Textbook: Gravitation | Science and Technology Class 10 (Maharashtra SSC Board) PDF Download

 Page 1


1
1. What are the effects of a force acting on an object? 
2. What types of forces are you familiar with? 
3. What do you know about the gravitational force? 
1. Gravitation 
 We have seen in the previous standard that the gravitational force is a universal force 
and it acts not only between two objects on the earth but also between any two objects in 
the universe. Let us now learn how this force was discovered.
Gravitation
As we have learnt, the phenomenon of gravitation was discovered by Sir Isaac 
Newton. As the story goes, he discovered the force by seeing an apple fall from a tree on 
the ground. He wondered why all apples fall vertically downward and not at an angle to 
the vertical. Why do they not fly off in a horizontal direction? 
After much thought, he came to the conclusion that the earth must be attracting the 
apple towards itself and this attractive force must be directed towards the center of the 
earth. The direction from the apple on the tree to the center of the earth is the vertical 
direction at the position of the apple and thus, the apple falls vertically downwards.  
 Figure 1.1 on the left shows an apple tree 
on the earth. The force on an apple on the tree 
is towards the center of the earth i.e. along the 
perpendicular from the position of the apple 
to the surface of the earth. The Figure also 
shows the gravitational force between the 
earth and the moon. The distances in the 
figure are not according to scale.
Ø Gravitation  Ø    Circular motion and centripetal force 
Ø 	 Kepler’s laws 		 Ø    Newton’s universal law of gravitation 
Ø   Acceleration due to the gravitational force of the Earth 
Ø Free fall                    Ø 	Escape velocity 
 Newton thought that if the force of 
gravitation acts on apples on the tree at 
different heights from the surface of the earth, 
can it also act on objects at even greater 
heights, much farther away from the earth, 
like for example, the moon? Can it act on 
even farther objects like the other planets and 
the Sun?
Use of ICT : Collect videos and ppts about the gravitational force of different planets.
Force and Motion
 We have seen that a force is necessary to change the speed as well as the direction of 
motion of an object. 
1.1 Concept of the gravitational force and 
the gravitational force between the earth 
and the moon.
What are Newton’s laws of motion?
Moon
Gravitational
force
Earth
Falling 
apple
Can you recall?
Can you recall?
.
Page 2


1
1. What are the effects of a force acting on an object? 
2. What types of forces are you familiar with? 
3. What do you know about the gravitational force? 
1. Gravitation 
 We have seen in the previous standard that the gravitational force is a universal force 
and it acts not only between two objects on the earth but also between any two objects in 
the universe. Let us now learn how this force was discovered.
Gravitation
As we have learnt, the phenomenon of gravitation was discovered by Sir Isaac 
Newton. As the story goes, he discovered the force by seeing an apple fall from a tree on 
the ground. He wondered why all apples fall vertically downward and not at an angle to 
the vertical. Why do they not fly off in a horizontal direction? 
After much thought, he came to the conclusion that the earth must be attracting the 
apple towards itself and this attractive force must be directed towards the center of the 
earth. The direction from the apple on the tree to the center of the earth is the vertical 
direction at the position of the apple and thus, the apple falls vertically downwards.  
 Figure 1.1 on the left shows an apple tree 
on the earth. The force on an apple on the tree 
is towards the center of the earth i.e. along the 
perpendicular from the position of the apple 
to the surface of the earth. The Figure also 
shows the gravitational force between the 
earth and the moon. The distances in the 
figure are not according to scale.
Ø Gravitation  Ø    Circular motion and centripetal force 
Ø 	 Kepler’s laws 		 Ø    Newton’s universal law of gravitation 
Ø   Acceleration due to the gravitational force of the Earth 
Ø Free fall                    Ø 	Escape velocity 
 Newton thought that if the force of 
gravitation acts on apples on the tree at 
different heights from the surface of the earth, 
can it also act on objects at even greater 
heights, much farther away from the earth, 
like for example, the moon? Can it act on 
even farther objects like the other planets and 
the Sun?
Use of ICT : Collect videos and ppts about the gravitational force of different planets.
Force and Motion
 We have seen that a force is necessary to change the speed as well as the direction of 
motion of an object. 
1.1 Concept of the gravitational force and 
the gravitational force between the earth 
and the moon.
What are Newton’s laws of motion?
Moon
Gravitational
force
Earth
Falling 
apple
Can you recall?
Can you recall?
.
2
Introduction to scientist
Great Scientists: Sir Isaac Newton (1642-1727) was one of 
the greatest scientists of recent times. He was born in England. 
He gave his laws of motion, equations of motion and theory of 
gravity in his book Principia. Before this book was written, 
Kepler had given three laws describing planetary motions. 
However, the reason why planets move in the way described by 
Kepler’s laws was not known. Newton, with his theory of 
gravity, mathematically derived Kepler’s laws. 
Try this
 Tie a stone to one end of a string. Take the other end in your 
hand and rotate the string so that the stone moves along a circle 
as shown in figure 1.2 a. Are you applying any force on the 
stone? In which direction is this force acting?  How will you 
stop this force from acting? What will be the effect on the stone? 
As long as we are holding the string, we are pulling the 
stone towards us i.e. towards the centre of the circle and are 
applying a force towards it. The force stops acting if we release 
the string. In this case, the stone will fly off along a straight line 
which is the tangent to the circle at the position of the stone 
when the string is released, because that is the direction of its 
velocity at that instant of time (Figure 1.2 b). You may recall 
that we have performed a similar activity previously in which a 
5 rupee coin kept on a rotating circular disk flies off the disk 
along the tangent to the disk. Thus, a force acts on any object 
moving along a circle and it is directed towards the centre of the 
circle. This is called the Centripetal force.  ‘Centripetal’ means 
centre seeking, i.e. the object tries to go towards the centre of the 
circle because of this force. 
1.2 A stone tied to a string, 
moving along a circular 
path and its velocity in 
tangential direction
In addition to this, Newton did ground breaking work in several areas including 
light, heat, sound and mathematics. He invented a new branch of mathematics. This is 
called calculus and has wide ranging applications in physics and mathematics. He was 
the  first scientist to construct a reflecting telescope. 
You know that the moon, which is the natural satellite of the earth, goes round it in a 
definite orbit. The direction of motion of the moon as well as its speed constantly changes 
during this motion. Do you think some force is constantly acting on the moon? What must 
be the direction of this force? How would its motion have been if no such force acted on 
it? Do the other planets in the solar system revolve around the Sun in a similar fashion? Is 
similar force acting on them? What must be its direction?
From the above activity, example and questions it is clear that for the moon to go 
around the earth, there must be a force which is exerted on the moon and this force must 
be exerted  by the earth which attracts the moon towards itself. Similarly, the Sun must be 
attracting the planets, including the earth, towards itself.  
Circular motion and Centripetal force
b.
a.
Page 3


1
1. What are the effects of a force acting on an object? 
2. What types of forces are you familiar with? 
3. What do you know about the gravitational force? 
1. Gravitation 
 We have seen in the previous standard that the gravitational force is a universal force 
and it acts not only between two objects on the earth but also between any two objects in 
the universe. Let us now learn how this force was discovered.
Gravitation
As we have learnt, the phenomenon of gravitation was discovered by Sir Isaac 
Newton. As the story goes, he discovered the force by seeing an apple fall from a tree on 
the ground. He wondered why all apples fall vertically downward and not at an angle to 
the vertical. Why do they not fly off in a horizontal direction? 
After much thought, he came to the conclusion that the earth must be attracting the 
apple towards itself and this attractive force must be directed towards the center of the 
earth. The direction from the apple on the tree to the center of the earth is the vertical 
direction at the position of the apple and thus, the apple falls vertically downwards.  
 Figure 1.1 on the left shows an apple tree 
on the earth. The force on an apple on the tree 
is towards the center of the earth i.e. along the 
perpendicular from the position of the apple 
to the surface of the earth. The Figure also 
shows the gravitational force between the 
earth and the moon. The distances in the 
figure are not according to scale.
Ø Gravitation  Ø    Circular motion and centripetal force 
Ø 	 Kepler’s laws 		 Ø    Newton’s universal law of gravitation 
Ø   Acceleration due to the gravitational force of the Earth 
Ø Free fall                    Ø 	Escape velocity 
 Newton thought that if the force of 
gravitation acts on apples on the tree at 
different heights from the surface of the earth, 
can it also act on objects at even greater 
heights, much farther away from the earth, 
like for example, the moon? Can it act on 
even farther objects like the other planets and 
the Sun?
Use of ICT : Collect videos and ppts about the gravitational force of different planets.
Force and Motion
 We have seen that a force is necessary to change the speed as well as the direction of 
motion of an object. 
1.1 Concept of the gravitational force and 
the gravitational force between the earth 
and the moon.
What are Newton’s laws of motion?
Moon
Gravitational
force
Earth
Falling 
apple
Can you recall?
Can you recall?
.
2
Introduction to scientist
Great Scientists: Sir Isaac Newton (1642-1727) was one of 
the greatest scientists of recent times. He was born in England. 
He gave his laws of motion, equations of motion and theory of 
gravity in his book Principia. Before this book was written, 
Kepler had given three laws describing planetary motions. 
However, the reason why planets move in the way described by 
Kepler’s laws was not known. Newton, with his theory of 
gravity, mathematically derived Kepler’s laws. 
Try this
 Tie a stone to one end of a string. Take the other end in your 
hand and rotate the string so that the stone moves along a circle 
as shown in figure 1.2 a. Are you applying any force on the 
stone? In which direction is this force acting?  How will you 
stop this force from acting? What will be the effect on the stone? 
As long as we are holding the string, we are pulling the 
stone towards us i.e. towards the centre of the circle and are 
applying a force towards it. The force stops acting if we release 
the string. In this case, the stone will fly off along a straight line 
which is the tangent to the circle at the position of the stone 
when the string is released, because that is the direction of its 
velocity at that instant of time (Figure 1.2 b). You may recall 
that we have performed a similar activity previously in which a 
5 rupee coin kept on a rotating circular disk flies off the disk 
along the tangent to the disk. Thus, a force acts on any object 
moving along a circle and it is directed towards the centre of the 
circle. This is called the Centripetal force.  ‘Centripetal’ means 
centre seeking, i.e. the object tries to go towards the centre of the 
circle because of this force. 
1.2 A stone tied to a string, 
moving along a circular 
path and its velocity in 
tangential direction
In addition to this, Newton did ground breaking work in several areas including 
light, heat, sound and mathematics. He invented a new branch of mathematics. This is 
called calculus and has wide ranging applications in physics and mathematics. He was 
the  first scientist to construct a reflecting telescope. 
You know that the moon, which is the natural satellite of the earth, goes round it in a 
definite orbit. The direction of motion of the moon as well as its speed constantly changes 
during this motion. Do you think some force is constantly acting on the moon? What must 
be the direction of this force? How would its motion have been if no such force acted on 
it? Do the other planets in the solar system revolve around the Sun in a similar fashion? Is 
similar force acting on them? What must be its direction?
From the above activity, example and questions it is clear that for the moon to go 
around the earth, there must be a force which is exerted on the moon and this force must 
be exerted  by the earth which attracts the moon towards itself. Similarly, the Sun must be 
attracting the planets, including the earth, towards itself.  
Circular motion and Centripetal force
b.
a.
3
 An ellipse is the curve obtained when 
a cone is cut by an inclined plane. It has 
two focal points. The sum of the distances 
to the two focal points from every point 
on the curve is constant. F
1
 and F
2
 are two 
focal points of the ellipse shown in figure 
1.3.  If A, B and C are three points on the 
ellipse then,  
AF
1
 + AF
2
  = BF
1
 + BF
2
  = CF
1
 + CF
2
 
Kepler’s Laws
Planetary motion had been observed by astronomers since ancient times. Before 
Galileo, all observations of the planet’s positions were made with naked eyes. By the 16th 
century a lot of data were available about planetary positions and motion. Johannes 
Kepler, studied these data. He noticed that the motion of planets follows certain laws. He 
stated three laws describing planetary motion. These are known as Kepler’s laws which 
are given below. 
1.3 An ellipse
The straight lines AS and CS sweep equal area in equal interval of time i.e. area ASB 
and CSD are equal.  
1.4  The orbit of a planet moving 
around the Sun.  
 The orbit of a planet is an ellipse 
with the Sun at one of the foci. 
 Figure 1.4 shows the elliptical orbit of 
a planet revolving around the sun. The 
position of the Sun is indicated by S.
Kepler’s second law : 
  The line joining the planet and the 
Sun sweeps equal areas in equal intervals 
of time. 
 AB and CD are distances covered by 
the planet in equal time i.e. after equal in-
tervals of time, the positions of the planet 
starting from A and C are shown by B and 
D respectively.
Kepler’s third law  :  
 The square of its period of revolution around the Sun is directly proportional to 
the cube of the mean distance of a planet from the Sun.
Thus, if r is the average distance of the planet from the Sun and T is its period of 
revolution then,
 T
2
 a r
3 
 i.e.
Kepler’s first law : 
A
C
F
1
F
2
Kepler obtained these laws simply from the study of the positions of planets obtained 
by regular observations. He had no explanation as to why planets obey these laws. We 
will see below how these laws helped Newton in the formulation of his theory of gravitation.
Do you know ?
T
2
r
3
=  constant = K  ............. (1)
B
A
B
C D
S
E
F
1
Sun
Planet
2
3
Page 4


1
1. What are the effects of a force acting on an object? 
2. What types of forces are you familiar with? 
3. What do you know about the gravitational force? 
1. Gravitation 
 We have seen in the previous standard that the gravitational force is a universal force 
and it acts not only between two objects on the earth but also between any two objects in 
the universe. Let us now learn how this force was discovered.
Gravitation
As we have learnt, the phenomenon of gravitation was discovered by Sir Isaac 
Newton. As the story goes, he discovered the force by seeing an apple fall from a tree on 
the ground. He wondered why all apples fall vertically downward and not at an angle to 
the vertical. Why do they not fly off in a horizontal direction? 
After much thought, he came to the conclusion that the earth must be attracting the 
apple towards itself and this attractive force must be directed towards the center of the 
earth. The direction from the apple on the tree to the center of the earth is the vertical 
direction at the position of the apple and thus, the apple falls vertically downwards.  
 Figure 1.1 on the left shows an apple tree 
on the earth. The force on an apple on the tree 
is towards the center of the earth i.e. along the 
perpendicular from the position of the apple 
to the surface of the earth. The Figure also 
shows the gravitational force between the 
earth and the moon. The distances in the 
figure are not according to scale.
Ø Gravitation  Ø    Circular motion and centripetal force 
Ø 	 Kepler’s laws 		 Ø    Newton’s universal law of gravitation 
Ø   Acceleration due to the gravitational force of the Earth 
Ø Free fall                    Ø 	Escape velocity 
 Newton thought that if the force of 
gravitation acts on apples on the tree at 
different heights from the surface of the earth, 
can it also act on objects at even greater 
heights, much farther away from the earth, 
like for example, the moon? Can it act on 
even farther objects like the other planets and 
the Sun?
Use of ICT : Collect videos and ppts about the gravitational force of different planets.
Force and Motion
 We have seen that a force is necessary to change the speed as well as the direction of 
motion of an object. 
1.1 Concept of the gravitational force and 
the gravitational force between the earth 
and the moon.
What are Newton’s laws of motion?
Moon
Gravitational
force
Earth
Falling 
apple
Can you recall?
Can you recall?
.
2
Introduction to scientist
Great Scientists: Sir Isaac Newton (1642-1727) was one of 
the greatest scientists of recent times. He was born in England. 
He gave his laws of motion, equations of motion and theory of 
gravity in his book Principia. Before this book was written, 
Kepler had given three laws describing planetary motions. 
However, the reason why planets move in the way described by 
Kepler’s laws was not known. Newton, with his theory of 
gravity, mathematically derived Kepler’s laws. 
Try this
 Tie a stone to one end of a string. Take the other end in your 
hand and rotate the string so that the stone moves along a circle 
as shown in figure 1.2 a. Are you applying any force on the 
stone? In which direction is this force acting?  How will you 
stop this force from acting? What will be the effect on the stone? 
As long as we are holding the string, we are pulling the 
stone towards us i.e. towards the centre of the circle and are 
applying a force towards it. The force stops acting if we release 
the string. In this case, the stone will fly off along a straight line 
which is the tangent to the circle at the position of the stone 
when the string is released, because that is the direction of its 
velocity at that instant of time (Figure 1.2 b). You may recall 
that we have performed a similar activity previously in which a 
5 rupee coin kept on a rotating circular disk flies off the disk 
along the tangent to the disk. Thus, a force acts on any object 
moving along a circle and it is directed towards the centre of the 
circle. This is called the Centripetal force.  ‘Centripetal’ means 
centre seeking, i.e. the object tries to go towards the centre of the 
circle because of this force. 
1.2 A stone tied to a string, 
moving along a circular 
path and its velocity in 
tangential direction
In addition to this, Newton did ground breaking work in several areas including 
light, heat, sound and mathematics. He invented a new branch of mathematics. This is 
called calculus and has wide ranging applications in physics and mathematics. He was 
the  first scientist to construct a reflecting telescope. 
You know that the moon, which is the natural satellite of the earth, goes round it in a 
definite orbit. The direction of motion of the moon as well as its speed constantly changes 
during this motion. Do you think some force is constantly acting on the moon? What must 
be the direction of this force? How would its motion have been if no such force acted on 
it? Do the other planets in the solar system revolve around the Sun in a similar fashion? Is 
similar force acting on them? What must be its direction?
From the above activity, example and questions it is clear that for the moon to go 
around the earth, there must be a force which is exerted on the moon and this force must 
be exerted  by the earth which attracts the moon towards itself. Similarly, the Sun must be 
attracting the planets, including the earth, towards itself.  
Circular motion and Centripetal force
b.
a.
3
 An ellipse is the curve obtained when 
a cone is cut by an inclined plane. It has 
two focal points. The sum of the distances 
to the two focal points from every point 
on the curve is constant. F
1
 and F
2
 are two 
focal points of the ellipse shown in figure 
1.3.  If A, B and C are three points on the 
ellipse then,  
AF
1
 + AF
2
  = BF
1
 + BF
2
  = CF
1
 + CF
2
 
Kepler’s Laws
Planetary motion had been observed by astronomers since ancient times. Before 
Galileo, all observations of the planet’s positions were made with naked eyes. By the 16th 
century a lot of data were available about planetary positions and motion. Johannes 
Kepler, studied these data. He noticed that the motion of planets follows certain laws. He 
stated three laws describing planetary motion. These are known as Kepler’s laws which 
are given below. 
1.3 An ellipse
The straight lines AS and CS sweep equal area in equal interval of time i.e. area ASB 
and CSD are equal.  
1.4  The orbit of a planet moving 
around the Sun.  
 The orbit of a planet is an ellipse 
with the Sun at one of the foci. 
 Figure 1.4 shows the elliptical orbit of 
a planet revolving around the sun. The 
position of the Sun is indicated by S.
Kepler’s second law : 
  The line joining the planet and the 
Sun sweeps equal areas in equal intervals 
of time. 
 AB and CD are distances covered by 
the planet in equal time i.e. after equal in-
tervals of time, the positions of the planet 
starting from A and C are shown by B and 
D respectively.
Kepler’s third law  :  
 The square of its period of revolution around the Sun is directly proportional to 
the cube of the mean distance of a planet from the Sun.
Thus, if r is the average distance of the planet from the Sun and T is its period of 
revolution then,
 T
2
 a r
3 
 i.e.
Kepler’s first law : 
A
C
F
1
F
2
Kepler obtained these laws simply from the study of the positions of planets obtained 
by regular observations. He had no explanation as to why planets obey these laws. We 
will see below how these laws helped Newton in the formulation of his theory of gravitation.
Do you know ?
T
2
r
3
=  constant = K  ............. (1)
B
A
B
C D
S
E
F
1
Sun
Planet
2
3
4
From equation (2), it can be seen that 
the value of G is the gravitational force 
acting between two unit masses kept at a 
unit distance away from each other. Thus, 
in SI units, the value of G is equal to the 
gravitational force between two masses of 
1 kg kept 1 m apart. 
 Show that in SI units, the unit of G 
is Newton m
2
 kg
-2
. The value of G was 
first experimentally measured by Henry 
Cavendish. In SI units its value is 
6.673 x 10
-11
 N m
2
 kg
-2
.  
Use your brain power
If the area ESF in  figure 1.4 is equal to area ASB, what 
will you infer about EF?   
An introduction to scientists
Johannes Kepler (1571-1630) was a German astronomer 
and mathematician. He started working as a helper to the famous 
astronomer Tycho Brahe in Prague in 1600.  After the sudden 
death of Brahe in 1601, Kepler was appointed as the Royal 
mathematician in his place. Kepler used the observations of 
planetary positions made by Brahe to discover the laws of 
planetary motion.  He wrote several books. His work was later 
used by Newton in postulating his law of gravitation. 
1.5 Gravitational force between
 two objects 
 Figure 1.5 shows two objects with masses m
1 
and 
m
2
 kept at a distance d from each other. Mathematically, 
the gravitational force of attraction between these two 
bodies can be written as
m
1
m
2
   d
2
F a ....... (2)
m
1
m
2
   d
2
F =  G or
Newton’s universal law of gravitation
 All the above considerations including Kepler’s laws led Newton to formulate his 
theory of Universal gravity. According to this theory, every object in the Universe attracts 
every other object with a definite force. This force is directly proportional to the product 
of the masses of the two objects and is inversely proportional to the square of the distance 
between them.
 Here, G is the constant of proportionality and is called the Universal gravitational 
constant.
The above law means that if the mass of one object is doubled, the force between the 
two objects also doubles. Also, if the distance is doubled, the force decreases by a factor 
of 4. If the two bodies are spherical, the direction of the force is always along the line 
joining the centres of the two bodies and the distance between the centres is taken to be d. 
In case when the bodies are not spherical or have irregular shape, then the direction of 
force is along the line joining their centres of mass and d is taken to be the distance 
between the two centres of mass.
d  
Use your brain power
Page 5


1
1. What are the effects of a force acting on an object? 
2. What types of forces are you familiar with? 
3. What do you know about the gravitational force? 
1. Gravitation 
 We have seen in the previous standard that the gravitational force is a universal force 
and it acts not only between two objects on the earth but also between any two objects in 
the universe. Let us now learn how this force was discovered.
Gravitation
As we have learnt, the phenomenon of gravitation was discovered by Sir Isaac 
Newton. As the story goes, he discovered the force by seeing an apple fall from a tree on 
the ground. He wondered why all apples fall vertically downward and not at an angle to 
the vertical. Why do they not fly off in a horizontal direction? 
After much thought, he came to the conclusion that the earth must be attracting the 
apple towards itself and this attractive force must be directed towards the center of the 
earth. The direction from the apple on the tree to the center of the earth is the vertical 
direction at the position of the apple and thus, the apple falls vertically downwards.  
 Figure 1.1 on the left shows an apple tree 
on the earth. The force on an apple on the tree 
is towards the center of the earth i.e. along the 
perpendicular from the position of the apple 
to the surface of the earth. The Figure also 
shows the gravitational force between the 
earth and the moon. The distances in the 
figure are not according to scale.
Ø Gravitation  Ø    Circular motion and centripetal force 
Ø 	 Kepler’s laws 		 Ø    Newton’s universal law of gravitation 
Ø   Acceleration due to the gravitational force of the Earth 
Ø Free fall                    Ø 	Escape velocity 
 Newton thought that if the force of 
gravitation acts on apples on the tree at 
different heights from the surface of the earth, 
can it also act on objects at even greater 
heights, much farther away from the earth, 
like for example, the moon? Can it act on 
even farther objects like the other planets and 
the Sun?
Use of ICT : Collect videos and ppts about the gravitational force of different planets.
Force and Motion
 We have seen that a force is necessary to change the speed as well as the direction of 
motion of an object. 
1.1 Concept of the gravitational force and 
the gravitational force between the earth 
and the moon.
What are Newton’s laws of motion?
Moon
Gravitational
force
Earth
Falling 
apple
Can you recall?
Can you recall?
.
2
Introduction to scientist
Great Scientists: Sir Isaac Newton (1642-1727) was one of 
the greatest scientists of recent times. He was born in England. 
He gave his laws of motion, equations of motion and theory of 
gravity in his book Principia. Before this book was written, 
Kepler had given three laws describing planetary motions. 
However, the reason why planets move in the way described by 
Kepler’s laws was not known. Newton, with his theory of 
gravity, mathematically derived Kepler’s laws. 
Try this
 Tie a stone to one end of a string. Take the other end in your 
hand and rotate the string so that the stone moves along a circle 
as shown in figure 1.2 a. Are you applying any force on the 
stone? In which direction is this force acting?  How will you 
stop this force from acting? What will be the effect on the stone? 
As long as we are holding the string, we are pulling the 
stone towards us i.e. towards the centre of the circle and are 
applying a force towards it. The force stops acting if we release 
the string. In this case, the stone will fly off along a straight line 
which is the tangent to the circle at the position of the stone 
when the string is released, because that is the direction of its 
velocity at that instant of time (Figure 1.2 b). You may recall 
that we have performed a similar activity previously in which a 
5 rupee coin kept on a rotating circular disk flies off the disk 
along the tangent to the disk. Thus, a force acts on any object 
moving along a circle and it is directed towards the centre of the 
circle. This is called the Centripetal force.  ‘Centripetal’ means 
centre seeking, i.e. the object tries to go towards the centre of the 
circle because of this force. 
1.2 A stone tied to a string, 
moving along a circular 
path and its velocity in 
tangential direction
In addition to this, Newton did ground breaking work in several areas including 
light, heat, sound and mathematics. He invented a new branch of mathematics. This is 
called calculus and has wide ranging applications in physics and mathematics. He was 
the  first scientist to construct a reflecting telescope. 
You know that the moon, which is the natural satellite of the earth, goes round it in a 
definite orbit. The direction of motion of the moon as well as its speed constantly changes 
during this motion. Do you think some force is constantly acting on the moon? What must 
be the direction of this force? How would its motion have been if no such force acted on 
it? Do the other planets in the solar system revolve around the Sun in a similar fashion? Is 
similar force acting on them? What must be its direction?
From the above activity, example and questions it is clear that for the moon to go 
around the earth, there must be a force which is exerted on the moon and this force must 
be exerted  by the earth which attracts the moon towards itself. Similarly, the Sun must be 
attracting the planets, including the earth, towards itself.  
Circular motion and Centripetal force
b.
a.
3
 An ellipse is the curve obtained when 
a cone is cut by an inclined plane. It has 
two focal points. The sum of the distances 
to the two focal points from every point 
on the curve is constant. F
1
 and F
2
 are two 
focal points of the ellipse shown in figure 
1.3.  If A, B and C are three points on the 
ellipse then,  
AF
1
 + AF
2
  = BF
1
 + BF
2
  = CF
1
 + CF
2
 
Kepler’s Laws
Planetary motion had been observed by astronomers since ancient times. Before 
Galileo, all observations of the planet’s positions were made with naked eyes. By the 16th 
century a lot of data were available about planetary positions and motion. Johannes 
Kepler, studied these data. He noticed that the motion of planets follows certain laws. He 
stated three laws describing planetary motion. These are known as Kepler’s laws which 
are given below. 
1.3 An ellipse
The straight lines AS and CS sweep equal area in equal interval of time i.e. area ASB 
and CSD are equal.  
1.4  The orbit of a planet moving 
around the Sun.  
 The orbit of a planet is an ellipse 
with the Sun at one of the foci. 
 Figure 1.4 shows the elliptical orbit of 
a planet revolving around the sun. The 
position of the Sun is indicated by S.
Kepler’s second law : 
  The line joining the planet and the 
Sun sweeps equal areas in equal intervals 
of time. 
 AB and CD are distances covered by 
the planet in equal time i.e. after equal in-
tervals of time, the positions of the planet 
starting from A and C are shown by B and 
D respectively.
Kepler’s third law  :  
 The square of its period of revolution around the Sun is directly proportional to 
the cube of the mean distance of a planet from the Sun.
Thus, if r is the average distance of the planet from the Sun and T is its period of 
revolution then,
 T
2
 a r
3 
 i.e.
Kepler’s first law : 
A
C
F
1
F
2
Kepler obtained these laws simply from the study of the positions of planets obtained 
by regular observations. He had no explanation as to why planets obey these laws. We 
will see below how these laws helped Newton in the formulation of his theory of gravitation.
Do you know ?
T
2
r
3
=  constant = K  ............. (1)
B
A
B
C D
S
E
F
1
Sun
Planet
2
3
4
From equation (2), it can be seen that 
the value of G is the gravitational force 
acting between two unit masses kept at a 
unit distance away from each other. Thus, 
in SI units, the value of G is equal to the 
gravitational force between two masses of 
1 kg kept 1 m apart. 
 Show that in SI units, the unit of G 
is Newton m
2
 kg
-2
. The value of G was 
first experimentally measured by Henry 
Cavendish. In SI units its value is 
6.673 x 10
-11
 N m
2
 kg
-2
.  
Use your brain power
If the area ESF in  figure 1.4 is equal to area ASB, what 
will you infer about EF?   
An introduction to scientists
Johannes Kepler (1571-1630) was a German astronomer 
and mathematician. He started working as a helper to the famous 
astronomer Tycho Brahe in Prague in 1600.  After the sudden 
death of Brahe in 1601, Kepler was appointed as the Royal 
mathematician in his place. Kepler used the observations of 
planetary positions made by Brahe to discover the laws of 
planetary motion.  He wrote several books. His work was later 
used by Newton in postulating his law of gravitation. 
1.5 Gravitational force between
 two objects 
 Figure 1.5 shows two objects with masses m
1 
and 
m
2
 kept at a distance d from each other. Mathematically, 
the gravitational force of attraction between these two 
bodies can be written as
m
1
m
2
   d
2
F a ....... (2)
m
1
m
2
   d
2
F =  G or
Newton’s universal law of gravitation
 All the above considerations including Kepler’s laws led Newton to formulate his 
theory of Universal gravity. According to this theory, every object in the Universe attracts 
every other object with a definite force. This force is directly proportional to the product 
of the masses of the two objects and is inversely proportional to the square of the distance 
between them.
 Here, G is the constant of proportionality and is called the Universal gravitational 
constant.
The above law means that if the mass of one object is doubled, the force between the 
two objects also doubles. Also, if the distance is doubled, the force decreases by a factor 
of 4. If the two bodies are spherical, the direction of the force is always along the line 
joining the centres of the two bodies and the distance between the centres is taken to be d. 
In case when the bodies are not spherical or have irregular shape, then the direction of 
force is along the line joining their centres of mass and d is taken to be the distance 
between the two centres of mass.
d  
Use your brain power
5
 The distance travelled by the planet in one revolution =perimeter of the orbit 2 p r ;
r = distance of the planet from the Sun, Time taken = Period of revolution = T
Why did Newton assume inverse square dependence on distance in his law of  
gravitation? He was helped by Kepler’s third law in this as shown below.  
Uniform circular motion / Magnitude of centripetal force
Consider an object moving in a circle with constant speed. We have seen earlier that 
such a motion is possible only when the object is constantly acted upon by a force directed 
towards the centre of the circle. This force is called the centripetal force. If m is the mass 
of the object, v is its speed and r is the radius of the circle, then it can be shown that this 
force is equal to F = m v
2
/r.  
If a planet is revolving around the Sun in a circular 
orbit in uniform circular motion, then the centripetal force 
acting on the planet towards the Sun must be F = mv
2
/r,  
where, m is the mass of the planet, v is its speed and r is its 
distance from the Sun.  
The speed of the planet can be expressed in terms of the period of revolution T as 
follows.
 The centre of mass of an object is the point inside or outside the object at which the 
total mass of the object can be assumed to be concentrated. The centre of mass of a 
spherical object having uniform density is at its geometrical centre. The centre of mass 
of any object having uniform density is at its centroid.
 Thus, Newton concluded that the centripetal force which is the force acting on the 
planet and is responsible for its circular motion, must be inversely proportional to the 
square of the distance between the planet and the Sun. Newton identified this force with 
the force of gravity and hence postulated the inverse square law of gravitation. The 
gravitational force is much weaker than other forces in nature but it controls the Universe 
and decides its future.  This is possible because of the huge masses of planets, stars and 
other constituents of the Universe.
 Is there a gravitational force between two objects kept on a table or between you 
and your friend sitting next to you? If yes, why don’t the two move towards each other?
Use your brain power
Speed = 
distance travelled 
time taken
4 m p
 2 
    
 T
2
(
r
3
(
  F =                ´
T
2
r
3
= K
, multiplying and dividing by r
2  
we get,
4 m p
 2 
    
r
2 
K
F =          
mv
2
 
 r 
F =       
 
      = 
2 p r
T
) (
2
r
= 
4 m p
2
 r
T
2
m
v =                                   = 
     2pr 
          T 
   distance travelled 
        time taken  
.  According to Kepler’s third law,
4 m p
 2 
    
K
, But
= Constant  
r
2
1  
r
2
1  
r
2
\ F = constant ´ \ F   a