NCERT Textbook: Gravitation | Old & New NCERTs for IAS Preparation (Must Read) - UPSC PDF Download

 Page 1


In Chapters 8 and 9, we have learnt about
the motion of objects and force as the cause
of motion. We have learnt that a force is
needed to change the speed or the direction
of motion of an object. We always observe that
an object dropped from a height falls towards
the earth. We know that all the planets go
around the Sun. The moon goes around the
earth. In all these cases, there must be some
force acting on the objects, the planets and
on the moon. Isaac Newton could grasp that
the same force is responsible for all these.
This force is called the gravitational force.
In this chapter we shall learn about
gravitation and the universal law of
gravitation. We shall discuss the motion of
objects under the influence of gravitational
force on the earth. We shall study how the
weight of a body varies from place to place.
We shall also discuss the conditions for
objects to float in liquids.
10.1 Gravitation
We know that the moon goes around the
earth. An object when thrown upwards,
reaches a certain height and then falls
downwards. It is said that when Newton was
sitting under a tree, an apple fell on him. The
fall of the apple made Newton start thinking.
He thought that: if the earth can attract an
apple, can it not attract the moon? Is the force
the same in both cases? He conjectured  that
the same type of force is responsible in both
the cases. He argued that at each point of its
orbit, the moon falls towards the earth,
instead of going off in a straight line. So, it
must be attracted by the earth. But we do
not really see the moon falling towards the
earth.
Let us try to understand the motion of
the moon by recalling activity 8.11.
Activity _____________10.1
• Take a piece of thread.
• Tie a small stone at one end. Hold the
other end of the thread and whirl it
round, as shown in Fig. 10.1.
• Note the motion of the stone.
• Release the thread.
• Again, note the direction of motion of
the stone.
Fig. 10.1: A stone describing a circular path with a
velocity of constant magnitude.
Before the thread is released, the stone
moves in a circular path with a certain speed
and changes direction at every point. The
change in direction involves change in velocity
or acceleration. The force that causes this
acceleration and keeps the body moving along
the circular path is acting towards the centre.
This force is called the centripetal (meaning
‘centre-seeking’) force. In the absence of this
10 10
10 10 10
G G G G GRAVITATION RAVITATION RAVITATION RAVITATION RAVITATION
Chapter
Page 2


In Chapters 8 and 9, we have learnt about
the motion of objects and force as the cause
of motion. We have learnt that a force is
needed to change the speed or the direction
of motion of an object. We always observe that
an object dropped from a height falls towards
the earth. We know that all the planets go
around the Sun. The moon goes around the
earth. In all these cases, there must be some
force acting on the objects, the planets and
on the moon. Isaac Newton could grasp that
the same force is responsible for all these.
This force is called the gravitational force.
In this chapter we shall learn about
gravitation and the universal law of
gravitation. We shall discuss the motion of
objects under the influence of gravitational
force on the earth. We shall study how the
weight of a body varies from place to place.
We shall also discuss the conditions for
objects to float in liquids.
10.1 Gravitation
We know that the moon goes around the
earth. An object when thrown upwards,
reaches a certain height and then falls
downwards. It is said that when Newton was
sitting under a tree, an apple fell on him. The
fall of the apple made Newton start thinking.
He thought that: if the earth can attract an
apple, can it not attract the moon? Is the force
the same in both cases? He conjectured  that
the same type of force is responsible in both
the cases. He argued that at each point of its
orbit, the moon falls towards the earth,
instead of going off in a straight line. So, it
must be attracted by the earth. But we do
not really see the moon falling towards the
earth.
Let us try to understand the motion of
the moon by recalling activity 8.11.
Activity _____________10.1
• Take a piece of thread.
• Tie a small stone at one end. Hold the
other end of the thread and whirl it
round, as shown in Fig. 10.1.
• Note the motion of the stone.
• Release the thread.
• Again, note the direction of motion of
the stone.
Fig. 10.1: A stone describing a circular path with a
velocity of constant magnitude.
Before the thread is released, the stone
moves in a circular path with a certain speed
and changes direction at every point. The
change in direction involves change in velocity
or acceleration. The force that causes this
acceleration and keeps the body moving along
the circular path is acting towards the centre.
This force is called the centripetal (meaning
‘centre-seeking’) force. In the absence of this
10 10
10 10 10
G G G G GRAVITATION RAVITATION RAVITATION RAVITATION RAVITATION
Chapter
SCIENCE 132
10.1.1 UNIVERSAL LAW OF GRAVITATION
Every object in the universe attracts every
other object with a force which is proportional
to the product of their masses and inversely
proportional to the square of the distance
between them. The force is along the line
joining the centres of two objects.
force, the stone flies off along a straight line.
This straight line will be a tangent to the
circular path.
More to  know
Tangent to a circle
A straight line that meets the circle
at one and only one point is called a
tangent to the circle. Straight line
ABC is a tangent to the circle at
point B.
The motion of the moon around the earth
is due to the centripetal force. The centripetal
force is provided by the force of attraction of
the earth. If there were no such force, the
moon would pursue a uniform straight line
motion.
It is seen that a falling apple is attracted
towards the earth. Does the apple attract the
earth? If so, we do not see the earth moving
towards an apple. Why?
According to the third law of motion, the
apple does attract the earth. But according
to the second law of motion, for a given force,
acceleration is inversely proportional to the
mass of an object [Eq. (9.4)]. The mass of an
apple is negligibly small compared to that of
the earth. So, we do not see the earth moving
towards the apple. Extend the same argument
for why the earth does not move towards the
moon.
In our solar system, all the planets go
around the Sun. By arguing the same way,
we can say that there exists a force between
the Sun and the planets. From the above facts
Newton concluded that not only does the
earth attract an apple and the moon, but all
objects in the universe attract each other. This
force of attraction between objects is called
the gravitational force.
G
2
Mm
F=
d
Fig. 10.2: The gravitational force between two
uniform objects is directed along the line
joining their centres.
Let two objects A and B of masses M and
m lie at a distance d from each other as shown
in Fig. 10.2. Let the force of attraction between
two objects be F. According to the universal
law of gravitation, the force between two
objects is directly proportional to the product
of their masses. That is,
F
?
M × m (10.1)
And the force between two objects is inversely
proportional to the square of the distance
between them, that is,
?
2
1
F
d
(10.2)
Combining Eqs. (10.1) and (10.2), we get
F 
?
 
2
× Mm
d
(10.3)
or,  
G
2
M× m
F=
d
(10.4)
where G is the constant of proportionality and
is called the universal gravitation constant.
By multiplying crosswise, Eq. (10.4) gives
 F × d 
2
 = G M × m
Page 3


In Chapters 8 and 9, we have learnt about
the motion of objects and force as the cause
of motion. We have learnt that a force is
needed to change the speed or the direction
of motion of an object. We always observe that
an object dropped from a height falls towards
the earth. We know that all the planets go
around the Sun. The moon goes around the
earth. In all these cases, there must be some
force acting on the objects, the planets and
on the moon. Isaac Newton could grasp that
the same force is responsible for all these.
This force is called the gravitational force.
In this chapter we shall learn about
gravitation and the universal law of
gravitation. We shall discuss the motion of
objects under the influence of gravitational
force on the earth. We shall study how the
weight of a body varies from place to place.
We shall also discuss the conditions for
objects to float in liquids.
10.1 Gravitation
We know that the moon goes around the
earth. An object when thrown upwards,
reaches a certain height and then falls
downwards. It is said that when Newton was
sitting under a tree, an apple fell on him. The
fall of the apple made Newton start thinking.
He thought that: if the earth can attract an
apple, can it not attract the moon? Is the force
the same in both cases? He conjectured  that
the same type of force is responsible in both
the cases. He argued that at each point of its
orbit, the moon falls towards the earth,
instead of going off in a straight line. So, it
must be attracted by the earth. But we do
not really see the moon falling towards the
earth.
Let us try to understand the motion of
the moon by recalling activity 8.11.
Activity _____________10.1
• Take a piece of thread.
• Tie a small stone at one end. Hold the
other end of the thread and whirl it
round, as shown in Fig. 10.1.
• Note the motion of the stone.
• Release the thread.
• Again, note the direction of motion of
the stone.
Fig. 10.1: A stone describing a circular path with a
velocity of constant magnitude.
Before the thread is released, the stone
moves in a circular path with a certain speed
and changes direction at every point. The
change in direction involves change in velocity
or acceleration. The force that causes this
acceleration and keeps the body moving along
the circular path is acting towards the centre.
This force is called the centripetal (meaning
‘centre-seeking’) force. In the absence of this
10 10
10 10 10
G G G G GRAVITATION RAVITATION RAVITATION RAVITATION RAVITATION
Chapter
SCIENCE 132
10.1.1 UNIVERSAL LAW OF GRAVITATION
Every object in the universe attracts every
other object with a force which is proportional
to the product of their masses and inversely
proportional to the square of the distance
between them. The force is along the line
joining the centres of two objects.
force, the stone flies off along a straight line.
This straight line will be a tangent to the
circular path.
More to  know
Tangent to a circle
A straight line that meets the circle
at one and only one point is called a
tangent to the circle. Straight line
ABC is a tangent to the circle at
point B.
The motion of the moon around the earth
is due to the centripetal force. The centripetal
force is provided by the force of attraction of
the earth. If there were no such force, the
moon would pursue a uniform straight line
motion.
It is seen that a falling apple is attracted
towards the earth. Does the apple attract the
earth? If so, we do not see the earth moving
towards an apple. Why?
According to the third law of motion, the
apple does attract the earth. But according
to the second law of motion, for a given force,
acceleration is inversely proportional to the
mass of an object [Eq. (9.4)]. The mass of an
apple is negligibly small compared to that of
the earth. So, we do not see the earth moving
towards the apple. Extend the same argument
for why the earth does not move towards the
moon.
In our solar system, all the planets go
around the Sun. By arguing the same way,
we can say that there exists a force between
the Sun and the planets. From the above facts
Newton concluded that not only does the
earth attract an apple and the moon, but all
objects in the universe attract each other. This
force of attraction between objects is called
the gravitational force.
G
2
Mm
F=
d
Fig. 10.2: The gravitational force between two
uniform objects is directed along the line
joining their centres.
Let two objects A and B of masses M and
m lie at a distance d from each other as shown
in Fig. 10.2. Let the force of attraction between
two objects be F. According to the universal
law of gravitation, the force between two
objects is directly proportional to the product
of their masses. That is,
F
?
M × m (10.1)
And the force between two objects is inversely
proportional to the square of the distance
between them, that is,
?
2
1
F
d
(10.2)
Combining Eqs. (10.1) and (10.2), we get
F 
?
 
2
× Mm
d
(10.3)
or,  
G
2
M× m
F=
d
(10.4)
where G is the constant of proportionality and
is called the universal gravitation constant.
By multiplying crosswise, Eq. (10.4) gives
 F × d 
2
 = G M × m
GRAVITATION 133
Isaac Newton was born
in Woolsthorpe near
Grantham,  England.
He  is generally
regarded as the most
original and
influential theorist in
the history of science.
He was born in a poor
farming  family. But he
was not good at
farming. He was sent
to  study at  Cambridge
University in 1661. In
1665 a plague broke
out in Cambridge and so Newton took a year
off. It was during this year that the incident of
the apple falling on him is said to have
occurred. This incident prompted Newton to
explore the possibility of connecting gravity
with the force that kept the moon in its orbit.
This led him to the universal law of
gravitation.  It is remarkable that many great
scientists before him knew of gravity but failed
to realise it.
Newton formulated the well-known laws of
motion.  He worked on theories of light and
colour .  He designed an astronomical telescope
to carry out astronomical observations.
Newton was also a great mathematician. He
invented a new branch of mathematics, called
calculus. He used it to prove that for objects
outside a sphere of uniform density, the sphere
behaves as if the whole of its mass is
concentrated at its centre. Newton
transformed the structure of physical
science with his three laws of motion and the
universal law of gravitation. As the keystone
of the scientific revolution of the seventeenth
century, Newton’s work combined the
contributions of Copernicus, Kepler,  Galileo,
and others into a new powerful synthesis.
It is remarkable that though the
gravitational theory could not be verified at
that time, there was hardly any doubt about
its correctness. This is because Newton based
his theory on sound scientific reasoning and
backed it with mathematics.  This made the
theory simple and elegant.  These qualities are
now recognised as essential requirements of a
good scientific theory.
Isaac Newton
(1642 – 1727)
How did Newton guess the
inverse-square rule?
There has always been a great interest
in the motion of planets. By the 16th
century, a lot of data on the motion of
planets had been collected by many
astronomers. Based on these data
Johannes Kepler derived three laws,
which govern the motion of planets.
These are called Kepler’s laws. These are:
1. The orbit of a planet is an ellipse with
the Sun at one of the foci, as shown in
the figure given below. In this figure O
is the position of the Sun.
2. The line joining the planet and the Sun
sweep equal areas in equal intervals
of time. Thus, if the time of travel from
A to B is the same as that from C to D,
then the areas OAB and OCD are
equal.
3. The cube of the mean distance of a
planet from the Sun is proportional to
the square of its orbital period T. Or,
r
3
/T
2
 = constant.
It is important to note that Kepler
could not give a theory to explain
the motion of planets. It was Newton
who showed that the cause of the
planetary motion is the gravitational
force that the Sun exerts on them. Newton
used the third law
of Kepler to
calculate the
gravitational force
of attraction. The
gravitational force
of the earth is
weakened by distance. A simple argument
goes like this. We can assume that the
planetary orbits are circular. Suppose the
orbital velocity is v and the radius of the
orbit is r. Then the force acting on an
orbiting planet is given by F
?
v
2
/r.
If T denotes the period, then v = 2pr/T,
so that v
2 
?
 r
2
/T
2
.
We can rewrite this as v
2 
?
 (1/r) ×
( r
3
/T
2
). Since r
3
/T
2
 is constant by Kepler’s
third law, we have v
2  
?
 1/r. Combining
this with F
 
?
 v
2
/ r, we get, F
 
?
 1/ r
2
.
A
B
C
D
O
Page 4


In Chapters 8 and 9, we have learnt about
the motion of objects and force as the cause
of motion. We have learnt that a force is
needed to change the speed or the direction
of motion of an object. We always observe that
an object dropped from a height falls towards
the earth. We know that all the planets go
around the Sun. The moon goes around the
earth. In all these cases, there must be some
force acting on the objects, the planets and
on the moon. Isaac Newton could grasp that
the same force is responsible for all these.
This force is called the gravitational force.
In this chapter we shall learn about
gravitation and the universal law of
gravitation. We shall discuss the motion of
objects under the influence of gravitational
force on the earth. We shall study how the
weight of a body varies from place to place.
We shall also discuss the conditions for
objects to float in liquids.
10.1 Gravitation
We know that the moon goes around the
earth. An object when thrown upwards,
reaches a certain height and then falls
downwards. It is said that when Newton was
sitting under a tree, an apple fell on him. The
fall of the apple made Newton start thinking.
He thought that: if the earth can attract an
apple, can it not attract the moon? Is the force
the same in both cases? He conjectured  that
the same type of force is responsible in both
the cases. He argued that at each point of its
orbit, the moon falls towards the earth,
instead of going off in a straight line. So, it
must be attracted by the earth. But we do
not really see the moon falling towards the
earth.
Let us try to understand the motion of
the moon by recalling activity 8.11.
Activity _____________10.1
• Take a piece of thread.
• Tie a small stone at one end. Hold the
other end of the thread and whirl it
round, as shown in Fig. 10.1.
• Note the motion of the stone.
• Release the thread.
• Again, note the direction of motion of
the stone.
Fig. 10.1: A stone describing a circular path with a
velocity of constant magnitude.
Before the thread is released, the stone
moves in a circular path with a certain speed
and changes direction at every point. The
change in direction involves change in velocity
or acceleration. The force that causes this
acceleration and keeps the body moving along
the circular path is acting towards the centre.
This force is called the centripetal (meaning
‘centre-seeking’) force. In the absence of this
10 10
10 10 10
G G G G GRAVITATION RAVITATION RAVITATION RAVITATION RAVITATION
Chapter
SCIENCE 132
10.1.1 UNIVERSAL LAW OF GRAVITATION
Every object in the universe attracts every
other object with a force which is proportional
to the product of their masses and inversely
proportional to the square of the distance
between them. The force is along the line
joining the centres of two objects.
force, the stone flies off along a straight line.
This straight line will be a tangent to the
circular path.
More to  know
Tangent to a circle
A straight line that meets the circle
at one and only one point is called a
tangent to the circle. Straight line
ABC is a tangent to the circle at
point B.
The motion of the moon around the earth
is due to the centripetal force. The centripetal
force is provided by the force of attraction of
the earth. If there were no such force, the
moon would pursue a uniform straight line
motion.
It is seen that a falling apple is attracted
towards the earth. Does the apple attract the
earth? If so, we do not see the earth moving
towards an apple. Why?
According to the third law of motion, the
apple does attract the earth. But according
to the second law of motion, for a given force,
acceleration is inversely proportional to the
mass of an object [Eq. (9.4)]. The mass of an
apple is negligibly small compared to that of
the earth. So, we do not see the earth moving
towards the apple. Extend the same argument
for why the earth does not move towards the
moon.
In our solar system, all the planets go
around the Sun. By arguing the same way,
we can say that there exists a force between
the Sun and the planets. From the above facts
Newton concluded that not only does the
earth attract an apple and the moon, but all
objects in the universe attract each other. This
force of attraction between objects is called
the gravitational force.
G
2
Mm
F=
d
Fig. 10.2: The gravitational force between two
uniform objects is directed along the line
joining their centres.
Let two objects A and B of masses M and
m lie at a distance d from each other as shown
in Fig. 10.2. Let the force of attraction between
two objects be F. According to the universal
law of gravitation, the force between two
objects is directly proportional to the product
of their masses. That is,
F
?
M × m (10.1)
And the force between two objects is inversely
proportional to the square of the distance
between them, that is,
?
2
1
F
d
(10.2)
Combining Eqs. (10.1) and (10.2), we get
F 
?
 
2
× Mm
d
(10.3)
or,  
G
2
M× m
F=
d
(10.4)
where G is the constant of proportionality and
is called the universal gravitation constant.
By multiplying crosswise, Eq. (10.4) gives
 F × d 
2
 = G M × m
GRAVITATION 133
Isaac Newton was born
in Woolsthorpe near
Grantham,  England.
He  is generally
regarded as the most
original and
influential theorist in
the history of science.
He was born in a poor
farming  family. But he
was not good at
farming. He was sent
to  study at  Cambridge
University in 1661. In
1665 a plague broke
out in Cambridge and so Newton took a year
off. It was during this year that the incident of
the apple falling on him is said to have
occurred. This incident prompted Newton to
explore the possibility of connecting gravity
with the force that kept the moon in its orbit.
This led him to the universal law of
gravitation.  It is remarkable that many great
scientists before him knew of gravity but failed
to realise it.
Newton formulated the well-known laws of
motion.  He worked on theories of light and
colour .  He designed an astronomical telescope
to carry out astronomical observations.
Newton was also a great mathematician. He
invented a new branch of mathematics, called
calculus. He used it to prove that for objects
outside a sphere of uniform density, the sphere
behaves as if the whole of its mass is
concentrated at its centre. Newton
transformed the structure of physical
science with his three laws of motion and the
universal law of gravitation. As the keystone
of the scientific revolution of the seventeenth
century, Newton’s work combined the
contributions of Copernicus, Kepler,  Galileo,
and others into a new powerful synthesis.
It is remarkable that though the
gravitational theory could not be verified at
that time, there was hardly any doubt about
its correctness. This is because Newton based
his theory on sound scientific reasoning and
backed it with mathematics.  This made the
theory simple and elegant.  These qualities are
now recognised as essential requirements of a
good scientific theory.
Isaac Newton
(1642 – 1727)
How did Newton guess the
inverse-square rule?
There has always been a great interest
in the motion of planets. By the 16th
century, a lot of data on the motion of
planets had been collected by many
astronomers. Based on these data
Johannes Kepler derived three laws,
which govern the motion of planets.
These are called Kepler’s laws. These are:
1. The orbit of a planet is an ellipse with
the Sun at one of the foci, as shown in
the figure given below. In this figure O
is the position of the Sun.
2. The line joining the planet and the Sun
sweep equal areas in equal intervals
of time. Thus, if the time of travel from
A to B is the same as that from C to D,
then the areas OAB and OCD are
equal.
3. The cube of the mean distance of a
planet from the Sun is proportional to
the square of its orbital period T. Or,
r
3
/T
2
 = constant.
It is important to note that Kepler
could not give a theory to explain
the motion of planets. It was Newton
who showed that the cause of the
planetary motion is the gravitational
force that the Sun exerts on them. Newton
used the third law
of Kepler to
calculate the
gravitational force
of attraction. The
gravitational force
of the earth is
weakened by distance. A simple argument
goes like this. We can assume that the
planetary orbits are circular. Suppose the
orbital velocity is v and the radius of the
orbit is r. Then the force acting on an
orbiting planet is given by F
?
v
2
/r.
If T denotes the period, then v = 2pr/T,
so that v
2 
?
 r
2
/T
2
.
We can rewrite this as v
2 
?
 (1/r) ×
( r
3
/T
2
). Since r
3
/T
2
 is constant by Kepler’s
third law, we have v
2  
?
 1/r. Combining
this with F
 
?
 v
2
/ r, we get, F
 
?
 1/ r
2
.
A
B
C
D
O
SCIENCE 134
From Eq. (10.4), the force exerted by
the earth on the moon is
G
2
M × m
F=
d
11 2 -2 24 22
82
6.7 10 N m kg 6 10 kg 7.4 10 kg
(3.84 10 m)
-
×××××
=
×
= 2.01 × 10
20
 N.
Thus, the force exerted by the earth on
the moon is 2.01 × 10
20
 N.
uestions
1. State the universal law of
gravitation.
2. Write the formula to find the
magnitude of the gravitational
force between the earth and an
object on the surface of the earth.
10.1.2 IMPORTANCE OF THE UNIVERSAL
LAW OF GRAVITATION
The universal law of gravitation successfully
explained several phenomena which were
believed to be unconnected:
(i) the force that binds us to the earth;
(ii) the motion of the moon around the
earth;
(iii) the motion of planets around the Sun;
and
(iv) the tides due to the moon and the Sun.
10.2 Free Fall
Let us try to understand the meaning of free
fall by performing this activity.
Activity _____________10.2
• Take a stone.
• Throw it upwards.
• It reaches a certain height and then it
starts falling down.
We have learnt that the earth attracts
objects towards it. This is due to the
gravitational force. Whenever objects fall
towards the earth under this force alone, we
say that the objects are in free fall. Is there
or
2
G =
×
Fd
Mm
(10.5)
The SI unit of G can be obtained by
substituting the units of force, distance and
mass in Eq. (10.5) as N m
2
 kg
–2
.
The value of G was found out by
Henry Cavendish (1731 – 1810) by using a
sensitive balance. The accepted value of G is
6.673 × 10
–11
 N m
2
 kg
–2
.
We know that there exists a force of
attraction between any two objects. Compute
the value of this force between you and your
friend sitting closeby. Conclude how you do
not experience this force!
The law is universal in the sense that
it is applicable to all bodies, whether
the bodies are big or small, whether
they are celestial or terrestrial.
Inverse-square
Saying that F is inversely
proportional to the square of d
means, for example, that if d gets
bigger by a factor of 6, F becomes
1
36
times smaller.
Example 10.1 The mass of the earth is
6 × 10
24
 kg and that of the moon is
7.4 × 10
22
 kg. If the distance between
the earth and the moon is 3.84×10
5 
km,
calculate the force exerted by the earth
on the moon. G = 6.7 × 10
–11
 N m
2
 kg
-2
.
Solution:
The mass of the earth, M = 6 × 10
24
 kg
The mass of the moon,
m = 7.4 × 10
22
 kg
The distance between the earth and the
moon,
d = 3.84 × 10
5 
km
= 3.84 × 10
5 
× 1000 m
= 3.84 × 10
8
 m
G = 6.7 × 10
–11
 N m
2
 kg
–2
More to  know
Q
Page 5


In Chapters 8 and 9, we have learnt about
the motion of objects and force as the cause
of motion. We have learnt that a force is
needed to change the speed or the direction
of motion of an object. We always observe that
an object dropped from a height falls towards
the earth. We know that all the planets go
around the Sun. The moon goes around the
earth. In all these cases, there must be some
force acting on the objects, the planets and
on the moon. Isaac Newton could grasp that
the same force is responsible for all these.
This force is called the gravitational force.
In this chapter we shall learn about
gravitation and the universal law of
gravitation. We shall discuss the motion of
objects under the influence of gravitational
force on the earth. We shall study how the
weight of a body varies from place to place.
We shall also discuss the conditions for
objects to float in liquids.
10.1 Gravitation
We know that the moon goes around the
earth. An object when thrown upwards,
reaches a certain height and then falls
downwards. It is said that when Newton was
sitting under a tree, an apple fell on him. The
fall of the apple made Newton start thinking.
He thought that: if the earth can attract an
apple, can it not attract the moon? Is the force
the same in both cases? He conjectured  that
the same type of force is responsible in both
the cases. He argued that at each point of its
orbit, the moon falls towards the earth,
instead of going off in a straight line. So, it
must be attracted by the earth. But we do
not really see the moon falling towards the
earth.
Let us try to understand the motion of
the moon by recalling activity 8.11.
Activity _____________10.1
• Take a piece of thread.
• Tie a small stone at one end. Hold the
other end of the thread and whirl it
round, as shown in Fig. 10.1.
• Note the motion of the stone.
• Release the thread.
• Again, note the direction of motion of
the stone.
Fig. 10.1: A stone describing a circular path with a
velocity of constant magnitude.
Before the thread is released, the stone
moves in a circular path with a certain speed
and changes direction at every point. The
change in direction involves change in velocity
or acceleration. The force that causes this
acceleration and keeps the body moving along
the circular path is acting towards the centre.
This force is called the centripetal (meaning
‘centre-seeking’) force. In the absence of this
10 10
10 10 10
G G G G GRAVITATION RAVITATION RAVITATION RAVITATION RAVITATION
Chapter
SCIENCE 132
10.1.1 UNIVERSAL LAW OF GRAVITATION
Every object in the universe attracts every
other object with a force which is proportional
to the product of their masses and inversely
proportional to the square of the distance
between them. The force is along the line
joining the centres of two objects.
force, the stone flies off along a straight line.
This straight line will be a tangent to the
circular path.
More to  know
Tangent to a circle
A straight line that meets the circle
at one and only one point is called a
tangent to the circle. Straight line
ABC is a tangent to the circle at
point B.
The motion of the moon around the earth
is due to the centripetal force. The centripetal
force is provided by the force of attraction of
the earth. If there were no such force, the
moon would pursue a uniform straight line
motion.
It is seen that a falling apple is attracted
towards the earth. Does the apple attract the
earth? If so, we do not see the earth moving
towards an apple. Why?
According to the third law of motion, the
apple does attract the earth. But according
to the second law of motion, for a given force,
acceleration is inversely proportional to the
mass of an object [Eq. (9.4)]. The mass of an
apple is negligibly small compared to that of
the earth. So, we do not see the earth moving
towards the apple. Extend the same argument
for why the earth does not move towards the
moon.
In our solar system, all the planets go
around the Sun. By arguing the same way,
we can say that there exists a force between
the Sun and the planets. From the above facts
Newton concluded that not only does the
earth attract an apple and the moon, but all
objects in the universe attract each other. This
force of attraction between objects is called
the gravitational force.
G
2
Mm
F=
d
Fig. 10.2: The gravitational force between two
uniform objects is directed along the line
joining their centres.
Let two objects A and B of masses M and
m lie at a distance d from each other as shown
in Fig. 10.2. Let the force of attraction between
two objects be F. According to the universal
law of gravitation, the force between two
objects is directly proportional to the product
of their masses. That is,
F
?
M × m (10.1)
And the force between two objects is inversely
proportional to the square of the distance
between them, that is,
?
2
1
F
d
(10.2)
Combining Eqs. (10.1) and (10.2), we get
F 
?
 
2
× Mm
d
(10.3)
or,  
G
2
M× m
F=
d
(10.4)
where G is the constant of proportionality and
is called the universal gravitation constant.
By multiplying crosswise, Eq. (10.4) gives
 F × d 
2
 = G M × m
GRAVITATION 133
Isaac Newton was born
in Woolsthorpe near
Grantham,  England.
He  is generally
regarded as the most
original and
influential theorist in
the history of science.
He was born in a poor
farming  family. But he
was not good at
farming. He was sent
to  study at  Cambridge
University in 1661. In
1665 a plague broke
out in Cambridge and so Newton took a year
off. It was during this year that the incident of
the apple falling on him is said to have
occurred. This incident prompted Newton to
explore the possibility of connecting gravity
with the force that kept the moon in its orbit.
This led him to the universal law of
gravitation.  It is remarkable that many great
scientists before him knew of gravity but failed
to realise it.
Newton formulated the well-known laws of
motion.  He worked on theories of light and
colour .  He designed an astronomical telescope
to carry out astronomical observations.
Newton was also a great mathematician. He
invented a new branch of mathematics, called
calculus. He used it to prove that for objects
outside a sphere of uniform density, the sphere
behaves as if the whole of its mass is
concentrated at its centre. Newton
transformed the structure of physical
science with his three laws of motion and the
universal law of gravitation. As the keystone
of the scientific revolution of the seventeenth
century, Newton’s work combined the
contributions of Copernicus, Kepler,  Galileo,
and others into a new powerful synthesis.
It is remarkable that though the
gravitational theory could not be verified at
that time, there was hardly any doubt about
its correctness. This is because Newton based
his theory on sound scientific reasoning and
backed it with mathematics.  This made the
theory simple and elegant.  These qualities are
now recognised as essential requirements of a
good scientific theory.
Isaac Newton
(1642 – 1727)
How did Newton guess the
inverse-square rule?
There has always been a great interest
in the motion of planets. By the 16th
century, a lot of data on the motion of
planets had been collected by many
astronomers. Based on these data
Johannes Kepler derived three laws,
which govern the motion of planets.
These are called Kepler’s laws. These are:
1. The orbit of a planet is an ellipse with
the Sun at one of the foci, as shown in
the figure given below. In this figure O
is the position of the Sun.
2. The line joining the planet and the Sun
sweep equal areas in equal intervals
of time. Thus, if the time of travel from
A to B is the same as that from C to D,
then the areas OAB and OCD are
equal.
3. The cube of the mean distance of a
planet from the Sun is proportional to
the square of its orbital period T. Or,
r
3
/T
2
 = constant.
It is important to note that Kepler
could not give a theory to explain
the motion of planets. It was Newton
who showed that the cause of the
planetary motion is the gravitational
force that the Sun exerts on them. Newton
used the third law
of Kepler to
calculate the
gravitational force
of attraction. The
gravitational force
of the earth is
weakened by distance. A simple argument
goes like this. We can assume that the
planetary orbits are circular. Suppose the
orbital velocity is v and the radius of the
orbit is r. Then the force acting on an
orbiting planet is given by F
?
v
2
/r.
If T denotes the period, then v = 2pr/T,
so that v
2 
?
 r
2
/T
2
.
We can rewrite this as v
2 
?
 (1/r) ×
( r
3
/T
2
). Since r
3
/T
2
 is constant by Kepler’s
third law, we have v
2  
?
 1/r. Combining
this with F
 
?
 v
2
/ r, we get, F
 
?
 1/ r
2
.
A
B
C
D
O
SCIENCE 134
From Eq. (10.4), the force exerted by
the earth on the moon is
G
2
M × m
F=
d
11 2 -2 24 22
82
6.7 10 N m kg 6 10 kg 7.4 10 kg
(3.84 10 m)
-
×××××
=
×
= 2.01 × 10
20
 N.
Thus, the force exerted by the earth on
the moon is 2.01 × 10
20
 N.
uestions
1. State the universal law of
gravitation.
2. Write the formula to find the
magnitude of the gravitational
force between the earth and an
object on the surface of the earth.
10.1.2 IMPORTANCE OF THE UNIVERSAL
LAW OF GRAVITATION
The universal law of gravitation successfully
explained several phenomena which were
believed to be unconnected:
(i) the force that binds us to the earth;
(ii) the motion of the moon around the
earth;
(iii) the motion of planets around the Sun;
and
(iv) the tides due to the moon and the Sun.
10.2 Free Fall
Let us try to understand the meaning of free
fall by performing this activity.
Activity _____________10.2
• Take a stone.
• Throw it upwards.
• It reaches a certain height and then it
starts falling down.
We have learnt that the earth attracts
objects towards it. This is due to the
gravitational force. Whenever objects fall
towards the earth under this force alone, we
say that the objects are in free fall. Is there
or
2
G =
×
Fd
Mm
(10.5)
The SI unit of G can be obtained by
substituting the units of force, distance and
mass in Eq. (10.5) as N m
2
 kg
–2
.
The value of G was found out by
Henry Cavendish (1731 – 1810) by using a
sensitive balance. The accepted value of G is
6.673 × 10
–11
 N m
2
 kg
–2
.
We know that there exists a force of
attraction between any two objects. Compute
the value of this force between you and your
friend sitting closeby. Conclude how you do
not experience this force!
The law is universal in the sense that
it is applicable to all bodies, whether
the bodies are big or small, whether
they are celestial or terrestrial.
Inverse-square
Saying that F is inversely
proportional to the square of d
means, for example, that if d gets
bigger by a factor of 6, F becomes
1
36
times smaller.
Example 10.1 The mass of the earth is
6 × 10
24
 kg and that of the moon is
7.4 × 10
22
 kg. If the distance between
the earth and the moon is 3.84×10
5 
km,
calculate the force exerted by the earth
on the moon. G = 6.7 × 10
–11
 N m
2
 kg
-2
.
Solution:
The mass of the earth, M = 6 × 10
24
 kg
The mass of the moon,
m = 7.4 × 10
22
 kg
The distance between the earth and the
moon,
d = 3.84 × 10
5 
km
= 3.84 × 10
5 
× 1000 m
= 3.84 × 10
8
 m
G = 6.7 × 10
–11
 N m
2
 kg
–2
More to  know
Q
GRAVITATION 135
calculations, we can take g to be more or less
constant on or near the earth. But for objects
far from the earth, the acceleration due to
gravitational force of earth is given by
Eq. (10.7).
10.2.1 TO CALCULATE THE VALUE OF g
To calculate the value of g, we should put
the values of G, M and R in Eq. (10.9),
namely, universal gravitational constant,
G = 6.7 × 10
–11
 N m
2
 kg
-2
, mass of the earth,
M = 6 × 10
24
 kg, and radius of the earth,
R = 6.4 × 10
6
 m.
G
2
M
g=
R
-11 2 -2 24
62
6.7 10 N m kg 6 10 kg
=
(6.4 10 m)
×××
×
      = 9.8 m s
–2
.
Thus, the value of acceleration due to gravity
of the earth, g = 9.8 m s
–2
.
10.2.2 MOTION OF OBJECTS UNDER THE
INFLUENCE OF GRAVITATIONAL
FORCE OF THE EARTH
Let us do an activity to understand whether
all objects hollow or solid, big or small, will
fall from a height at the same rate.
Activity _____________10.3
• Take a sheet of paper and a stone. Drop
them simultaneously from the first floor
of a building. Observe whether both of
them reach the ground simultaneously.
• We see that paper reaches the ground
little later than the stone. This happens
because of air resistance. The air offers
resistance due to friction to the motion
of the falling objects. The resistance
offered by air to the paper is more than
the resistance offered to the stone. If
we do the experiment in a glass jar from
which air has been sucked out, the
paper and the stone would fall at the
same rate.
any change in the velocity of falling objects?
While falling, there is no change in the
direction of motion of the objects. But due to
the earth’s attraction, there will be a change
in the magnitude of the velocity. Any change
in velocity involves acceleration. Whenever an
object falls towards the earth, an acceleration
is involved. This acceleration is due to the
earth’s gravitational force. Therefore, this
acceleration is called the acceleration due to
the gravitational force of the earth (or
acceleration due to gravity). It is denoted by
g. The unit of g is the same as that of
acceleration, that is, m s
–2
.
We know from the second law of motion
that force is the product of mass and
acceleration. Let the mass of the stone in
activity 10.2 be m. We already know that there
is acceleration involved in falling objects due
to the gravitational force and is denoted by g.
Therefore the magnitude of the gravitational
force F will be equal to the product of mass
and acceleration due to the gravitational
force, that is,
F = m g (10.6)
From Eqs. (10.4) and (10.6) we have
2
=G
× Mm
mg
d
or
G
2
M
g=
d
(10.7)
where M is the mass of the earth, and d is
the distance between the object and the earth.
Let an object be on or near the surface of
the earth. The distance d in Eq. (10.7) will be
equal to R, the radius of the earth. Thus, for
objects on or near the surface of the earth,
G
2
M×m
mg =
R
(10.8)
  
G
2
M
g=
R
(10.9)
The earth is not a perfect sphere. As the
radius of the earth increases from the poles
to the equator, the value of g becomes greater
at the poles than at the equator. For most