Page 1
CHAPTER SEVEN
GRAVITATION
7.1 INTRODUCTION
Early in our lives, we become aware of the tendency of all
material objects to be attracted towards the earth. Anything
thrown up falls down towards the earth, going uphill is lot
more tiring than going downhill, raindrops from the clouds
above fall towards the earth and there are many other such
phenomena. Historically it was the Italian Physicist Galileo
(15641642) who recognised the fact that all bodies,
irrespective of their masses, are accelerated towards the earth
with a constant acceleration. It is said that he made a public
demonstration of this fact. To find the truth, he certainly
did experiments with bodies rolling down inclined planes and
arrived at a value of the acceleration due to gravity which is
close to the more accurate value obtained later.
A seemingly unrelated phenomenon, observation of stars,
planets and their motion has been the subject of attention
in many countries since the earliest of times. Observations
since early times recognised stars which appeared in the
sky with positions unchanged year after year. The more
interesting objects are the planets which seem to have regular
motions against the background of stars. The earliest
recorded model for planetary motions proposed by Ptolemy
about 2000 years ago was a ‘geocentric’ model in which all
celestial objects, stars, the sun and the planets, all revolved
around the earth. The only motion that was thought to be
possible for celestial objects was motion in a circle.
Complicated schemes of motion were put forward by Ptolemy
in order to describe the observed motion of the planets. The
planets were described as moving in circles with the centre
of the circles themselves moving in larger circles. Similar
theories were also advanced by Indian astronomers some
400 years later. However a more elegant model in which the
Sun was the centre around which the planets revolved – the
‘heliocentric’ model – was already mentioned by Aryabhatta
(5
th
century A.D.) in his treatise. A thousand years later, a
Polish monk named Nicolas Copernicus (14731543)
7.1 Introduction
7.2 Kepler’s laws
7.3 Universal law of
gravitation
7.4 The gravitational
constant
7.5 Acceleration due to
gravity of the earth
7.6 Acceleration due to
gravity below and above
the surface of earth
7.7 Gravitational potential
energy
7.8 Escape speed
7.9 Earth satellites
7.10 Energy of an orbiting
satellite
Summary
Points to ponder
Exercises
Rationalised202324
Page 2
CHAPTER SEVEN
GRAVITATION
7.1 INTRODUCTION
Early in our lives, we become aware of the tendency of all
material objects to be attracted towards the earth. Anything
thrown up falls down towards the earth, going uphill is lot
more tiring than going downhill, raindrops from the clouds
above fall towards the earth and there are many other such
phenomena. Historically it was the Italian Physicist Galileo
(15641642) who recognised the fact that all bodies,
irrespective of their masses, are accelerated towards the earth
with a constant acceleration. It is said that he made a public
demonstration of this fact. To find the truth, he certainly
did experiments with bodies rolling down inclined planes and
arrived at a value of the acceleration due to gravity which is
close to the more accurate value obtained later.
A seemingly unrelated phenomenon, observation of stars,
planets and their motion has been the subject of attention
in many countries since the earliest of times. Observations
since early times recognised stars which appeared in the
sky with positions unchanged year after year. The more
interesting objects are the planets which seem to have regular
motions against the background of stars. The earliest
recorded model for planetary motions proposed by Ptolemy
about 2000 years ago was a ‘geocentric’ model in which all
celestial objects, stars, the sun and the planets, all revolved
around the earth. The only motion that was thought to be
possible for celestial objects was motion in a circle.
Complicated schemes of motion were put forward by Ptolemy
in order to describe the observed motion of the planets. The
planets were described as moving in circles with the centre
of the circles themselves moving in larger circles. Similar
theories were also advanced by Indian astronomers some
400 years later. However a more elegant model in which the
Sun was the centre around which the planets revolved – the
‘heliocentric’ model – was already mentioned by Aryabhatta
(5
th
century A.D.) in his treatise. A thousand years later, a
Polish monk named Nicolas Copernicus (14731543)
7.1 Introduction
7.2 Kepler’s laws
7.3 Universal law of
gravitation
7.4 The gravitational
constant
7.5 Acceleration due to
gravity of the earth
7.6 Acceleration due to
gravity below and above
the surface of earth
7.7 Gravitational potential
energy
7.8 Escape speed
7.9 Earth satellites
7.10 Energy of an orbiting
satellite
Summary
Points to ponder
Exercises
Rationalised202324
128 PHYSICS
B
A
C
P
S
S'
2b
2a
proposed a definitive model in which the planets
moved in circles around a fixed central sun. His
theory was discredited by the church, but
notable amongst its supporters was Galileo who
had to face prosecution from the state for his
beliefs.
It was around the same time as Galileo, a
nobleman called Tycho Brahe (15461601)
hailing from Denmark, spent his entire lifetime
recording observations of the planets with the
naked eye. His compiled data were analysed
later by his assistant Johannes Kepler (1571
1640). He could extract from the data three
elegant laws that now go by the name of Kepler’s
laws. These laws were known to Newton and
enabled him to make a great scientific leap in
proposing his universal law of gravitation.
7.2 KEPLER’S LAWS
The three laws of Kepler can be stated as
follows:
1. Law of orbits : All planets move in elliptical
orbits with the Sun situated at one of the foci
Fig. 7.1(a) An ellipse traced out by a planet around
the sun. The closest point is P and the
farthest point is A, P is called the
perihelion and A the aphelion. The
semimajor axis is half the distance AP.
Fig. 7.1(b) Drawing an ellipse. A string has its ends
fixed at F
1
and F
2
. The tip of a pencil holds
the string taut and is moved around.
of the ellipse (Fig. 7.1a). This law was a
deviation from the Copernican model which
allowed only circular orbits. The ellipse, of
which the circle is a special case, is a closed
curve which can be drawn very simply as
follows.
Select two points F
1
and F
2
. Take a length
of a string and fix its ends at F
1
and F
2
by
pins. With the tip of a pencil stretch the string
taut and then draw a curve by moving the
pencil keeping the string taut throughout.(Fig.
7.1(b)) The closed curve you get is called an
ellipse. Clearly for any point T on the ellipse,
the sum of the distances from F
1
and F
2
is a
constant. F
1
,
F
2
are called the focii. Join the
points F
1
and
F
2
and extend
the line to
intersect the ellipse at points P and A as shown
in Fig. 7.1(b). The midpoint of the line PA is
the centre of the ellipse O and the length PO =
AO is called the semimajor axis of the ellipse.
For a circle, the two focii merge into one and
the semimajor axis becomes the radius of the
circle.
2. Law of areas : The line that joins any planet
to the sun sweeps equal areas in equal
intervals of time (Fig. 7.2). This law comes from
the observations that planets appear to move
slower when they are farther from the sun
than when they are nearer.
Fig. 7.2 The planet P moves around the sun in an
elliptical orbit. The shaded area is the area
?A swept out in a small interval of time ?t.
Rationalised202324
Page 3
CHAPTER SEVEN
GRAVITATION
7.1 INTRODUCTION
Early in our lives, we become aware of the tendency of all
material objects to be attracted towards the earth. Anything
thrown up falls down towards the earth, going uphill is lot
more tiring than going downhill, raindrops from the clouds
above fall towards the earth and there are many other such
phenomena. Historically it was the Italian Physicist Galileo
(15641642) who recognised the fact that all bodies,
irrespective of their masses, are accelerated towards the earth
with a constant acceleration. It is said that he made a public
demonstration of this fact. To find the truth, he certainly
did experiments with bodies rolling down inclined planes and
arrived at a value of the acceleration due to gravity which is
close to the more accurate value obtained later.
A seemingly unrelated phenomenon, observation of stars,
planets and their motion has been the subject of attention
in many countries since the earliest of times. Observations
since early times recognised stars which appeared in the
sky with positions unchanged year after year. The more
interesting objects are the planets which seem to have regular
motions against the background of stars. The earliest
recorded model for planetary motions proposed by Ptolemy
about 2000 years ago was a ‘geocentric’ model in which all
celestial objects, stars, the sun and the planets, all revolved
around the earth. The only motion that was thought to be
possible for celestial objects was motion in a circle.
Complicated schemes of motion were put forward by Ptolemy
in order to describe the observed motion of the planets. The
planets were described as moving in circles with the centre
of the circles themselves moving in larger circles. Similar
theories were also advanced by Indian astronomers some
400 years later. However a more elegant model in which the
Sun was the centre around which the planets revolved – the
‘heliocentric’ model – was already mentioned by Aryabhatta
(5
th
century A.D.) in his treatise. A thousand years later, a
Polish monk named Nicolas Copernicus (14731543)
7.1 Introduction
7.2 Kepler’s laws
7.3 Universal law of
gravitation
7.4 The gravitational
constant
7.5 Acceleration due to
gravity of the earth
7.6 Acceleration due to
gravity below and above
the surface of earth
7.7 Gravitational potential
energy
7.8 Escape speed
7.9 Earth satellites
7.10 Energy of an orbiting
satellite
Summary
Points to ponder
Exercises
Rationalised202324
128 PHYSICS
B
A
C
P
S
S'
2b
2a
proposed a definitive model in which the planets
moved in circles around a fixed central sun. His
theory was discredited by the church, but
notable amongst its supporters was Galileo who
had to face prosecution from the state for his
beliefs.
It was around the same time as Galileo, a
nobleman called Tycho Brahe (15461601)
hailing from Denmark, spent his entire lifetime
recording observations of the planets with the
naked eye. His compiled data were analysed
later by his assistant Johannes Kepler (1571
1640). He could extract from the data three
elegant laws that now go by the name of Kepler’s
laws. These laws were known to Newton and
enabled him to make a great scientific leap in
proposing his universal law of gravitation.
7.2 KEPLER’S LAWS
The three laws of Kepler can be stated as
follows:
1. Law of orbits : All planets move in elliptical
orbits with the Sun situated at one of the foci
Fig. 7.1(a) An ellipse traced out by a planet around
the sun. The closest point is P and the
farthest point is A, P is called the
perihelion and A the aphelion. The
semimajor axis is half the distance AP.
Fig. 7.1(b) Drawing an ellipse. A string has its ends
fixed at F
1
and F
2
. The tip of a pencil holds
the string taut and is moved around.
of the ellipse (Fig. 7.1a). This law was a
deviation from the Copernican model which
allowed only circular orbits. The ellipse, of
which the circle is a special case, is a closed
curve which can be drawn very simply as
follows.
Select two points F
1
and F
2
. Take a length
of a string and fix its ends at F
1
and F
2
by
pins. With the tip of a pencil stretch the string
taut and then draw a curve by moving the
pencil keeping the string taut throughout.(Fig.
7.1(b)) The closed curve you get is called an
ellipse. Clearly for any point T on the ellipse,
the sum of the distances from F
1
and F
2
is a
constant. F
1
,
F
2
are called the focii. Join the
points F
1
and
F
2
and extend
the line to
intersect the ellipse at points P and A as shown
in Fig. 7.1(b). The midpoint of the line PA is
the centre of the ellipse O and the length PO =
AO is called the semimajor axis of the ellipse.
For a circle, the two focii merge into one and
the semimajor axis becomes the radius of the
circle.
2. Law of areas : The line that joins any planet
to the sun sweeps equal areas in equal
intervals of time (Fig. 7.2). This law comes from
the observations that planets appear to move
slower when they are farther from the sun
than when they are nearer.
Fig. 7.2 The planet P moves around the sun in an
elliptical orbit. The shaded area is the area
?A swept out in a small interval of time ?t.
Rationalised202324
GRAVITATION 129
?
3. Law of periods : The square of the time period
of revolution of a planet is proportional to the
cube of the semimajor axis of the ellipse traced
out by the planet.
Table 7.1 gives the approximate time periods
of revolution of eight* planets around the sun
along with values of their semimajor axes.
Table 7.1Data from measurement of
planetary motions given below
confirm Kepler’s Law of Periods
(a = Semimajor axis in units of 10
10
m.
T = Time period of revolution of the planet
in years(y).
Q = The quotient ( T
2
/a
3
) in units of
10
34
y
2
m
3
.)
Planet a T Q
Mercury 5.79 0.24 2.95
Venus 10.8 0.615 3.00
Earth 15.0 1 2.96
Mars 22.8 1.88 2.98
Jupiter 77.8 11.9 3.01
Saturn 143 29.5 2.98
Uranus 287 84 2.98
Neptune 450 165 2.99
The law of areas can be understood as a
consequence of conservation of angular
momentum whch is valid for any central
force . A central force is such that the force
on the planet is along the vector joining the
Sun and the planet. Let the Sun be at the
origin and let the position and momentum
of the planet be denoted by r and p
respectively. Then the area swept out by the
planet of mass m in time interval ?t is (Fig.
7.2) ?A given by
?A
= ½ (r × v?t) (7.1)
Hence
?A
/?t =½ (r × p)/m, (since v = p/m)
= L / (2 m) (7.2)
where v is the velocity, L is the angular
momentum equal to ( r × p). For a central
force, which is directed along r, L is a constant
as the planet goes around. Hence, ?A /?t is a
constant according to the last equation. This is
the law of areas. Gravitation is a central force
and hence the law of areas follows.
Example 7.1 Let the speed of the planet
at the perihelion P in Fig. 7.1(a) be v
P
and
the Sunplanet distance SP be r
P
. Relate
{r
P
, v
P
} to the corresponding quantities at
the aphelion {r
A,
v
A
}. Will the planet take
equal times to traverse BAC and CPB ?
Answer The magnitude of the angular
momentum at P is L
p
= m
p
r
p
v
p
, since inspection
tells us that r
p
and v
p
are mutually
perpendicular. Similarly, L
A
= m
p
r
A
v
A
. From
angular momentum conservation
m
p
r
p
v
p
= m
p
r
A
v
A
or
v
v
p
A
=
r
r
A
p
?
Since r
A
> r
p
,
v
p
> v
A
.
The area SBAC bounded by the ellipse and
the radius vectors SB and SC is larger than SBPC
in Fig. 7.1. From Kepler’s second law, equal areas
are swept in equal times. Hence the planet will
take a longer time to traverse BAC than CPB.
7.3 UNIVERSAL LAW OF GRAVITATION
Legend has it that observing an apple falling from
a tree, Newton was inspired to arrive at an
universal law of gravitation that led to an
explanation of terrestrial gravitation as well as
of Kepler’s laws. Newton’s reasoning was that
the moon revolving in an orbit of radius R
m
was
subject to a centripetal acceleration due to
earth’s gravity of magnitude
2 2
2
4
m
m
m
R V
a
R T
p
= =
(7.3)
where V is the speed of the moon related to the
time period T by the relation 2 /
m
V R T p = . The
time period T is about 27.3 days and R
m
was
already known then to be about 3.84 × 10
8
m. If
we substitute these numbers in Eq. (7.3), we get
a value of a
m
much smaller than the value of
acceleration due to gravity g on the surface of
the earth, arising also due to earth’s gravitational
attraction.
Rationalised202324
Page 4
CHAPTER SEVEN
GRAVITATION
7.1 INTRODUCTION
Early in our lives, we become aware of the tendency of all
material objects to be attracted towards the earth. Anything
thrown up falls down towards the earth, going uphill is lot
more tiring than going downhill, raindrops from the clouds
above fall towards the earth and there are many other such
phenomena. Historically it was the Italian Physicist Galileo
(15641642) who recognised the fact that all bodies,
irrespective of their masses, are accelerated towards the earth
with a constant acceleration. It is said that he made a public
demonstration of this fact. To find the truth, he certainly
did experiments with bodies rolling down inclined planes and
arrived at a value of the acceleration due to gravity which is
close to the more accurate value obtained later.
A seemingly unrelated phenomenon, observation of stars,
planets and their motion has been the subject of attention
in many countries since the earliest of times. Observations
since early times recognised stars which appeared in the
sky with positions unchanged year after year. The more
interesting objects are the planets which seem to have regular
motions against the background of stars. The earliest
recorded model for planetary motions proposed by Ptolemy
about 2000 years ago was a ‘geocentric’ model in which all
celestial objects, stars, the sun and the planets, all revolved
around the earth. The only motion that was thought to be
possible for celestial objects was motion in a circle.
Complicated schemes of motion were put forward by Ptolemy
in order to describe the observed motion of the planets. The
planets were described as moving in circles with the centre
of the circles themselves moving in larger circles. Similar
theories were also advanced by Indian astronomers some
400 years later. However a more elegant model in which the
Sun was the centre around which the planets revolved – the
‘heliocentric’ model – was already mentioned by Aryabhatta
(5
th
century A.D.) in his treatise. A thousand years later, a
Polish monk named Nicolas Copernicus (14731543)
7.1 Introduction
7.2 Kepler’s laws
7.3 Universal law of
gravitation
7.4 The gravitational
constant
7.5 Acceleration due to
gravity of the earth
7.6 Acceleration due to
gravity below and above
the surface of earth
7.7 Gravitational potential
energy
7.8 Escape speed
7.9 Earth satellites
7.10 Energy of an orbiting
satellite
Summary
Points to ponder
Exercises
Rationalised202324
128 PHYSICS
B
A
C
P
S
S'
2b
2a
proposed a definitive model in which the planets
moved in circles around a fixed central sun. His
theory was discredited by the church, but
notable amongst its supporters was Galileo who
had to face prosecution from the state for his
beliefs.
It was around the same time as Galileo, a
nobleman called Tycho Brahe (15461601)
hailing from Denmark, spent his entire lifetime
recording observations of the planets with the
naked eye. His compiled data were analysed
later by his assistant Johannes Kepler (1571
1640). He could extract from the data three
elegant laws that now go by the name of Kepler’s
laws. These laws were known to Newton and
enabled him to make a great scientific leap in
proposing his universal law of gravitation.
7.2 KEPLER’S LAWS
The three laws of Kepler can be stated as
follows:
1. Law of orbits : All planets move in elliptical
orbits with the Sun situated at one of the foci
Fig. 7.1(a) An ellipse traced out by a planet around
the sun. The closest point is P and the
farthest point is A, P is called the
perihelion and A the aphelion. The
semimajor axis is half the distance AP.
Fig. 7.1(b) Drawing an ellipse. A string has its ends
fixed at F
1
and F
2
. The tip of a pencil holds
the string taut and is moved around.
of the ellipse (Fig. 7.1a). This law was a
deviation from the Copernican model which
allowed only circular orbits. The ellipse, of
which the circle is a special case, is a closed
curve which can be drawn very simply as
follows.
Select two points F
1
and F
2
. Take a length
of a string and fix its ends at F
1
and F
2
by
pins. With the tip of a pencil stretch the string
taut and then draw a curve by moving the
pencil keeping the string taut throughout.(Fig.
7.1(b)) The closed curve you get is called an
ellipse. Clearly for any point T on the ellipse,
the sum of the distances from F
1
and F
2
is a
constant. F
1
,
F
2
are called the focii. Join the
points F
1
and
F
2
and extend
the line to
intersect the ellipse at points P and A as shown
in Fig. 7.1(b). The midpoint of the line PA is
the centre of the ellipse O and the length PO =
AO is called the semimajor axis of the ellipse.
For a circle, the two focii merge into one and
the semimajor axis becomes the radius of the
circle.
2. Law of areas : The line that joins any planet
to the sun sweeps equal areas in equal
intervals of time (Fig. 7.2). This law comes from
the observations that planets appear to move
slower when they are farther from the sun
than when they are nearer.
Fig. 7.2 The planet P moves around the sun in an
elliptical orbit. The shaded area is the area
?A swept out in a small interval of time ?t.
Rationalised202324
GRAVITATION 129
?
3. Law of periods : The square of the time period
of revolution of a planet is proportional to the
cube of the semimajor axis of the ellipse traced
out by the planet.
Table 7.1 gives the approximate time periods
of revolution of eight* planets around the sun
along with values of their semimajor axes.
Table 7.1Data from measurement of
planetary motions given below
confirm Kepler’s Law of Periods
(a = Semimajor axis in units of 10
10
m.
T = Time period of revolution of the planet
in years(y).
Q = The quotient ( T
2
/a
3
) in units of
10
34
y
2
m
3
.)
Planet a T Q
Mercury 5.79 0.24 2.95
Venus 10.8 0.615 3.00
Earth 15.0 1 2.96
Mars 22.8 1.88 2.98
Jupiter 77.8 11.9 3.01
Saturn 143 29.5 2.98
Uranus 287 84 2.98
Neptune 450 165 2.99
The law of areas can be understood as a
consequence of conservation of angular
momentum whch is valid for any central
force . A central force is such that the force
on the planet is along the vector joining the
Sun and the planet. Let the Sun be at the
origin and let the position and momentum
of the planet be denoted by r and p
respectively. Then the area swept out by the
planet of mass m in time interval ?t is (Fig.
7.2) ?A given by
?A
= ½ (r × v?t) (7.1)
Hence
?A
/?t =½ (r × p)/m, (since v = p/m)
= L / (2 m) (7.2)
where v is the velocity, L is the angular
momentum equal to ( r × p). For a central
force, which is directed along r, L is a constant
as the planet goes around. Hence, ?A /?t is a
constant according to the last equation. This is
the law of areas. Gravitation is a central force
and hence the law of areas follows.
Example 7.1 Let the speed of the planet
at the perihelion P in Fig. 7.1(a) be v
P
and
the Sunplanet distance SP be r
P
. Relate
{r
P
, v
P
} to the corresponding quantities at
the aphelion {r
A,
v
A
}. Will the planet take
equal times to traverse BAC and CPB ?
Answer The magnitude of the angular
momentum at P is L
p
= m
p
r
p
v
p
, since inspection
tells us that r
p
and v
p
are mutually
perpendicular. Similarly, L
A
= m
p
r
A
v
A
. From
angular momentum conservation
m
p
r
p
v
p
= m
p
r
A
v
A
or
v
v
p
A
=
r
r
A
p
?
Since r
A
> r
p
,
v
p
> v
A
.
The area SBAC bounded by the ellipse and
the radius vectors SB and SC is larger than SBPC
in Fig. 7.1. From Kepler’s second law, equal areas
are swept in equal times. Hence the planet will
take a longer time to traverse BAC than CPB.
7.3 UNIVERSAL LAW OF GRAVITATION
Legend has it that observing an apple falling from
a tree, Newton was inspired to arrive at an
universal law of gravitation that led to an
explanation of terrestrial gravitation as well as
of Kepler’s laws. Newton’s reasoning was that
the moon revolving in an orbit of radius R
m
was
subject to a centripetal acceleration due to
earth’s gravity of magnitude
2 2
2
4
m
m
m
R V
a
R T
p
= =
(7.3)
where V is the speed of the moon related to the
time period T by the relation 2 /
m
V R T p = . The
time period T is about 27.3 days and R
m
was
already known then to be about 3.84 × 10
8
m. If
we substitute these numbers in Eq. (7.3), we get
a value of a
m
much smaller than the value of
acceleration due to gravity g on the surface of
the earth, arising also due to earth’s gravitational
attraction.
Rationalised202324
130 PHYSICS
?
This clearly shows that the force due to
earth’s gravity decreases with distance. If one
assumes that the gravitational force due to the
earth decreases in proportion to the inverse
square of the distance from the centre of the
earth, we will have a
m
a
2
m
R

; g a
2
E
R

and we get
2
2
m
m E
R g
a R
=
3600 (7.4)
in agreement with a value of g 9.8 m s
2
and
the value of a
m
from Eq. (7.3). These observations
led Newton to propose the following Universal Law
of Gravitation :
Every body in the universe attracts every other
body with a force which is directly proportional
to the product of their masses and inversely
proportional to the square of the distance
between them.
The quotation is essentially from Newton’s
famous treatise called ‘Mathematical Principles
of Natural Philosophy’ (Principia for short).
Stated Mathematically, Newton’s gravitation
law reads : The force F on a point mass m
2
due
to another point mass m
1
has the magnitude
1 2
2
 
m m
G
r
= F
(7.5)
Equation (7.5) can be expressed in vector form as
?
( )
? 1 2 1 2
2 2
– –
m m m m
G G
r r
= = F r r
? 1 2
3
–
m m
G = r
r
where G is the universal gravitational constant,
?
r
is the unit vector from m
1
to m
2
and r = r
2
– r
1
as shown in Fig. 7.3.
The gravitational force is attractive, i.e., the
force F is along – r. The force on point mass m
1
due to m
2
is of course – F by Newton’s third law.
Thus, the gravitational force F
12
on the body 1
due to 2 and F
21
on the body 2 due to 1 are related
as F
12
= – F
21
.
Before we can apply Eq. (7.5) to objects under
consideration, we have to be careful since the
law refers to point masses whereas we deal with
extended objects which have finite size. If we have
a collection of point masses, the force on any
one of them is the vector sum of the gravitational
forces exerted by the other point masses as
shown in Fig 7.4.
Fig. 7.4 Gravitational force on point mass m
1
is the
vector sum of the gravitational forces exerted
by m
2
, m
3
and m
4
.
The total force on m
1
is
2 1
1 2
21
Gm m
r
= F
? 3 1
21
2
31
Gm m
r
+ r
? ? 4 1
31 41
2
41
Gm m
r
+ r r
Example 7.2 Three equal masses of m kg
each are fixed at the vertices of an
equilateral triangle ABC.
(a) What is the force acting on a mass 2m
placed at the centroid G of the triangle?
(b) What is the force if the mass at the
vertex A is doubled ?
Take AG = BG = CG = 1 m (see Fig. 7.5)
Answer (a) The angle between GC and the
positive xaxis is 30° and so is the angle between
GB and the negative xaxis. The individual forces
in vector notation are
Fig. 7.3 Gravitational force on m
1
due to m
2
is along
r where the vector r is (r
2
– r
1
).
O
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Page 5
CHAPTER SEVEN
GRAVITATION
7.1 INTRODUCTION
Early in our lives, we become aware of the tendency of all
material objects to be attracted towards the earth. Anything
thrown up falls down towards the earth, going uphill is lot
more tiring than going downhill, raindrops from the clouds
above fall towards the earth and there are many other such
phenomena. Historically it was the Italian Physicist Galileo
(15641642) who recognised the fact that all bodies,
irrespective of their masses, are accelerated towards the earth
with a constant acceleration. It is said that he made a public
demonstration of this fact. To find the truth, he certainly
did experiments with bodies rolling down inclined planes and
arrived at a value of the acceleration due to gravity which is
close to the more accurate value obtained later.
A seemingly unrelated phenomenon, observation of stars,
planets and their motion has been the subject of attention
in many countries since the earliest of times. Observations
since early times recognised stars which appeared in the
sky with positions unchanged year after year. The more
interesting objects are the planets which seem to have regular
motions against the background of stars. The earliest
recorded model for planetary motions proposed by Ptolemy
about 2000 years ago was a ‘geocentric’ model in which all
celestial objects, stars, the sun and the planets, all revolved
around the earth. The only motion that was thought to be
possible for celestial objects was motion in a circle.
Complicated schemes of motion were put forward by Ptolemy
in order to describe the observed motion of the planets. The
planets were described as moving in circles with the centre
of the circles themselves moving in larger circles. Similar
theories were also advanced by Indian astronomers some
400 years later. However a more elegant model in which the
Sun was the centre around which the planets revolved – the
‘heliocentric’ model – was already mentioned by Aryabhatta
(5
th
century A.D.) in his treatise. A thousand years later, a
Polish monk named Nicolas Copernicus (14731543)
7.1 Introduction
7.2 Kepler’s laws
7.3 Universal law of
gravitation
7.4 The gravitational
constant
7.5 Acceleration due to
gravity of the earth
7.6 Acceleration due to
gravity below and above
the surface of earth
7.7 Gravitational potential
energy
7.8 Escape speed
7.9 Earth satellites
7.10 Energy of an orbiting
satellite
Summary
Points to ponder
Exercises
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128 PHYSICS
B
A
C
P
S
S'
2b
2a
proposed a definitive model in which the planets
moved in circles around a fixed central sun. His
theory was discredited by the church, but
notable amongst its supporters was Galileo who
had to face prosecution from the state for his
beliefs.
It was around the same time as Galileo, a
nobleman called Tycho Brahe (15461601)
hailing from Denmark, spent his entire lifetime
recording observations of the planets with the
naked eye. His compiled data were analysed
later by his assistant Johannes Kepler (1571
1640). He could extract from the data three
elegant laws that now go by the name of Kepler’s
laws. These laws were known to Newton and
enabled him to make a great scientific leap in
proposing his universal law of gravitation.
7.2 KEPLER’S LAWS
The three laws of Kepler can be stated as
follows:
1. Law of orbits : All planets move in elliptical
orbits with the Sun situated at one of the foci
Fig. 7.1(a) An ellipse traced out by a planet around
the sun. The closest point is P and the
farthest point is A, P is called the
perihelion and A the aphelion. The
semimajor axis is half the distance AP.
Fig. 7.1(b) Drawing an ellipse. A string has its ends
fixed at F
1
and F
2
. The tip of a pencil holds
the string taut and is moved around.
of the ellipse (Fig. 7.1a). This law was a
deviation from the Copernican model which
allowed only circular orbits. The ellipse, of
which the circle is a special case, is a closed
curve which can be drawn very simply as
follows.
Select two points F
1
and F
2
. Take a length
of a string and fix its ends at F
1
and F
2
by
pins. With the tip of a pencil stretch the string
taut and then draw a curve by moving the
pencil keeping the string taut throughout.(Fig.
7.1(b)) The closed curve you get is called an
ellipse. Clearly for any point T on the ellipse,
the sum of the distances from F
1
and F
2
is a
constant. F
1
,
F
2
are called the focii. Join the
points F
1
and
F
2
and extend
the line to
intersect the ellipse at points P and A as shown
in Fig. 7.1(b). The midpoint of the line PA is
the centre of the ellipse O and the length PO =
AO is called the semimajor axis of the ellipse.
For a circle, the two focii merge into one and
the semimajor axis becomes the radius of the
circle.
2. Law of areas : The line that joins any planet
to the sun sweeps equal areas in equal
intervals of time (Fig. 7.2). This law comes from
the observations that planets appear to move
slower when they are farther from the sun
than when they are nearer.
Fig. 7.2 The planet P moves around the sun in an
elliptical orbit. The shaded area is the area
?A swept out in a small interval of time ?t.
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GRAVITATION 129
?
3. Law of periods : The square of the time period
of revolution of a planet is proportional to the
cube of the semimajor axis of the ellipse traced
out by the planet.
Table 7.1 gives the approximate time periods
of revolution of eight* planets around the sun
along with values of their semimajor axes.
Table 7.1Data from measurement of
planetary motions given below
confirm Kepler’s Law of Periods
(a = Semimajor axis in units of 10
10
m.
T = Time period of revolution of the planet
in years(y).
Q = The quotient ( T
2
/a
3
) in units of
10
34
y
2
m
3
.)
Planet a T Q
Mercury 5.79 0.24 2.95
Venus 10.8 0.615 3.00
Earth 15.0 1 2.96
Mars 22.8 1.88 2.98
Jupiter 77.8 11.9 3.01
Saturn 143 29.5 2.98
Uranus 287 84 2.98
Neptune 450 165 2.99
The law of areas can be understood as a
consequence of conservation of angular
momentum whch is valid for any central
force . A central force is such that the force
on the planet is along the vector joining the
Sun and the planet. Let the Sun be at the
origin and let the position and momentum
of the planet be denoted by r and p
respectively. Then the area swept out by the
planet of mass m in time interval ?t is (Fig.
7.2) ?A given by
?A
= ½ (r × v?t) (7.1)
Hence
?A
/?t =½ (r × p)/m, (since v = p/m)
= L / (2 m) (7.2)
where v is the velocity, L is the angular
momentum equal to ( r × p). For a central
force, which is directed along r, L is a constant
as the planet goes around. Hence, ?A /?t is a
constant according to the last equation. This is
the law of areas. Gravitation is a central force
and hence the law of areas follows.
Example 7.1 Let the speed of the planet
at the perihelion P in Fig. 7.1(a) be v
P
and
the Sunplanet distance SP be r
P
. Relate
{r
P
, v
P
} to the corresponding quantities at
the aphelion {r
A,
v
A
}. Will the planet take
equal times to traverse BAC and CPB ?
Answer The magnitude of the angular
momentum at P is L
p
= m
p
r
p
v
p
, since inspection
tells us that r
p
and v
p
are mutually
perpendicular. Similarly, L
A
= m
p
r
A
v
A
. From
angular momentum conservation
m
p
r
p
v
p
= m
p
r
A
v
A
or
v
v
p
A
=
r
r
A
p
?
Since r
A
> r
p
,
v
p
> v
A
.
The area SBAC bounded by the ellipse and
the radius vectors SB and SC is larger than SBPC
in Fig. 7.1. From Kepler’s second law, equal areas
are swept in equal times. Hence the planet will
take a longer time to traverse BAC than CPB.
7.3 UNIVERSAL LAW OF GRAVITATION
Legend has it that observing an apple falling from
a tree, Newton was inspired to arrive at an
universal law of gravitation that led to an
explanation of terrestrial gravitation as well as
of Kepler’s laws. Newton’s reasoning was that
the moon revolving in an orbit of radius R
m
was
subject to a centripetal acceleration due to
earth’s gravity of magnitude
2 2
2
4
m
m
m
R V
a
R T
p
= =
(7.3)
where V is the speed of the moon related to the
time period T by the relation 2 /
m
V R T p = . The
time period T is about 27.3 days and R
m
was
already known then to be about 3.84 × 10
8
m. If
we substitute these numbers in Eq. (7.3), we get
a value of a
m
much smaller than the value of
acceleration due to gravity g on the surface of
the earth, arising also due to earth’s gravitational
attraction.
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130 PHYSICS
?
This clearly shows that the force due to
earth’s gravity decreases with distance. If one
assumes that the gravitational force due to the
earth decreases in proportion to the inverse
square of the distance from the centre of the
earth, we will have a
m
a
2
m
R

; g a
2
E
R

and we get
2
2
m
m E
R g
a R
=
3600 (7.4)
in agreement with a value of g 9.8 m s
2
and
the value of a
m
from Eq. (7.3). These observations
led Newton to propose the following Universal Law
of Gravitation :
Every body in the universe attracts every other
body with a force which is directly proportional
to the product of their masses and inversely
proportional to the square of the distance
between them.
The quotation is essentially from Newton’s
famous treatise called ‘Mathematical Principles
of Natural Philosophy’ (Principia for short).
Stated Mathematically, Newton’s gravitation
law reads : The force F on a point mass m
2
due
to another point mass m
1
has the magnitude
1 2
2
 
m m
G
r
= F
(7.5)
Equation (7.5) can be expressed in vector form as
?
( )
? 1 2 1 2
2 2
– –
m m m m
G G
r r
= = F r r
? 1 2
3
–
m m
G = r
r
where G is the universal gravitational constant,
?
r
is the unit vector from m
1
to m
2
and r = r
2
– r
1
as shown in Fig. 7.3.
The gravitational force is attractive, i.e., the
force F is along – r. The force on point mass m
1
due to m
2
is of course – F by Newton’s third law.
Thus, the gravitational force F
12
on the body 1
due to 2 and F
21
on the body 2 due to 1 are related
as F
12
= – F
21
.
Before we can apply Eq. (7.5) to objects under
consideration, we have to be careful since the
law refers to point masses whereas we deal with
extended objects which have finite size. If we have
a collection of point masses, the force on any
one of them is the vector sum of the gravitational
forces exerted by the other point masses as
shown in Fig 7.4.
Fig. 7.4 Gravitational force on point mass m
1
is the
vector sum of the gravitational forces exerted
by m
2
, m
3
and m
4
.
The total force on m
1
is
2 1
1 2
21
Gm m
r
= F
? 3 1
21
2
31
Gm m
r
+ r
? ? 4 1
31 41
2
41
Gm m
r
+ r r
Example 7.2 Three equal masses of m kg
each are fixed at the vertices of an
equilateral triangle ABC.
(a) What is the force acting on a mass 2m
placed at the centroid G of the triangle?
(b) What is the force if the mass at the
vertex A is doubled ?
Take AG = BG = CG = 1 m (see Fig. 7.5)
Answer (a) The angle between GC and the
positive xaxis is 30° and so is the angle between
GB and the negative xaxis. The individual forces
in vector notation are
Fig. 7.3 Gravitational force on m
1
due to m
2
is along
r where the vector r is (r
2
– r
1
).
O
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GRAVITATION 131
cases, a simple law results when you do that :
(1) The force of attraction between a hollow
spherical shell of uniform density and a
point mass situated outside is just as if
the entire mass of the shell is
concentrated at the centre of the shell.
Qualitatively this can be understood as
follows: Gravitational forces caused by the
various regions of the shell have components
along the line joining the point mass to the
centre as well as along a direction
prependicular to this line. The components
prependicular to this line cancel out when
summing over all regions of the shell leaving
only a resultant force along the line joining
the point to the centre. The magnitude of
this force works out to be as stated above.
(2) The force of attraction due to a hollow
spherical shell of uniform density, on a
point mass situated inside it is zero.
Qualitatively, we can again understand this
result. Various regions of the spherical shell
attract the point mass inside it in various
directions. These forces cancel each other
completely.
7.4 THE GRAVITATIONAL CONSTANT
The value of the gravitational constant G entering
the Universal law of gravitation can be
determined experimentally and this was first done
by English scientist Henry Cavendish in 1798.
The apparatus used by him is schematically
shown in Fig.7.6
Fig. 7.6 Schematic drawing of Cavendish’s
experiment. S
1
and S
2
are large spheres
which are kept on either side (shown
shades) of the masses at A and B. When
the big spheres are taken to the other side
of the masses (shown by dotted circles),
the bar AB rotates a little since the torque
reverses direction. The angle of rotation can
be measured experimentally.
Fig. 7.5 Three equal masses are placed at the three
vertices of the ? ABC. A mass 2m is placed
at the centroid G.
( )
GA
2
ˆ
1
Gm m
= F j
( )
( ) GB
2
ˆ ˆ
cos 30 sin 30
1
Gm m
? ?
=   F i j
( )
( ) GC
2
ˆ ˆ
cos 30 sin 30
1
Gm m
? ?
= +  F i j
From the principle of superposition and the law
of vector addition, the resultant gravitational
force F
R
on (2m) is
F
R
= F
GA
+ F
GB
+ F
GC
( )
? ?
  + = 30 sin
ˆ
30 cos
ˆ
2
ˆ
2
2 2
R
j i j F Gm Gm
( ) 0 30 sin
ˆ
30 cos
ˆ
2
2
=  +
? ?
j i Gm
Alternatively, one expects on the basis of
symmetry that the resultant force ought to be
zero.
(b) Now if the mass at vertex A is doubled
then
?
For the gravitational force between an extended
object (like the earth) and a point mass, Eq. (7.5) is not
directly applicable. Each point mass in the extended
object will exert a force on the given point mass and
these force will not all be in the same direction. We
have to add up these forces vectorially for all the point
masses in the extended object to get the total force.
This is easily done using calculus. For two special
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