1. Universal Gravitation
Chin-Sung Lin

2. Isaac Newton

3. Newton & Physics

4. Universal Gravitation
Newton’s Law of Universal Gravitation states that gravity is an attractive force acting between all pairs of massive objects.
Gravity depends on:
Masses of the two objects
Distance between the objects

5. Universal Gravitation - Apple

6. Universal Gravitation - Moon

7. Universal Gravitation - Moon

8. Universal Gravitation
Newton’s question:
Can gravity be the force keeping the Moon in its orbit?
Newton’s approximation: Moon is on a circular orbit
Even if its orbit were perfectly circular, the Moon would still be accelerated

9. The Moon’s Orbital Speed
radius of orbit: r = 3.8 x 108 m
Circumference: 2pr = ???? m
orbital period: T = 27.3 days = ???? sec
orbital speed: v = (2pr)/T = ??? m/sec

10. The Moon’s Orbital Speed
radius of orbit: r = 3.8 x 108 m
Circumference: 2pr = 2.4 x 109 m
orbital period: T = 27.3 days = 2.4 x 106 sec
orbital speed: v = (2pr)/T = 103 m/sec = 1 km/s

11. The Moon’s Centripetal Acceleration
The centripetal acceleration of the moon: orbital speed: v = 103 m/s orbital radius: r = 3.8 x 108 m centripetal acceleration: Ac = v2 / r = ???? m/s2

12. The Moon’s Centripetal Acceleration
The centripetal acceleration of the moon: orbital speed: v = 103 m/s orbital radius: r = 3.8 x 108 m centripetal acceleration: Ac = v2 / r Ac = (103 m/s)2 / (3.8 x 108 m) = m/s2

13. The Moon’s Centripetal Acceleration
At the surface of Earth (r = radius of Earth)
a = 9.8 m/s2
At the orbit of the Moon (r = 60x radius of Earth)
a = m/s2
What’s relation between them?

14. The Moon’s Centripetal Acceleration
At the surface of Earth (r = radius of Earth)
a = 9.8 m/s2
At the orbit of the Moon (r = 60x radius of Earth)
a = m/s2
9.8 m/s2 / m/s = / 1 = / 1

15. Bottom Line The Moon’s Centripetal Acceleration r 2r 3r 4r 5r 6r 60r g1 4 9 16 25 36
3600
Figure 5-2 The force on a 0.1-kilogram mass at various distances from Earth. Notice that the force decreases as the square of the distance.

16. Bottom Line The Moon’s Centripetal Acceleration
If the acceleration due to gravity is inverse proportional to the square of the distance, then it provides the right acceleration to keep the Moon on its orbit (“to keep it falling”)

17. Bottom Line The Moon’s Centripetal Acceleration
If the acceleration due to gravity is inverse proportional to the square of the distance, then it provides the right acceleration to keep the Moon on its orbit (“to keep it falling”)
The moon is falling as the apple does
!!! Triumph for Newton !!!

18. Bottom Line Ac ~ 1/r2 Gravity’s Inverse Square Law
The acceleration due to gravity is inverse proportional to the square of the distance
Ac ~ 1/r2
The gravity is inverse proportional to the square of the distance
Fg = Fc = m Ac Fg ~Ac Fg ~ 1/r2

19. Bottom Line Fg ~ 1/r2 Gravity’s Inverse Square Law
Gravity is reduced as the inverse square of its distance from its source increased
Fg ~ 1/r2 r 2r 3r 4r 5r 6r
60r Fg 1 4 9 16 25 36
3600

20. Bottom Line Fg ~ 1/r2 Gravity’s Inverse Square Law
Figure 5-1 If the sphere’s radius is doubled, the sphere’s surface increases by a factor of 4.

21. Gravity’s Inverse Square Law
Bottom Line

22. Bottom Line Gravity’s Inverse Square Law
Gravity decreases with altitude, since greater altitude means greater distance from the Earth's centre
If all other things being equal, on the top of Mount Everest (8,850 metres), weight decreases about 0.28%

23. Bottom Line Gravity’s Inverse Square Law
Astronauts in orbit are NOT weightless
At an altitude of 400 km, a typical orbit of the Space Shuttle, gravity is still nearly 90% as strong as at the Earth's surface

24. Bottom Line Gravity’s Inverse Square Law Earth's surface 6.38 x 106 m
Location
Distance from
Earth's center (m)
Value of g
(m/s2)
Earth's surface
6.38 x 106 m
9.8
1000 km above
7.38 x 106 m
7.33
2000 km above
8.38 x 106 m
5.68
3000 km above
9.38 x 106 m
4.53
4000 km above
1.04 x 107 m
3.70
5000 km above
1.14 x 107 m
3.08
6000 km above
1.24 x 107 m
2.60
7000 km above
1.34 x 107 m
2.23
8000 km above
1.44 x 107 m
1.93
9000 km above
1.54 x 107 m
1.69
10000 km above
1.64 x 107 m
1.49
50000 km above
5.64 x 107 m
0.13

25. Bottom Line Law of Universal Gravitation Newton’s discovery
Newton didn’t discover gravity. In stead, he discovered that the gravity is universal
Everything pulls everything in a beautifully simple way that involves only mass and distance
Figure 5-1 If the sphere’s radius is doubled, the sphere’s surface increases by a factor of 4.

26. Bottom Line Fg = G m1 m2 / d2 Law of Universal Gravitation
Universal gravitation formula
Fg = G m1 m2 / d2
Fg: gravitational force between objects
G: universal gravitational constant
m1: mass of one object
m2: mass of the other object
d: distance between their centers of mass
Figure 5-1 If the sphere’s radius is doubled, the sphere’s surface increases by a factor of 4.

27. Bottom Line Law of Universal Gravitation d Fg m2 m1 p.83
p. 83: The attraction between these spheroidal friends depends on the distance between their centers.

28. Bottom Line Fg = G m1 m2 / d2 Law of Universal Gravitation
Gravity is always there
Though the gravity decreases rapidly with the distance, it never drop to zero
The gravitational influence of every object, however small or far, is exerted through all space
Figure 5-1 If the sphere’s radius is doubled, the sphere’s surface increases by a factor of 4.

29. Bottom Line Law of Universal Gravitation Example m1 m2 d F 2m1 3m2 2d
Mass 1
Mass 2
Distance
Relative Force m1 m2 d F
2m1
3m2 2d 3d
2m2
Figure 5-1 If the sphere’s radius is doubled, the sphere’s surface increases by a factor of 4.

30. Law of Universal Gravitation Example
Mass 1
Mass 2
Distance
Relative Force m1 m2 d F
2m1 2F
3m2 3F 6F 2d
F/4 3d
F/9
2m2
Figure 5-1 If the sphere’s radius is doubled, the sphere’s surface increases by a factor of 4.

31. Universal Gravitational Constant
The Universal Gravitational Constant (G) was first measured by Henry Cavendish 150 years after Newton’s discovery of universal gravitation

32. Henry Cavendish

33. Universal Gravitational Constant
Cavendish’s experiment
Use Torsion balance (Metal thread, 6-foot wooden rod and 2” diameter lead sphere)
Two 12”, 350 lb lead spheres
The reason why Cavendish measuring the G is to “Weight the Earth”
The measurement is accurate to 1% and his data was lasting for a century

34. Cavendish’s Experiment

35. Universal Gravitational Constant
G = Fg d2 / m1 m2 = 6.67 x N·m2/kg2
Fg = G m1 m2 / d2

36. Calculate the Mass of Earth
G = 6.67 x N·m2/kg2
Fg = G M m / r2
The force (Fg) that Earth exerts on a mass (m) of 1 kg at its surface is 9.8 newtons
The distance between the 1-kg mass and the center of Earth is Earth’s radius (r), 6.4 x 106 m

37. Calculate the Mass of Earth
G = 6.67 x N·m2/kg2
Fg = G M m / r2
9.8 N = 6.67 x N·m2/kg2 x 1 kg x M / (6.4 x 106 m)2
where M is the mass of Earth M = 6 x 1024 kg

38. Gravitational force is a
Universal Gravitational Force
G = x N·m2/kg2
Fg = G m1 m2 / d2
Gravitational force is a
VERY WEAK FORCE

39. Universal Gravitational Force
G = 6.67 x N·m2/kg2
Gravity is is the weakest of the presently known four fundamental forces

40. ∞ Universal Gravitational Force Force Strong Electro-magnetic Weak
Gravity
Strength 1
1/137
10-6
6x10-39
Range
10-15 m ∞
10-18 m

41. Universal Gravitation Example
Calculate the force of gravity between two students with mass 55 kg and 45kg, and they are 1 meter away from each other

42. Universal Gravitation Example
Calculate the force of gravity between two students with mass 55 kg and 45kg, and they are 1 meter away from each other
Fg = G m1 m2 / d2
Fg = (6.67 x N·m2/kg2)(55 kg)(45 kg)/(1 m)2
= 1.65 x 10-7 N

43. Universal Gravitation Example
Calculate the force of gravity between Earth (mass = 6.0 x 1024 kg) and the moon (mass = 7.4 x kg). The Earth-moon distance is 3.8 x 108 m

44. Universal Gravitation Example
Calculate the force of gravity between Earth (mass = 6.0 x 1024 kg) and the moon (mass = 7.4 x kg). The Earth-moon distance is 3.8 x 108 m
Fg = G m1 m2 / d2
Fg = (6.67 x N·m2/kg2)(6.0 x 1024 kg)
(7.4 x 1022 kg)/(3.8 x 108 m)2
= 2.1 x 1020 N

45. Acceleration Due to Gravity
Law of Universal Gravitation:
Fg = G m M / r2
Weight
Fg = m g
Acceleration due to gravity
g = G M / r2
Fg: gravitational force / weight
G: univ. gravitational constant
M: mass of Earth
m: mass of the object
r: radius of Earth
g: acceleration due to gravity

46. Universal Gravitation Example
Calculate the acceleration due to gravity of Earth (mass = 6.0 x 1024 kg, radius = 6.37 × 106 m )

47. Universal Gravitation Example
Calculate the acceleration due to gravity of Earth (mass = 6.0 x 1024 kg, radius = 6.37 × 106 m )
g = G M / r2
g = (6.67 x N·m2/kg2)(5.98 x 1024 kg)/(6.37 x 106 m)2
= 9.83 m/s2

48. Universal Gravitation Example
In The Little Prince, the Prince visits a small asteroid called B612. If asteroid B612 has a radius of only m and a mass of 1.00 x 104 kg, what is the acceleration due to gravity on asteroid B612?

49. Universal Gravitation Example
In The Little Prince, the Prince visits a small asteroid called B612. If asteroid B612 has a radius of only m and a mass of 1.00 x 104 kg, what is the acceleration due to gravity on asteroid B612?
g = G M / r2
g = (6.67 x N·m2/kg2)(1.00 x 104 kg)/(20.0 m)2
= 1.67 x 10-9 m/s2

50. Universal Gravitation Example
The planet Saturn has a mass that is 95 times as massive as Earth and a radius that is 9.4 times Earth’s radius. If an object is 1000 N on the surface of Earth, what is the weight of the same object on the surface of Saturn?

51. Universal Gravitation Example
The planet Saturn has a mass that is 95 times as massive as Earth and a radius that is 9.4 times Earth’s radius. If an object is 1000 N on the surface of Earth, what is the weight of the same object on the surface of Saturn?
Fg = G m M / r2 Fg ~ M / r2
Fg = 1000 N x 95 / (9.4)2 = 1075 N

52. Relative Weight on Each Planet

53. p. 85: The lunar rover would collapse under its own weight if used on Earth. (NASA)

54. Defined the World Isaac Newton’s Influence
People could uncover the workings of the physical universe
Moons, planets, stars, and galaxies have such a beautifully simple rule to govern them
Phenomena of the world might also be described by equally simple and universal laws

55. Summary Isaac Newton Universal gravitation – Apple and Moon?
Moon’s centripetal acceleration
Gravity’s inverse square law
Law of universal gravitation
Universal gravitational constant – Henry Cavendish
Calculate the mass of Earth
Weak gravitational force
Acceleration due to gravity
Newton’s influence

56. Gravitational Interaction
Chin-Sung Lin

57. Gravitational Field Force Field
A force field exerts a force on objects in its vicinity
Magnetic Field
A magnetic field is a force field that surrounds a magnet and exerts a magnetic force on magnetic substances
Electric Field
An electric field is a force field surrounding electric charges

58. Gravitational Field Gravitational Field
A gravitational field is a force field
that surrounds massive objects

59. Earth’s Gravitational Field

60. Earth’s Gravitational Field
Earth’s gravitational field is
represented by imaginary field lines
Where the field lines are closer together, the gravitational field is stronger
The direction of the field at any point is along the line the point lies on
Arrows show the field direction
Any mass in the vicinity of Earth will be accelerated in the direction of the field line at that location

61. Strength of Gravitational Field
Strength of the gravitational field is the force per unit mass exerted by Earth on any object
Gravitational Field
g = Fg / m = (G m M / r2) / m = G M / r2
F: weight of the object
G: universal gravitational constant (6.67 x N·m2/kg2)
m: mass of the object
M: mass of Earth (5.98 x 1024 kg)
r: Earth’s radius (6.37 x 106 m)

62. Gravitational Field Inside a Planet

63. Gravitational Field Inside a PlanetC r

64. Gravitational Field Inside a Planet
Cancellation of gravitational force
If Earth were of uniform density, the gravity of the entire surrounding shell of inner radius equal to your radial distance from the center will completely cancel out P C r

65. Gravitational Field Inside a PlanetB P
Cancellation of gravitational force
The gravity of area A and area B on P completely cancel out

66. Gravitational Field Inside a PlanetC r
Cancellation of gravitational force
You are pulled only by the mass within this shell – below you
At Earth’s center, the whole Earth is the shell and complete cancellation occurs

67. Gravitational Field Inside a Planet
Strength of the gravitational field is proportional to M / r2
g = GM/r2 = GDV/r2 = GD(4/3)πr3/r2 = (4/3)GDπr
g ~ r
G: universal gravitational constant (6.67 x N·m2/kg2)
M: mass of Earth (5.98 x 1024 kg)
D: density of Earth
V: volume of Earth
r: distance to the Earth’s center

68. Gravitational Field Inside a Planet
a = g
a = g/2
a = 0

69. Gravitational Field Inside a Planet
Without air drag, the trip would take nearly 45 minutes. The gravitational field strength is steadily decreasing as you continue toward the center
At the center of Earth, you are pulled in every direction equally, so that the net force is zero and the gravitational field is zero