NCERT Textbook - Gravitation Class 11 Notes | EduRev

 Page 1


CHAPTER EIGHT
GRAVITATION
8.1 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
(1564-1642) 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
8.1 Introduction
8.2 Kepler’s laws
8.3 Universal law of
gravitation
8.4 The gravitational
constant
8.5 Acceleration due to
gravity of the earth
8.6 Acceleration due to
gravity below and above
the surface of earth
8.7 Gravitational potential
energy
8.8 Escape speed
8.9 Earth satellites
8.10 Energy of an orbiting
satellite
8.11 Geostationary and polar
satellites
8.12 Weightlessness
Summary
Points to ponder
Exercises
Additional exercises
2020-21
Page 2


CHAPTER EIGHT
GRAVITATION
8.1 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
(1564-1642) 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
8.1 Introduction
8.2 Kepler’s laws
8.3 Universal law of
gravitation
8.4 The gravitational
constant
8.5 Acceleration due to
gravity of the earth
8.6 Acceleration due to
gravity below and above
the surface of earth
8.7 Gravitational potential
energy
8.8 Escape speed
8.9 Earth satellites
8.10 Energy of an orbiting
satellite
8.11 Geostationary and polar
satellites
8.12 Weightlessness
Summary
Points to ponder
Exercises
Additional exercises
2020-21
184 PHYSICS
B
A
C
P
S
S'
2b
2a
Copernicus (1473-1543) 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 (1546-1601)
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.
8.2  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. 8.1(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. 8.1(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. 8.1a). 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. 8.1(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. 8.1(b). The
midpoint of the line PA is the centre of the ellipse
O and the length PO = AO  is called the semi-
major axis of the ellipse. For a circle, the two
focii  merge into one  and the semi-major 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. 8.2).  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. 8.2 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.
3. Law of periods : The square of the time period
of revolution of a planet is proportional to the
cube of the semi-major axis of the ellipse traced
out by  the planet.
Table 8.1 gives the approximate time periods
of revolution of eight* planets around the sun
along with values of their semi-major axes.
* Refer to information given in the Box on Page 182
2020-21
Page 3


CHAPTER EIGHT
GRAVITATION
8.1 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
(1564-1642) 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
8.1 Introduction
8.2 Kepler’s laws
8.3 Universal law of
gravitation
8.4 The gravitational
constant
8.5 Acceleration due to
gravity of the earth
8.6 Acceleration due to
gravity below and above
the surface of earth
8.7 Gravitational potential
energy
8.8 Escape speed
8.9 Earth satellites
8.10 Energy of an orbiting
satellite
8.11 Geostationary and polar
satellites
8.12 Weightlessness
Summary
Points to ponder
Exercises
Additional exercises
2020-21
184 PHYSICS
B
A
C
P
S
S'
2b
2a
Copernicus (1473-1543) 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 (1546-1601)
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.
8.2  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. 8.1(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. 8.1(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. 8.1a). 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. 8.1(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. 8.1(b). The
midpoint of the line PA is the centre of the ellipse
O and the length PO = AO  is called the semi-
major axis of the ellipse. For a circle, the two
focii  merge into one  and the semi-major 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. 8.2).  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. 8.2 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.
3. Law of periods : The square of the time period
of revolution of a planet is proportional to the
cube of the semi-major axis of the ellipse traced
out by  the planet.
Table 8.1 gives the approximate time periods
of revolution of eight* planets around the sun
along with values of their semi-major axes.
* Refer to information given in the Box on Page 182
2020-21
GRAVITATION 185
t
Table 8.1Data from measurement of
planetary motions given below
confirm Kepler’s Law of Periods
(a = Semi-major  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
Pluto* 590 248 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. 8.2) ?A given by
?A
 = ½  (r × v?t) (8.1)
 Hence
?A
/?t   =½ (r × p)/m, (since  v = p/m)
                       =    L / (2 m) (8.2)
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 8.1  Let  the speed of the planet
at  the  perihelion P in Fig. 8.1(a) be v
P 
 and
the Sun-planet 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
t
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. 8.1. 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.
8.3  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
= =
(8.3)
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. (8.3), 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.
Johannes Kepler
(1571–1630)  was a
scientist of German
origin. He formulated
the three laws of
planetary motion based
on the painstaking
observations of Tycho
Brahe and coworkers. Kepler himself was an
assistant to Brahe and it took him sixteen long
years to arrive at the three planetary laws. He
is also known as the founder of geometrical
optics, being the first to describe what happens
to light after it enters a telescope.
* Refer to information given in the Box on Page 182
2020-21
Page 4


CHAPTER EIGHT
GRAVITATION
8.1 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
(1564-1642) 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
8.1 Introduction
8.2 Kepler’s laws
8.3 Universal law of
gravitation
8.4 The gravitational
constant
8.5 Acceleration due to
gravity of the earth
8.6 Acceleration due to
gravity below and above
the surface of earth
8.7 Gravitational potential
energy
8.8 Escape speed
8.9 Earth satellites
8.10 Energy of an orbiting
satellite
8.11 Geostationary and polar
satellites
8.12 Weightlessness
Summary
Points to ponder
Exercises
Additional exercises
2020-21
184 PHYSICS
B
A
C
P
S
S'
2b
2a
Copernicus (1473-1543) 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 (1546-1601)
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.
8.2  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. 8.1(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. 8.1(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. 8.1a). 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. 8.1(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. 8.1(b). The
midpoint of the line PA is the centre of the ellipse
O and the length PO = AO  is called the semi-
major axis of the ellipse. For a circle, the two
focii  merge into one  and the semi-major 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. 8.2).  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. 8.2 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.
3. Law of periods : The square of the time period
of revolution of a planet is proportional to the
cube of the semi-major axis of the ellipse traced
out by  the planet.
Table 8.1 gives the approximate time periods
of revolution of eight* planets around the sun
along with values of their semi-major axes.
* Refer to information given in the Box on Page 182
2020-21
GRAVITATION 185
t
Table 8.1Data from measurement of
planetary motions given below
confirm Kepler’s Law of Periods
(a = Semi-major  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
Pluto* 590 248 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. 8.2) ?A given by
?A
 = ½  (r × v?t) (8.1)
 Hence
?A
/?t   =½ (r × p)/m, (since  v = p/m)
                       =    L / (2 m) (8.2)
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 8.1  Let  the speed of the planet
at  the  perihelion P in Fig. 8.1(a) be v
P 
 and
the Sun-planet 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
t
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. 8.1. 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.
8.3  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
= =
(8.3)
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. (8.3), 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.
Johannes Kepler
(1571–1630)  was a
scientist of German
origin. He formulated
the three laws of
planetary motion based
on the painstaking
observations of Tycho
Brahe and coworkers. Kepler himself was an
assistant to Brahe and it took him sixteen long
years to arrive at the three planetary laws. He
is also known as the founder of geometrical
optics, being the first to describe what happens
to light after it enters a telescope.
* Refer to information given in the Box on Page 182
2020-21
186 PHYSICS
Central Forces
We know the time rate of change of the angular momentum of a single particle about the origin
is
d
dt
= ×
l
r F
The angular momentum of the particle is conserved, if the torque = × r F t t t t due to the
force F on it vanishes. This happens either when F is zero or when F is along r. We are
interested in forces which satisfy the latter condition. Central forces satisfy this condition.
A ‘central’ force is always directed towards or away from a fixed point, i.e., along the position
vector of the point of application of the force with respect to the fixed point. (See Figure below.)
Further, the magnitude of a central force F depends on r, the distance of the point of application
of the force from the fixed point; F = F(r).
In the motion under a central force the angular momentum is always conserved. Two important
results follow from this:
(1) The motion of a particle under the central force is always confined to a plane.
(2) The position vector of the particle with respect to the centre of the force (i.e. the fixed point)
has a constant areal velocity. In other words the position vector sweeps out equal areas in
equal times as the particle moves under the influence of the central force.
Try to prove both these results. You may need to know that the areal velocity is given by :
dA/dt = ½ r v sin a.
An immediate application of the above discussion can be made to the motion of a planet
under the gravitational force of the sun. For convenience the sun may be taken to be so heavy
that it is at rest. The gravitational force of the sun on the planet is directed towards the sun.
This force also satisfies the requirement F = F(r), since F = G m
1
m
2
/r
2
 where m
1
 and m
2
 are
respectively the masses of the planet and the sun and G is the universal constant of gravitation.
The two results (1) and (2) described above, therefore, apply to the motion of the planet. In fact,
the result (2) is the well-known second law of Kepler.
Tr is the trejectory of the particle under the central force. At a position P, the force is directed
along OP, O is the centre of the force taken as the origin. In time ?t, the particle moves from P to P',
arc PP' = ?s = v ?t. The tangent PQ  at P to the trajectory gives the direction of the velocity at P. The
area swept in ?t is the area of sector POP' ( ) sin r a ˜ PP'/2 = (r v sin a) ?t/2.)
2020-21
Page 5


CHAPTER EIGHT
GRAVITATION
8.1 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
(1564-1642) 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
8.1 Introduction
8.2 Kepler’s laws
8.3 Universal law of
gravitation
8.4 The gravitational
constant
8.5 Acceleration due to
gravity of the earth
8.6 Acceleration due to
gravity below and above
the surface of earth
8.7 Gravitational potential
energy
8.8 Escape speed
8.9 Earth satellites
8.10 Energy of an orbiting
satellite
8.11 Geostationary and polar
satellites
8.12 Weightlessness
Summary
Points to ponder
Exercises
Additional exercises
2020-21
184 PHYSICS
B
A
C
P
S
S'
2b
2a
Copernicus (1473-1543) 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 (1546-1601)
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.
8.2  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. 8.1(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. 8.1(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. 8.1a). 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. 8.1(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. 8.1(b). The
midpoint of the line PA is the centre of the ellipse
O and the length PO = AO  is called the semi-
major axis of the ellipse. For a circle, the two
focii  merge into one  and the semi-major 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. 8.2).  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. 8.2 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.
3. Law of periods : The square of the time period
of revolution of a planet is proportional to the
cube of the semi-major axis of the ellipse traced
out by  the planet.
Table 8.1 gives the approximate time periods
of revolution of eight* planets around the sun
along with values of their semi-major axes.
* Refer to information given in the Box on Page 182
2020-21
GRAVITATION 185
t
Table 8.1Data from measurement of
planetary motions given below
confirm Kepler’s Law of Periods
(a = Semi-major  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
Pluto* 590 248 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. 8.2) ?A given by
?A
 = ½  (r × v?t) (8.1)
 Hence
?A
/?t   =½ (r × p)/m, (since  v = p/m)
                       =    L / (2 m) (8.2)
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 8.1  Let  the speed of the planet
at  the  perihelion P in Fig. 8.1(a) be v
P 
 and
the Sun-planet 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
t
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. 8.1. 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.
8.3  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
= =
(8.3)
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. (8.3), 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.
Johannes Kepler
(1571–1630)  was a
scientist of German
origin. He formulated
the three laws of
planetary motion based
on the painstaking
observations of Tycho
Brahe and coworkers. Kepler himself was an
assistant to Brahe and it took him sixteen long
years to arrive at the three planetary laws. He
is also known as the founder of geometrical
optics, being the first to describe what happens
to light after it enters a telescope.
* Refer to information given in the Box on Page 182
2020-21
186 PHYSICS
Central Forces
We know the time rate of change of the angular momentum of a single particle about the origin
is
d
dt
= ×
l
r F
The angular momentum of the particle is conserved, if the torque = × r F t t t t due to the
force F on it vanishes. This happens either when F is zero or when F is along r. We are
interested in forces which satisfy the latter condition. Central forces satisfy this condition.
A ‘central’ force is always directed towards or away from a fixed point, i.e., along the position
vector of the point of application of the force with respect to the fixed point. (See Figure below.)
Further, the magnitude of a central force F depends on r, the distance of the point of application
of the force from the fixed point; F = F(r).
In the motion under a central force the angular momentum is always conserved. Two important
results follow from this:
(1) The motion of a particle under the central force is always confined to a plane.
(2) The position vector of the particle with respect to the centre of the force (i.e. the fixed point)
has a constant areal velocity. In other words the position vector sweeps out equal areas in
equal times as the particle moves under the influence of the central force.
Try to prove both these results. You may need to know that the areal velocity is given by :
dA/dt = ½ r v sin a.
An immediate application of the above discussion can be made to the motion of a planet
under the gravitational force of the sun. For convenience the sun may be taken to be so heavy
that it is at rest. The gravitational force of the sun on the planet is directed towards the sun.
This force also satisfies the requirement F = F(r), since F = G m
1
m
2
/r
2
 where m
1
 and m
2
 are
respectively the masses of the planet and the sun and G is the universal constant of gravitation.
The two results (1) and (2) described above, therefore, apply to the motion of the planet. In fact,
the result (2) is the well-known second law of Kepler.
Tr is the trejectory of the particle under the central force. At a position P, the force is directed
along OP, O is the centre of the force taken as the origin. In time ?t, the particle moves from P to P',
arc PP' = ?s = v ?t. The tangent PQ  at P to the trajectory gives the direction of the velocity at P. The
area swept in ?t is the area of sector POP' ( ) sin r a ˜ PP'/2 = (r v sin a) ?t/2.)
2020-21
GRAVITATION 187
t
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 (8.4)
in agreement with a value of g  9.8  m s
-2
 and
the value of a
m
 from Eq. (8.3).  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
(8.5)
Equation (8.5) 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. 8.3.
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. (8.5) 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 8.4.
Fig. 8.4 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 8.2  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. 8.5)
Answer  (a) The angle between GC and the
positive x-axis is 30° and so is the angle between
GB and the negative x-axis. The individual forces
in vector notation are
Fig. 8.3 Gravitational force on m
1
 due to m
2
 is along
r where the vector r is (r
2
– r
1
).
O
2020-21