate the gravitational force between 5. 0 Mg & 6
Calculate the gravitational force between 5.0 Mg & 6.0 kg objects whose centers are 3.0 mm apart:

20. The gravitational force between 6. 0 Mg & 50. 0 kg objects is 2
The gravitational force between 6.0 Mg & 50.0 kg objects is 2.22 x 10-2 N. Calculate the distance between them.

21. mEmo d2
Fg = G
Fg = mog

22. Thus:
mEmo d2
mog = G

23. mEmo d2
mog = G
d = r, thus

24. GmE r2
g =

25. GmE r2
g = or

26. Mass of the Earth
gr2 G
mE =

27. Drill: rE = 6.37 x 103 km Solve for the mass of the Earth:

28. Centripetal Force
m4p2r T2
Fc =

29. Fg = Fc
msmp mp4p2r
r T2
G =

30. msmp mp4p2r
r T2
G =

31. ms p2r
r T2
G =

32. T2ms p2r3
G =

33. Therefore
( )
4p2
Gms
T2 = r3

34. Kepler’s Constant for Solar System
( )
4p2
Gms
k =

35. Earth’s mean orbital radius is 1. 50 x 107 km while its period 365
Earth’s mean orbital radius is 1.50 x 107 km while its period days. Calculate the Sun’s orbital constant in m3/s2:

36. The moon’s mean orbital radius is 4
The moon’s mean orbital radius is 4.00 x 105 km while its period 28 days. Calculate the Earth’s orbital constant in m3/s2:

37. or
( )
4p2
Gms
T2 = r3

38. Mass of the Sun
4p2r3
GT2
ms =

39. mEmo d2
Fg = G
Fc = mov2/r

40. Thus:
mEmo mov2
r r
G =

41. mEmo mov2
r r
G =
Thus:

42. mE v2 r
G =
Take sq. rt. of both sides

43. Orbital Velocity
GmE r
v =

44. v =
(g)r

45. Calculate the velocity of a object orbiting at 50. 0 m around a 5
Calculate the velocity of a object orbiting at 50.0 m around a 5.0 x 106 Mg object

46. r3
GmE
Ts= 2p

47. Calculate the period of a object orbiting at 50. 0 m around a 5
Calculate the period of a object orbiting at 50.0 m around a 5.0 x 106 Mg object

48. ( ) rE d
a = g

49. Calculate the velocity of a satelite orbiting at 620 km above the Earth’s surface:

50. The space in which the force of gravity is apparent
Gravitational Field
The space in which the force of gravity is apparent

51. Measure of Mass
Use a balance

52. Inertial Mass
minertial = Fnet/a

53. Gravitational Mass
Fgr2
GmE
mg =

54. Warped Space ?

55. Review

56. List Kepler’s Laws of Planetary Motion

57. What is the formula for Kepler’s 3rd Law

58. Planet. Orbital. Period Name. Radius (km). (s) Two 2. 0 x 108. 4
Planet Orbital Period Name Radius (km) (s) Two x x 106 Twit ? x 109 Solve for ?

59. What is the formula for the Universal Law of Gravitation

60. The masses of Earth & moon are 5. 98 x 1024 kg & 1
The masses of Earth & moon are 5.98 x 1024 kg & 1.0 x 1024 kg respectively. The average radius of the moon’s orbit is 4.0 x 106 km. Calculate the force of gravity between the Earth & the moon.

61. Orbital Velocity
GmE r
v =

62. Calculate the velocity of the moon.

63. Calculate the ac & Fc of a merry-go-round with a radius of 50
Calculate the ac & Fc of a merry-go-round with a radius of 50.0 m spinning at 1 revolution every 4 seconds.

64. A car is driven off a 2. 0 km cliff at 180 km /hr
A car is driven off a 2.0 km cliff at 180 km /hr. Calculate: tair, maximum vV, & dH:

65. A catapult launches a 250 kg rock at 100. 0 m/s at 37o from horizontal
A catapult launches a 250 kg rock at m/s at 37o from horizontal. Calculate: tup, dV, tair, & dH :

66. Drill:
A 3200 kg carousel with a diameter of 20.0 m is spinning at 1 revolution every 5 seconds.
Calculate: ac & Fc

67. Important Formulas

68. ( )2=( )3
Orbital Formula
TA rA
TB rB

69. Universal Law
of Gravitation
mAmB d2
Fg = G

70. Orbital Velocity
GmE r
v =

71. Orbital Velocity
v =
(g)r