Chapter 4 Gravitation

1
Chapter 4 Gravitation

• Physics Beyond 2000

2
Gravity

• Newton
• http//csep10.phys.utk.edu/astr161/lect/history/ne
  wtongrav.html
• http//www.britannica.com/bcom/eb/article/9/0,5716
  ,1091692106265,00.html
• http//www.nelsonitp.com/physics/guide/pages/gravi
  ty/g1.html

3
Gravity

• The moon is performing circular motion round the
  earth.
• The centripetal force comes from the gravity.

v
Fc
moon
earth
4
Gravity

• Newton found that the gravity on the moon is the
  same force making an apple fall.

W
Ground
5
Newtons Law of Gravitation

• Objects attract each other with gravitational
  force.
• In the diagram,
• m1 and m2 are the masses of the objects and r is
  the distance between them.

6
Newtons Law of Gravitation

• Every particle of matter attracts every other
  particle with a force whose magnitude is

G is a universal constant G 6.67 ? 10-11
m3kg-1s-2
Note that the law applies to particles only.
7
Example 1

• Find how small the gravitation is.

8
Shell Theorem

• Extends the formula
• to spherical objects like a ball, the earth,
  the sun and all planets.

9
Theorem 1a. Outside a uniform spherical shell.

• The shell attracts the external particle as if
  all the shells mass were concentrated at its
  centre.

O
10
Theorem 1b. Outside a uniform sphere.

• The sphere attracts the external particle as if
  all the spheres mass were concentrated at its
  centre.

m2
m1
F
F
O
r
11
Example 2 Outside a uniform sphere.

• The earth is almost a uniform sphere.

12
Theorem 2a. Inside a uniform spherical shell.

• The net gravitational force is zero on an object
  inside a uniform shell.

13
Theorem 2b. Inside a uniform sphere.
where m1 is the mass of the core with r the
distance from the centre to the mass m2
14
Example 3

• Inside a uniform sphere.

15
Gravitational Field

• A gravitational field is a region in which any
  mass will experience a gravitational force.
• A uniform gravitational field is a field in which
  the gravitational force in independent of the
  position.
• http//saturn.vcu.edu/rgowdy/mod/g33/s.htm

16
Field strength, g

• The gravitational field strength, g, is the
  gravitational force per unit mass on a test mass.

F is the gravitational force m is the mass of the
test mass
g is a vector, in the same direction of F. SI
unit of g is Nkg-1.
17
Field strength, g

• The gravitational field strength, g, is the
  gravitational force per unit mass on a test mass.

F is the gravitational force m is the mass of the
test mass
SI unit of g is Nkg-1.
18
Field strength, g, outside an isolated sphere of
mass M

• The gravitational field strength, g, outside an
  isolated sphere of mass M is

O
Prove it by placing a test mass m at a point X
with distance r from the centre of the isolated
sphere M.
19
Example 4

• The field strength of the earth at the position
  of the moon.

20
Field strength, g

• Unit of g is Nkg-1.
• g is also a measure of the acceleration of the
  test mass.
• g is also the acceleration due to gravity, unit
  is ms-2.

21
Field strength, g

• Field strength, g.
• Unit Nkg-1.
• A measure of the strength of the gravitational
  field.

• Acceleration due to gravity, g.
• Unit ms-2.
• A description of the motion of a test mass in
  free fall.

22
Field lines

• We can represent the field strength by drawing
  field lines.
• The field lines for a planet are radially inward.

planet
Radial field
23
Field lines

• We can represent the field strength by drawing
  field lines.
• The field lines for a uniform field are parallel.

Uniform field
earths surface
24
Field lines

• The density of the field lines indicates the
  relative field strength.

g1 10 Nkg-1
g2 5 Nkg-1
25
Field lines

• The arrow and the tangent to the field lines
  indicates the direction of the force acting on
  the test mass.

26
The earths gravitational field

• Mass of the earth Me ? 5.98 ? 1024 kg
• Radius of the earth Re ? 6.37 ? 106 m

O
Re
27
Gravity on the earths surface, go

• The gravitational field go near the earths
  surface is uniform and

The value of go ? 9.8 Nkg-1
28
Example 5

• The gravity on the earths surface, go.

29
Apparent Weight

• Use a spring-balance to measure the weight of a
  body.
• Depending on the case, the measured weight R (the
  apparent weight) is not equal to the
  gravitational force mgo.

R
mgo
30
Apparent Weight

• The reading on the spring-balance is affected by
  the following factors
• The density of the earth crust is not uniform.
• The earth is not a perfect sphere.
• The earth is rotating.

31
Apparent Weight

• The density of the earth crust is not uniform.
• Places have different density underneath. Thus
  the gravitational force is not uniform.

32
Apparent Weight

• 2. The earth is not a perfect sphere.
• Points at the poles are closer to the centre
  than points on the equators.
• rpole lt requator
• gpole gt gequator

N-pole
Equator
S-pole
33
Apparent Weight

• 3. The earth is rotating.
• Except at the pole, all points on earth are
  performing circular motion with the same angular
  velocity ?. However the radii of the circles may
  be different.

34
Apparent Weight

• 3. The earth is rotating.
• Consider a mass m is at point X with latitude ?.
• The radius of the circle is r Re.cos ? .

m
X
Y
r
Re
?
O
35
Apparent Weight

• 3. The earth is rotating.
• The net force on the mass m must be equal to the
  centripetal force.

m
Fc
X
r
Y
Re
?
O
Note that Fnet points to Y.
36
Apparent Weight R
R

• 3. The earth is rotating.
• The net force on the mass m must be equal to the
  centripetal force.
• So the apparent weight (normal reaction) R does
  not cancel the gravitational force mgo.

Fc
X
r
Y
m
mgo
?
O
37
Apparent Weight R
R

• 3. The earth is rotating.
• The apparent weight R is not equal to the
  gravitational force mgo in magnitude.

Fc
X
r
Y
m
mgo
?
O
38
Apparent weight R on the equator
mgo
R
The apparent field strength on the equator is
39
Apparent weight R at the poles
R
mgo
The apparent field strength at the poles is
40
Example 6

• Compare the apparent weights.

41
Apparent weight at latitude ?
R
Fc
X
r
Y
m
mgo
?
O
Note that the apparent weight R is not exactly
along the line through the centre of the earth.
42
Variation of g with height and depth

• Outside the earth at height h.

h height of the mass m from the earths surface
43
Variation of g with height and depth

• Outside the earth at height h.

where go is the field strength on the earths
surface.
44
Variation of g with height and depth

• Outside the earth at height h.

where go is the field strength on the
earths surface.
45
Variation of g with height and depth

• Outside the earth at height h close to the
  earths surface. hltltRe.

?
where go is the field strength on the
earths surface.
46
Variation of g with height and depth

• Below the earths surface.

Only the core with colour gives
the gravitational force.
g
r Re-d
47
Variation of g with height and depth

• Below the earths surface.

Find the mass Mr of
g
r Re-d
48
Variation of g with height and depth

• Below the earths surface.

g
r Re-d
49
Variation of g with height and depth

• Below the earths surface.

g
g ? r
r Re-d
50
Variation of g with height and depth

1. r lt Re , g ? r.
2. r gt Re ,

earth
g
go
r distance from the centre of the earth
0
Re
51
Gravitational potential energy Up

• Object inside a gravitational field has
  gravitational potential energy.
• When object falls towards the earth, it gains
  kinetic energy and loses gravitational potential
  energy.

This object possesses Up
earth
52
Zero potential energy

• By convention, the gravitational potential energy
  of the object is zero when its separation x from
  the centre of the earth is ?.

Up 0
earth
O
x? ?
53
Negative potential energy

• For separation less than r, the gravitational
  potential energy of the object is less than zero.
  So it is negative.

54
Gravitational potential energy Up

• Definition 1
• It is the negative of the work done by the
  gravitational force FG as the object moves from
  infinity to that point.

?
earth
FG
O
r
dx
55
Gravitational potential energy Up

• Definition 1

?
earth
FG
O
r
dx
56
Gravitational potential energy Up

• Definition 2
• It is the negative of the work done by the
  external force F to bring the object from that
  point to infinity.

?
Me
earth
F
O
r
dx
m
57
Gravitational potential energy Up

• Definition 2

?
Me
earth
F
O
r
dx
m
58
Gravitational potential energy Up
59
Example 7

• Conservation of kinetic and gravitational
  potential energy.

60
Example 8

• - Work done
• gravitational potential energy

61
Example 9

• Two particles are each in the others
  gravitational field.
• Thus each particle possesses gravitational energy.

62
System of three particles

• Each particle is in another two particles
  gravitational fields.
• Each particle possesses gravitational potential
  energy due to the other two particles.

Up of
63
System of three particles

• Up of

Up of
64
Example 10

• Up of the moon due to the earths gravitational
  field.

What is the Up of the earth due to the moons
gravitational field?
65
Escape speed ve

• Escape speed ve is the minimum projection speed
  required for any object to escape from the
  surface of a planet without return.

66
Escape speed ve

• Escape speed ve is the minimum projection speed
  required for any object to escape from the
  surface of a planet without return.

67
Escape speed ve

• On the surface of the planet, the body possesses
  both kinetic energy Uk and gravitational
  potential energy Up.

UP
68
Escape speed ve

• If the body is able to escape away, it means the
  body still possesses kinetic energy at infinity.
• Note that the gravitational energy of the body at
  infinity is zero.

69
Escape speed ve

• If there is not any loss of energy on projection,
  
• the total energy of the body at lift-off
• the total energy of the body at infinity

70
Escape speed ve
kinetic energy at infinity
?0
71
Escape speed ve
where go is the gravitational acceleration on the
surface of the earth.
72
Escape speed ve
So the escape speed from earth is
73
Escape speed ve
Example Find ve
74
Gravitational potential V

• Definition
• The gravitational potential at a point is the
  gravitational potential energy per unit test
  mass.

where U is the gravitational potential energy
of a mass m at the point
75
Gravitational potential V

• Definition
• The gravitational potential at a point is the
  gravitational potential energy per unit test
  mass.

unit of V is J kg-1
76
Gravitational potential V

• Example 12 to find the change in gravitational
  potential energy.
• ?U U Uo
• If ?U gt0, there is a gain in U.
• If ?U lt0, there is a loss in U.

77
Equipotentials

• Equipotentials are lines or surfaces on which
  all points have the same potential.
• The equipotentials are always perpendicular to
  the field lines.

78
Equipotentials

• The equipotentials around the earth are imaginary
  spherical shells centered at the earths centre.

79
Equipotentials

• The field is radial.

80
Equipotentials

• The equipotentials near the earths surface are
  parallel and evenly spaced surface.
• The field is uniform.

surface
81
Equipotentials

• Example 13 Earths equipotential.

82
Potential V and field strength g
r
83
Potential V and field strength g
If we consider the magnitude of g only,
r
84
Earth-moon system

• http//tycho.usno.navy.mil/vphase.html
• The potential is the sum of the potentials due to
  the earth and the moon.

Me
Mm
r
P
D-r
D
85
Earth-moon system
86
Earth-moon system
V
0
r
87
Earth-moon system
V
0
r
88
Earth-moon system
V
g
0
r
89
Earth-moon system
V
g0
0
r
X
g 0 at a point X between the earth and the
moon. X is a neutral point.
90
Earth-moon system
V
ggt0
0
r
X
g points to the centre of the earth if it is
positive.
91
Earth-moon system
V
glt0
0
r
X
g points to the centre of the moon if it is
negative.
92
Earth-moon system
Given Me 5.98 1024 kg Mm 7.35
1022 kg D 3.84 108 m
G 6.67 10-11 Nm2kg-2 Find the position X at
which g 0.
93
Earth-moon system
Given Me 5.98 1024 kg Mm 7.35
1022 kg D 3.84 108 m
G 6.67 10-11 Nm2kg-2 Find the position X at
which g 0.
Answer x 3.46 108 m
94
Earth-moon system

• Example 14 potential difference near the
  earths surface.

95
Orbital motion

• The description of the motion of a planet round
  the sun.

96
Orbital motion

• Keplers law
• The law of orbits.
• All planets move in elliptical orbits, with
  the sun at one focus.

97
Orbital motion

• Keplers law
• 2. The law of areas.
• The area swept out in a given time by the
  line joining any planet to the sun is always the
  same.

98
Orbital motion

• Keplers law
• 3. The law of periods.
• The square of the period T of any planet
  about the sun is proportional to the cube of
  their mean distance r from the sun.

99
Orbital motion

• Basically, we only study the simple case of
  circular orbit.

r
100
Orbital motion
A satellite of mass m performs circular motion
round the earth with speed vc . The radius of the
orbit is r.
101
Orbital motion
The centripetal force is provided by the
gravitational force.
102
Orbital motion
Show that
where Me is the mass of the earth
103
Orbital motion

• Example 15 find the speed of a satellite.

104
Proof of Keplers 3rd law in a circular orbit
satellite 2
r2
vc2
105
Proof of Keplers 3rd law in a circular orbit
Note that the proof is true for satellites round
the same planet.
satellite 2
r2
vc2
106
Keplers 3rd law

• Example 16 apply Keplers 3rd law.

107
Satellites

• Natural satellites e.g. moon.
• Artificial satellites
• e.g. communication satellites,
• weather satellites.

http//www.smgaels.org/physics/97/MGRAHLFS.HTM
http//weather.yahoo.com/graphics/satellite/US.htm
l
108
Geosynchronous satellites

• A geosynchronous satellite is above the earths
  equator.
• It rotates about the earth with the same angular
  speed as the earth and in the same direction.
• It seems stationary by observers on earth.

109
Geosynchronous satellites
110
Geosynchronous satellites
Find the radius of the orbit of a
geosynchornous satellite.
111
Geosynchronous satellites
rs 4.23107 m
112
Geosynchronous satellites
h 3.59107 m
113
Parking Orbit
Note that there is only one such orbit. It is
called a parking orbit.
114
Satellites Near the Earths surface

• Assume that the orbit is circular with radius r ?
  Re , the radius of the earth.
• The gravitational field strength go is almost a
  constant (9.8 N kg-1).
• The gravitational force provides the required
  centripetal force.

115
Satellites Near the Earths surface
Find vr
vr
satellite
r ? Re
earth
116
Energy and Satellite Motion

• Find v and the kinetic energy Uk of the satellite.

m
117
Energy and Satellite Motion

• The satellite in the orbit possesses both kinetic
  energy and gravitational energy.

m
118
Energy and Satellite Motion
Note that Uk gt 0
119
Energy and Satellite Motion

• Find Up the gravitational potential of the
  satellite.

m
120
Energy and Satellite Motion
Note that Up lt 0
121
Energy and Satellite Motion
Find U, the total energy of the satellite.
122
Energy and Satellite Motion
Note that U lt 0
123
Energy and Satellite Motion
U Up Uk -1 -2 1
124
Falling to the earth
The satellite may lose energy due to
air resistance. The total energy becomes more
negative and r becomes less.
125
Falling to the earth
The satellite follows a spiral path towards the
earth.
126
Falling to the earth
As r decreases, the kinetic energy of the
satellite increases and the satellite moves
faster.
127
Falling to the earth
Example 17 Loss of energy
128
Weightlessness in spacecraft
v
v
mg
The astronaut is weightless.
129
Weightlessness in spacecraft

• We fell our weight because there is normal
  reaction on us.

Normal reaction
ground
mg
130
Weightlessness in spacecraft

• If there is not any normal reaction on us, we
  feel weightless. e.g. free falling

mg
131
Weightlessness in spacecraft
v
The gravitational force mg on the astronaut
is the required centripetal force. He does not
require any normal reaction to act on him.
mg
132
Weightlessness in spacecraft
The astronaut is weightless.
v
mg
http//www.nasm.edu/galleries/gal109/NEWHTF/HTF611
A.HTM