NCERT Textbook - Laws of Motion Class 11 Notes | EduRev

 Page 1


CHAPTER FIVE
LAWS OF MOTION
5.1  INTRODUCTION
In the preceding Chapter, our concern was to describe the
motion of a particle in space quantitatively. We saw that
uniform motion needs the concept of velocity alone whereas
non-uniform motion requires the concept of acceleration in
addition.  So far, we have not asked the question as to what
governs the motion of bodies. In this chapter, we turn to this
basic question.
Let us first guess the answer based on our common
experience. To move a football at rest, someone must kick it.
To throw a stone upwards, one has to  give it an upward
push.  A breeze causes the branches of a tree to swing; a
strong wind can even move heavy objects. A boat moves in a
flowing river without anyone rowing it. Clearly, some external
agency is needed to provide force to move a body from rest.
Likewise, an external force is needed also to retard or stop
motion.  You can stop a ball rolling down an inclined plane by
applying a force against the direction of its motion.
In these examples, the external agency of force (hands,
wind, stream, etc) is in contact with the object. This is not
always necessary. A stone released from the top of a building
accelerates downward due to the gravitational pull of the
earth.  A bar magnet can attract an iron nail from a distance.
This shows that external agencies (e.g. gravitational and
magnetic forces )  can exert  force on a body even from a
distance.
In short, a force is required to put a stationary body in
motion or stop a moving body, and some external agency is
needed to provide this force. The external agency may or may
not be in contact with the body.
So far so good. But what if a body is moving uniformly (e.g.
a skater moving straight with constant speed on a horizontal
ice slab) ?  Is an external force required to keep a body in
uniform motion?
5.1 Introduction
5.2 Aristotle’s fallacy
5.3 The law of inertia
5.4 Newton’s first law of motion
5.5 Newton’s second law of
motion
5.6 Newton’s third law of motion
5.7 Conservation of momentum
5.8 Equilibrium of a particle
5.9 Common forces in mechanics
5.10 Circular motion
5.11 Solving problems in
mechanics
Summary
Points to ponder
Exercises
Additional exercises
2020-21
Page 2


CHAPTER FIVE
LAWS OF MOTION
5.1  INTRODUCTION
In the preceding Chapter, our concern was to describe the
motion of a particle in space quantitatively. We saw that
uniform motion needs the concept of velocity alone whereas
non-uniform motion requires the concept of acceleration in
addition.  So far, we have not asked the question as to what
governs the motion of bodies. In this chapter, we turn to this
basic question.
Let us first guess the answer based on our common
experience. To move a football at rest, someone must kick it.
To throw a stone upwards, one has to  give it an upward
push.  A breeze causes the branches of a tree to swing; a
strong wind can even move heavy objects. A boat moves in a
flowing river without anyone rowing it. Clearly, some external
agency is needed to provide force to move a body from rest.
Likewise, an external force is needed also to retard or stop
motion.  You can stop a ball rolling down an inclined plane by
applying a force against the direction of its motion.
In these examples, the external agency of force (hands,
wind, stream, etc) is in contact with the object. This is not
always necessary. A stone released from the top of a building
accelerates downward due to the gravitational pull of the
earth.  A bar magnet can attract an iron nail from a distance.
This shows that external agencies (e.g. gravitational and
magnetic forces )  can exert  force on a body even from a
distance.
In short, a force is required to put a stationary body in
motion or stop a moving body, and some external agency is
needed to provide this force. The external agency may or may
not be in contact with the body.
So far so good. But what if a body is moving uniformly (e.g.
a skater moving straight with constant speed on a horizontal
ice slab) ?  Is an external force required to keep a body in
uniform motion?
5.1 Introduction
5.2 Aristotle’s fallacy
5.3 The law of inertia
5.4 Newton’s first law of motion
5.5 Newton’s second law of
motion
5.6 Newton’s third law of motion
5.7 Conservation of momentum
5.8 Equilibrium of a particle
5.9 Common forces in mechanics
5.10 Circular motion
5.11 Solving problems in
mechanics
Summary
Points to ponder
Exercises
Additional exercises
2020-21
PHYSICS 90
5.2  ARISTOTLE’S  FALLACY
The question posed above appears to be simple.
However, it took ages to answer it. Indeed, the
correct answer to this question given by Galileo
in the seventeenth century was the foundation
of Newtonian mechanics, which signalled the
birth of modern science.
The Greek thinker, Aristotle (384 B.C– 322
B.C.), held the view that if a body is moving,
something external is required to keep it moving.
According to this view, for example, an arrow
shot from a bow keeps flying since the air behind
the arrow keeps pushing it. The view was part of
an elaborate framework of ideas developed by
Aristotle on the motion of bodies in the universe.
Most of the Aristotelian ideas on motion are now
known to be wrong and need not concern us.
For our purpose here, the Aristotelian law of
motion may be phrased thus: An external force
is required  to keep a body in motion.
Aristotelian law of motion is flawed, as we shall
see.  However, it is a natural view that anyone
would hold from common experience. Even a
small child playing with a simple (non-electric)
toy-car on a floor knows intuitively that it needs
to constantly drag the string attached to the toy-
car with some force to keep it going.  If it releases
the string, it comes to rest. This experience is
common to most terrestrial motion. External
forces seem to be needed to keep bodies in
motion. Left to themselves, all bodies eventually
come to rest.
What is the flaw in Aristotle’s argument? The
answer is: a moving toy car comes to rest because
the external force of friction on the car by the floor
opposes its motion. To counter this force, the child
has to apply an external force on the car in the
direction of motion.  When the car is in uniform
motion, there is no net external force acting on it:
the force by the child cancels the force ( friction)
by the floor.  The corollary is: if there were no friction,
the child would not be required to apply any force
to keep the toy car in uniform motion.
The opposing forces such as friction (solids)
and viscous forces (for fluids) are always present
in the natural world.  This explains why forces
by external agencies are necessary to overcome
the frictional forces to keep bodies in uniform
motion. Now we understand  where Aristotle
went wrong.  He coded this practical experience
in the form of a basic argument.  To get at the
true law of nature for forces and motion, one has
to imagine a world in which uniform motion is
possible with no frictional forces opposing. This
is what Galileo did.
5.3  THE LAW OF INERTIA
Galileo studied motion of objects on an inclined
plane.  Objects (i) moving down an inclined plane
accelerate, while those (ii) moving up retard.
(iii) Motion on a horizontal plane  is an
intermediate situation.  Galileo concluded that
an object moving on a frictionless horizontal
plane must neither have acceleration nor
retardation, i.e. it should move with constant
velocity (Fig. 5.1(a)).
(i) (ii) (iii)
Fig. 5.1(a)
Another experiment by Galileo leading to the
same conclusion involves a double inclined plane.
A ball released from rest on one of the planes rolls
down and climbs up the other. If the planes are
smooth, the final height of the ball is nearly the
same as the initial height (a little less but never
greater). In the ideal situation, when friction is
absent, the final height of the ball is the same
as its initial height.
If the slope of the second plane is decreased
and the experiment repeated, the ball will still
reach the same height, but in doing so, it will
travel a longer distance.  In the limiting case, when
the slope of the second plane is zero (i.e. is a
horizontal) the ball travels an infinite distance.
In other words, its motion never ceases. This is,
of course, an idealised situation (Fig. 5.1(b)).
Fig. 5.1(b) The law of inertia was inferred by Galileo
from observations of motion of a ball on a
double inclined plane.
2020-21
Page 3


CHAPTER FIVE
LAWS OF MOTION
5.1  INTRODUCTION
In the preceding Chapter, our concern was to describe the
motion of a particle in space quantitatively. We saw that
uniform motion needs the concept of velocity alone whereas
non-uniform motion requires the concept of acceleration in
addition.  So far, we have not asked the question as to what
governs the motion of bodies. In this chapter, we turn to this
basic question.
Let us first guess the answer based on our common
experience. To move a football at rest, someone must kick it.
To throw a stone upwards, one has to  give it an upward
push.  A breeze causes the branches of a tree to swing; a
strong wind can even move heavy objects. A boat moves in a
flowing river without anyone rowing it. Clearly, some external
agency is needed to provide force to move a body from rest.
Likewise, an external force is needed also to retard or stop
motion.  You can stop a ball rolling down an inclined plane by
applying a force against the direction of its motion.
In these examples, the external agency of force (hands,
wind, stream, etc) is in contact with the object. This is not
always necessary. A stone released from the top of a building
accelerates downward due to the gravitational pull of the
earth.  A bar magnet can attract an iron nail from a distance.
This shows that external agencies (e.g. gravitational and
magnetic forces )  can exert  force on a body even from a
distance.
In short, a force is required to put a stationary body in
motion or stop a moving body, and some external agency is
needed to provide this force. The external agency may or may
not be in contact with the body.
So far so good. But what if a body is moving uniformly (e.g.
a skater moving straight with constant speed on a horizontal
ice slab) ?  Is an external force required to keep a body in
uniform motion?
5.1 Introduction
5.2 Aristotle’s fallacy
5.3 The law of inertia
5.4 Newton’s first law of motion
5.5 Newton’s second law of
motion
5.6 Newton’s third law of motion
5.7 Conservation of momentum
5.8 Equilibrium of a particle
5.9 Common forces in mechanics
5.10 Circular motion
5.11 Solving problems in
mechanics
Summary
Points to ponder
Exercises
Additional exercises
2020-21
PHYSICS 90
5.2  ARISTOTLE’S  FALLACY
The question posed above appears to be simple.
However, it took ages to answer it. Indeed, the
correct answer to this question given by Galileo
in the seventeenth century was the foundation
of Newtonian mechanics, which signalled the
birth of modern science.
The Greek thinker, Aristotle (384 B.C– 322
B.C.), held the view that if a body is moving,
something external is required to keep it moving.
According to this view, for example, an arrow
shot from a bow keeps flying since the air behind
the arrow keeps pushing it. The view was part of
an elaborate framework of ideas developed by
Aristotle on the motion of bodies in the universe.
Most of the Aristotelian ideas on motion are now
known to be wrong and need not concern us.
For our purpose here, the Aristotelian law of
motion may be phrased thus: An external force
is required  to keep a body in motion.
Aristotelian law of motion is flawed, as we shall
see.  However, it is a natural view that anyone
would hold from common experience. Even a
small child playing with a simple (non-electric)
toy-car on a floor knows intuitively that it needs
to constantly drag the string attached to the toy-
car with some force to keep it going.  If it releases
the string, it comes to rest. This experience is
common to most terrestrial motion. External
forces seem to be needed to keep bodies in
motion. Left to themselves, all bodies eventually
come to rest.
What is the flaw in Aristotle’s argument? The
answer is: a moving toy car comes to rest because
the external force of friction on the car by the floor
opposes its motion. To counter this force, the child
has to apply an external force on the car in the
direction of motion.  When the car is in uniform
motion, there is no net external force acting on it:
the force by the child cancels the force ( friction)
by the floor.  The corollary is: if there were no friction,
the child would not be required to apply any force
to keep the toy car in uniform motion.
The opposing forces such as friction (solids)
and viscous forces (for fluids) are always present
in the natural world.  This explains why forces
by external agencies are necessary to overcome
the frictional forces to keep bodies in uniform
motion. Now we understand  where Aristotle
went wrong.  He coded this practical experience
in the form of a basic argument.  To get at the
true law of nature for forces and motion, one has
to imagine a world in which uniform motion is
possible with no frictional forces opposing. This
is what Galileo did.
5.3  THE LAW OF INERTIA
Galileo studied motion of objects on an inclined
plane.  Objects (i) moving down an inclined plane
accelerate, while those (ii) moving up retard.
(iii) Motion on a horizontal plane  is an
intermediate situation.  Galileo concluded that
an object moving on a frictionless horizontal
plane must neither have acceleration nor
retardation, i.e. it should move with constant
velocity (Fig. 5.1(a)).
(i) (ii) (iii)
Fig. 5.1(a)
Another experiment by Galileo leading to the
same conclusion involves a double inclined plane.
A ball released from rest on one of the planes rolls
down and climbs up the other. If the planes are
smooth, the final height of the ball is nearly the
same as the initial height (a little less but never
greater). In the ideal situation, when friction is
absent, the final height of the ball is the same
as its initial height.
If the slope of the second plane is decreased
and the experiment repeated, the ball will still
reach the same height, but in doing so, it will
travel a longer distance.  In the limiting case, when
the slope of the second plane is zero (i.e. is a
horizontal) the ball travels an infinite distance.
In other words, its motion never ceases. This is,
of course, an idealised situation (Fig. 5.1(b)).
Fig. 5.1(b) The law of inertia was inferred by Galileo
from observations of motion of a ball on a
double inclined plane.
2020-21
LAWS OF MOTION 91
In practice, the ball does come to a stop after
moving a finite distance on the horizontal plane,
because of the opposing force of friction which
can never be totally eliminated.  However, if there
were no friction, the ball would continue  to move
with a constant velocity on the horizontal plane.
Galileo thus, arrived at a new insight on
motion that had eluded Aristotle and those who
followed him.  The state of rest and the state of
uniform linear motion (motion with constant
velocity) are equivalent. In both cases, there is
no net force acting on the body.  It is incorrect to
assume that a net force is needed to keep a body
in uniform motion. To maintain a body in
uniform motion, we need to apply an external
force to ecounter the frictional force, so that
the two forces sum up to zero net external
force.
To summarise, if the net external force is zero,
a body at rest continues to remain at rest and a
body in motion continues to move with a uniform
velocity.  This property of the body is called
inertia. Inertia means ‘resistance to  change’.
A body does not change its state of rest or
uniform motion, unless an external force
compels it to change that state.
5.4  NEWTON’S FIRST LAW OF MOTION
Galileo’s simple, but revolutionary ideas
dethroned Aristotelian mechanics. A new
mechanics had to be developed. This task was
Ideas on Motion in Ancient Indian Science
Ancient Indian thinkers had arrived at an elaborate system of ideas on motion. Force, the cause of
motion, was thought to be of different kinds : force due to continuous pressure (nodan), as the force
of wind on a sailing vessel; impact (abhighat), as when a potter’s rod strikes the wheel; persistent
tendency (sanskara) to move in a straight line(vega) or restoration of shape in an elastic body;
transmitted force by a string, rod, etc. The notion of (vega) in the Vaisesika theory of motion perhaps
comes closest to the concept of inertia.  Vega, the tendency to move in a straight line, was thought to
be opposed by contact with objects including atmosphere, a parallel to the ideas of friction and air
resistance.  It was correctly summarised that the different kinds of motion (translational, rotational
and vibrational) of an extended body arise from only the translational motion of its constituent
particles. A falling leaf in the wind may have downward motion as a whole (patan) and also rotational
and vibrational motion (bhraman, spandan), but each particle of the leaf at an instant only has a
definite (small) displacement. There was considerable focus in Indian thought on measurement of
motion and units of length and time.  It was known that the position of a particle in space can be
indicated by distance measured along three axes.  Bhaskara (1150 A.D.) had introduced the concept
of ‘instantaneous motion’ (tatkaliki gati), which anticipated the modern notion of instantaneous
velocity using Differential Calculus. The difference between a wave and a current (of water) was clearly
understood; a current is a motion of particles of water under gravity and fluidity while a wave results
from the transmission of vibrations of water particles.
accomplished almost single-handedly by Isaac
Newton, one of the greatest scientists of all times.
Newton built on Galileo’s ideas and laid the
foundation of mechanics in terms of three laws
of  motion that go by his name.  Galileo’s law of
inertia was his starting point which he
formulated as the first law of motion:
Every body continues to be in its state
of rest or of uniform motion in a straight
line unless compelled by some external
force to act otherwise.
The state of rest or uniform linear motion both
imply zero acceleration. The first law of motion  can,
therefore, be simply expressed as:
If the net external force on a body is zero, its
acceleration is zero.  Acceleration can be non
zero only if there is a net external force on
the body.
Two kinds of situations are encountered in the
application of this law in practice. In some
examples, we know that the net external force
on the object is zero. In that case we can
conclude that the acceleration of the object is
zero.  For example, a spaceship out in
interstellar space, far from all other objects and
with all its rockets turned off, has no net
external force acting on it.  Its acceleration,
according to the first law, must be zero.  If it is
in motion, it must continue to move with a
uniform velocity.
2020-21
Page 4


CHAPTER FIVE
LAWS OF MOTION
5.1  INTRODUCTION
In the preceding Chapter, our concern was to describe the
motion of a particle in space quantitatively. We saw that
uniform motion needs the concept of velocity alone whereas
non-uniform motion requires the concept of acceleration in
addition.  So far, we have not asked the question as to what
governs the motion of bodies. In this chapter, we turn to this
basic question.
Let us first guess the answer based on our common
experience. To move a football at rest, someone must kick it.
To throw a stone upwards, one has to  give it an upward
push.  A breeze causes the branches of a tree to swing; a
strong wind can even move heavy objects. A boat moves in a
flowing river without anyone rowing it. Clearly, some external
agency is needed to provide force to move a body from rest.
Likewise, an external force is needed also to retard or stop
motion.  You can stop a ball rolling down an inclined plane by
applying a force against the direction of its motion.
In these examples, the external agency of force (hands,
wind, stream, etc) is in contact with the object. This is not
always necessary. A stone released from the top of a building
accelerates downward due to the gravitational pull of the
earth.  A bar magnet can attract an iron nail from a distance.
This shows that external agencies (e.g. gravitational and
magnetic forces )  can exert  force on a body even from a
distance.
In short, a force is required to put a stationary body in
motion or stop a moving body, and some external agency is
needed to provide this force. The external agency may or may
not be in contact with the body.
So far so good. But what if a body is moving uniformly (e.g.
a skater moving straight with constant speed on a horizontal
ice slab) ?  Is an external force required to keep a body in
uniform motion?
5.1 Introduction
5.2 Aristotle’s fallacy
5.3 The law of inertia
5.4 Newton’s first law of motion
5.5 Newton’s second law of
motion
5.6 Newton’s third law of motion
5.7 Conservation of momentum
5.8 Equilibrium of a particle
5.9 Common forces in mechanics
5.10 Circular motion
5.11 Solving problems in
mechanics
Summary
Points to ponder
Exercises
Additional exercises
2020-21
PHYSICS 90
5.2  ARISTOTLE’S  FALLACY
The question posed above appears to be simple.
However, it took ages to answer it. Indeed, the
correct answer to this question given by Galileo
in the seventeenth century was the foundation
of Newtonian mechanics, which signalled the
birth of modern science.
The Greek thinker, Aristotle (384 B.C– 322
B.C.), held the view that if a body is moving,
something external is required to keep it moving.
According to this view, for example, an arrow
shot from a bow keeps flying since the air behind
the arrow keeps pushing it. The view was part of
an elaborate framework of ideas developed by
Aristotle on the motion of bodies in the universe.
Most of the Aristotelian ideas on motion are now
known to be wrong and need not concern us.
For our purpose here, the Aristotelian law of
motion may be phrased thus: An external force
is required  to keep a body in motion.
Aristotelian law of motion is flawed, as we shall
see.  However, it is a natural view that anyone
would hold from common experience. Even a
small child playing with a simple (non-electric)
toy-car on a floor knows intuitively that it needs
to constantly drag the string attached to the toy-
car with some force to keep it going.  If it releases
the string, it comes to rest. This experience is
common to most terrestrial motion. External
forces seem to be needed to keep bodies in
motion. Left to themselves, all bodies eventually
come to rest.
What is the flaw in Aristotle’s argument? The
answer is: a moving toy car comes to rest because
the external force of friction on the car by the floor
opposes its motion. To counter this force, the child
has to apply an external force on the car in the
direction of motion.  When the car is in uniform
motion, there is no net external force acting on it:
the force by the child cancels the force ( friction)
by the floor.  The corollary is: if there were no friction,
the child would not be required to apply any force
to keep the toy car in uniform motion.
The opposing forces such as friction (solids)
and viscous forces (for fluids) are always present
in the natural world.  This explains why forces
by external agencies are necessary to overcome
the frictional forces to keep bodies in uniform
motion. Now we understand  where Aristotle
went wrong.  He coded this practical experience
in the form of a basic argument.  To get at the
true law of nature for forces and motion, one has
to imagine a world in which uniform motion is
possible with no frictional forces opposing. This
is what Galileo did.
5.3  THE LAW OF INERTIA
Galileo studied motion of objects on an inclined
plane.  Objects (i) moving down an inclined plane
accelerate, while those (ii) moving up retard.
(iii) Motion on a horizontal plane  is an
intermediate situation.  Galileo concluded that
an object moving on a frictionless horizontal
plane must neither have acceleration nor
retardation, i.e. it should move with constant
velocity (Fig. 5.1(a)).
(i) (ii) (iii)
Fig. 5.1(a)
Another experiment by Galileo leading to the
same conclusion involves a double inclined plane.
A ball released from rest on one of the planes rolls
down and climbs up the other. If the planes are
smooth, the final height of the ball is nearly the
same as the initial height (a little less but never
greater). In the ideal situation, when friction is
absent, the final height of the ball is the same
as its initial height.
If the slope of the second plane is decreased
and the experiment repeated, the ball will still
reach the same height, but in doing so, it will
travel a longer distance.  In the limiting case, when
the slope of the second plane is zero (i.e. is a
horizontal) the ball travels an infinite distance.
In other words, its motion never ceases. This is,
of course, an idealised situation (Fig. 5.1(b)).
Fig. 5.1(b) The law of inertia was inferred by Galileo
from observations of motion of a ball on a
double inclined plane.
2020-21
LAWS OF MOTION 91
In practice, the ball does come to a stop after
moving a finite distance on the horizontal plane,
because of the opposing force of friction which
can never be totally eliminated.  However, if there
were no friction, the ball would continue  to move
with a constant velocity on the horizontal plane.
Galileo thus, arrived at a new insight on
motion that had eluded Aristotle and those who
followed him.  The state of rest and the state of
uniform linear motion (motion with constant
velocity) are equivalent. In both cases, there is
no net force acting on the body.  It is incorrect to
assume that a net force is needed to keep a body
in uniform motion. To maintain a body in
uniform motion, we need to apply an external
force to ecounter the frictional force, so that
the two forces sum up to zero net external
force.
To summarise, if the net external force is zero,
a body at rest continues to remain at rest and a
body in motion continues to move with a uniform
velocity.  This property of the body is called
inertia. Inertia means ‘resistance to  change’.
A body does not change its state of rest or
uniform motion, unless an external force
compels it to change that state.
5.4  NEWTON’S FIRST LAW OF MOTION
Galileo’s simple, but revolutionary ideas
dethroned Aristotelian mechanics. A new
mechanics had to be developed. This task was
Ideas on Motion in Ancient Indian Science
Ancient Indian thinkers had arrived at an elaborate system of ideas on motion. Force, the cause of
motion, was thought to be of different kinds : force due to continuous pressure (nodan), as the force
of wind on a sailing vessel; impact (abhighat), as when a potter’s rod strikes the wheel; persistent
tendency (sanskara) to move in a straight line(vega) or restoration of shape in an elastic body;
transmitted force by a string, rod, etc. The notion of (vega) in the Vaisesika theory of motion perhaps
comes closest to the concept of inertia.  Vega, the tendency to move in a straight line, was thought to
be opposed by contact with objects including atmosphere, a parallel to the ideas of friction and air
resistance.  It was correctly summarised that the different kinds of motion (translational, rotational
and vibrational) of an extended body arise from only the translational motion of its constituent
particles. A falling leaf in the wind may have downward motion as a whole (patan) and also rotational
and vibrational motion (bhraman, spandan), but each particle of the leaf at an instant only has a
definite (small) displacement. There was considerable focus in Indian thought on measurement of
motion and units of length and time.  It was known that the position of a particle in space can be
indicated by distance measured along three axes.  Bhaskara (1150 A.D.) had introduced the concept
of ‘instantaneous motion’ (tatkaliki gati), which anticipated the modern notion of instantaneous
velocity using Differential Calculus. The difference between a wave and a current (of water) was clearly
understood; a current is a motion of particles of water under gravity and fluidity while a wave results
from the transmission of vibrations of water particles.
accomplished almost single-handedly by Isaac
Newton, one of the greatest scientists of all times.
Newton built on Galileo’s ideas and laid the
foundation of mechanics in terms of three laws
of  motion that go by his name.  Galileo’s law of
inertia was his starting point which he
formulated as the first law of motion:
Every body continues to be in its state
of rest or of uniform motion in a straight
line unless compelled by some external
force to act otherwise.
The state of rest or uniform linear motion both
imply zero acceleration. The first law of motion  can,
therefore, be simply expressed as:
If the net external force on a body is zero, its
acceleration is zero.  Acceleration can be non
zero only if there is a net external force on
the body.
Two kinds of situations are encountered in the
application of this law in practice. In some
examples, we know that the net external force
on the object is zero. In that case we can
conclude that the acceleration of the object is
zero.  For example, a spaceship out in
interstellar space, far from all other objects and
with all its rockets turned off, has no net
external force acting on it.  Its acceleration,
according to the first law, must be zero.  If it is
in motion, it must continue to move with a
uniform velocity.
2020-21
PHYSICS 92
More often, however, we do not know all the
forces to begin with.  In that case, if we know
that an object is unaccelerated (i.e. it is either
at rest or in uniform linear motion), we can infer
from the first law that the net external force on
the object must be zero. Gravity is everywhere.
For terrestrial phenomena, in particular, every
object experiences gravitational force due to the
earth.  Also objects in motion generally experience
friction, viscous drag, etc. If then, on earth, an
object is at rest or in uniform linear motion, it is
not because there are no forces acting on it, but
because the various external forces cancel out
i.e. add up to zero net external force.
Consider a book at rest on a horizontal surface
Fig. (5.2(a)).  It is subject to two external forces :
the force due to gravity (i.e. its weight W) acting
downward and the upward force on the book by
the table, the normal force R . R is a self-adjusting
force. This is an example of the kind of situation
mentioned above. The forces are not quite known
fully but the state of motion is known. We observe
the book to be at rest.  Therefore, we conclude
from the first law that the magnitude of R equals
that of W. A statement often encountered is :
“Since W = R, forces cancel and, therefore, the book
is at rest”. This is incorrect reasoning. The correct
statement is : “Since the book is observed to be at
rest, the net external force on it must be zero,
according to the first law. This implies that the
Galileo Galilei, born in Pisa, Italy in 1564 was a key figure in the scientific revolution
in Europe about four centuries ago.  Galileo proposed the concept of acceleration.
From experiments on motion of bodies on inclined planes or falling freely, he
contradicted the Aristotelian notion that a force was required to keep a body in
motion, and that heavier bodies fall faster than lighter bodies under gravity.  He
thus arrived at the law of inertia that was the starting point of the subsequent
epochal work of Isaac Newton.
Galileo’s discoveries in astronomy were equally revolutionary.  In 1609, he designed
his own telescope (invented earlier in Holland) and used it to make a number of
startling  observations :  mountains  and depressions on  the  surface of the moon;
dark spots on the sun; the moons of Jupiter and the phases of Venus.  He concluded
that the Milky Way derived its luminosity because of a large number of stars not visible to the naked eye.
In his masterpiece of scientific reasoning : Dialogue on the Two Chief World Systems, Galileo advocated
the heliocentric theory of the solar system proposed by Copernicus, which eventually got universal
acceptance.
With Galileo came a turning point in the very method of scientific inquiry. Science was no longer
merely observations of nature and inferences from them. Science meant devising and doing experiments
to verify or refute theories. Science meant measurement of quantities and a search for mathematical
relations between them. Not undeservedly, many regard Galileo as the father of modern science.
normal force R  must be equal and opposite to the
weight W ”.
Fig. 5.2 (a) a book at rest on the table, and (b) a car
moving with uniform velocity. The net force
is zero in each case.
Consider the motion of a car starting from
rest, picking up speed and then moving on a
smooth straight road with uniform speed (Fig.
(5.2(b)).  When the car is stationary, there is no
net force acting on it. During pick-up, it
accelerates. This must happen due to a net
external force. Note, it has to be an external force.
The acceleration of the car cannot be accounted
for by any internal force.  This might sound
surprising, but it is true.  The only conceivable
external force along the road is the force of
friction.  It is the frictional force that accelerates
the car as a whole.  (You will learn about friction
in section 5.9).  When the car moves with
constant velocity, there is no net external force.
Galileo Galilei (1564 - 1642)
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CHAPTER FIVE
LAWS OF MOTION
5.1  INTRODUCTION
In the preceding Chapter, our concern was to describe the
motion of a particle in space quantitatively. We saw that
uniform motion needs the concept of velocity alone whereas
non-uniform motion requires the concept of acceleration in
addition.  So far, we have not asked the question as to what
governs the motion of bodies. In this chapter, we turn to this
basic question.
Let us first guess the answer based on our common
experience. To move a football at rest, someone must kick it.
To throw a stone upwards, one has to  give it an upward
push.  A breeze causes the branches of a tree to swing; a
strong wind can even move heavy objects. A boat moves in a
flowing river without anyone rowing it. Clearly, some external
agency is needed to provide force to move a body from rest.
Likewise, an external force is needed also to retard or stop
motion.  You can stop a ball rolling down an inclined plane by
applying a force against the direction of its motion.
In these examples, the external agency of force (hands,
wind, stream, etc) is in contact with the object. This is not
always necessary. A stone released from the top of a building
accelerates downward due to the gravitational pull of the
earth.  A bar magnet can attract an iron nail from a distance.
This shows that external agencies (e.g. gravitational and
magnetic forces )  can exert  force on a body even from a
distance.
In short, a force is required to put a stationary body in
motion or stop a moving body, and some external agency is
needed to provide this force. The external agency may or may
not be in contact with the body.
So far so good. But what if a body is moving uniformly (e.g.
a skater moving straight with constant speed on a horizontal
ice slab) ?  Is an external force required to keep a body in
uniform motion?
5.1 Introduction
5.2 Aristotle’s fallacy
5.3 The law of inertia
5.4 Newton’s first law of motion
5.5 Newton’s second law of
motion
5.6 Newton’s third law of motion
5.7 Conservation of momentum
5.8 Equilibrium of a particle
5.9 Common forces in mechanics
5.10 Circular motion
5.11 Solving problems in
mechanics
Summary
Points to ponder
Exercises
Additional exercises
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5.2  ARISTOTLE’S  FALLACY
The question posed above appears to be simple.
However, it took ages to answer it. Indeed, the
correct answer to this question given by Galileo
in the seventeenth century was the foundation
of Newtonian mechanics, which signalled the
birth of modern science.
The Greek thinker, Aristotle (384 B.C– 322
B.C.), held the view that if a body is moving,
something external is required to keep it moving.
According to this view, for example, an arrow
shot from a bow keeps flying since the air behind
the arrow keeps pushing it. The view was part of
an elaborate framework of ideas developed by
Aristotle on the motion of bodies in the universe.
Most of the Aristotelian ideas on motion are now
known to be wrong and need not concern us.
For our purpose here, the Aristotelian law of
motion may be phrased thus: An external force
is required  to keep a body in motion.
Aristotelian law of motion is flawed, as we shall
see.  However, it is a natural view that anyone
would hold from common experience. Even a
small child playing with a simple (non-electric)
toy-car on a floor knows intuitively that it needs
to constantly drag the string attached to the toy-
car with some force to keep it going.  If it releases
the string, it comes to rest. This experience is
common to most terrestrial motion. External
forces seem to be needed to keep bodies in
motion. Left to themselves, all bodies eventually
come to rest.
What is the flaw in Aristotle’s argument? The
answer is: a moving toy car comes to rest because
the external force of friction on the car by the floor
opposes its motion. To counter this force, the child
has to apply an external force on the car in the
direction of motion.  When the car is in uniform
motion, there is no net external force acting on it:
the force by the child cancels the force ( friction)
by the floor.  The corollary is: if there were no friction,
the child would not be required to apply any force
to keep the toy car in uniform motion.
The opposing forces such as friction (solids)
and viscous forces (for fluids) are always present
in the natural world.  This explains why forces
by external agencies are necessary to overcome
the frictional forces to keep bodies in uniform
motion. Now we understand  where Aristotle
went wrong.  He coded this practical experience
in the form of a basic argument.  To get at the
true law of nature for forces and motion, one has
to imagine a world in which uniform motion is
possible with no frictional forces opposing. This
is what Galileo did.
5.3  THE LAW OF INERTIA
Galileo studied motion of objects on an inclined
plane.  Objects (i) moving down an inclined plane
accelerate, while those (ii) moving up retard.
(iii) Motion on a horizontal plane  is an
intermediate situation.  Galileo concluded that
an object moving on a frictionless horizontal
plane must neither have acceleration nor
retardation, i.e. it should move with constant
velocity (Fig. 5.1(a)).
(i) (ii) (iii)
Fig. 5.1(a)
Another experiment by Galileo leading to the
same conclusion involves a double inclined plane.
A ball released from rest on one of the planes rolls
down and climbs up the other. If the planes are
smooth, the final height of the ball is nearly the
same as the initial height (a little less but never
greater). In the ideal situation, when friction is
absent, the final height of the ball is the same
as its initial height.
If the slope of the second plane is decreased
and the experiment repeated, the ball will still
reach the same height, but in doing so, it will
travel a longer distance.  In the limiting case, when
the slope of the second plane is zero (i.e. is a
horizontal) the ball travels an infinite distance.
In other words, its motion never ceases. This is,
of course, an idealised situation (Fig. 5.1(b)).
Fig. 5.1(b) The law of inertia was inferred by Galileo
from observations of motion of a ball on a
double inclined plane.
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LAWS OF MOTION 91
In practice, the ball does come to a stop after
moving a finite distance on the horizontal plane,
because of the opposing force of friction which
can never be totally eliminated.  However, if there
were no friction, the ball would continue  to move
with a constant velocity on the horizontal plane.
Galileo thus, arrived at a new insight on
motion that had eluded Aristotle and those who
followed him.  The state of rest and the state of
uniform linear motion (motion with constant
velocity) are equivalent. In both cases, there is
no net force acting on the body.  It is incorrect to
assume that a net force is needed to keep a body
in uniform motion. To maintain a body in
uniform motion, we need to apply an external
force to ecounter the frictional force, so that
the two forces sum up to zero net external
force.
To summarise, if the net external force is zero,
a body at rest continues to remain at rest and a
body in motion continues to move with a uniform
velocity.  This property of the body is called
inertia. Inertia means ‘resistance to  change’.
A body does not change its state of rest or
uniform motion, unless an external force
compels it to change that state.
5.4  NEWTON’S FIRST LAW OF MOTION
Galileo’s simple, but revolutionary ideas
dethroned Aristotelian mechanics. A new
mechanics had to be developed. This task was
Ideas on Motion in Ancient Indian Science
Ancient Indian thinkers had arrived at an elaborate system of ideas on motion. Force, the cause of
motion, was thought to be of different kinds : force due to continuous pressure (nodan), as the force
of wind on a sailing vessel; impact (abhighat), as when a potter’s rod strikes the wheel; persistent
tendency (sanskara) to move in a straight line(vega) or restoration of shape in an elastic body;
transmitted force by a string, rod, etc. The notion of (vega) in the Vaisesika theory of motion perhaps
comes closest to the concept of inertia.  Vega, the tendency to move in a straight line, was thought to
be opposed by contact with objects including atmosphere, a parallel to the ideas of friction and air
resistance.  It was correctly summarised that the different kinds of motion (translational, rotational
and vibrational) of an extended body arise from only the translational motion of its constituent
particles. A falling leaf in the wind may have downward motion as a whole (patan) and also rotational
and vibrational motion (bhraman, spandan), but each particle of the leaf at an instant only has a
definite (small) displacement. There was considerable focus in Indian thought on measurement of
motion and units of length and time.  It was known that the position of a particle in space can be
indicated by distance measured along three axes.  Bhaskara (1150 A.D.) had introduced the concept
of ‘instantaneous motion’ (tatkaliki gati), which anticipated the modern notion of instantaneous
velocity using Differential Calculus. The difference between a wave and a current (of water) was clearly
understood; a current is a motion of particles of water under gravity and fluidity while a wave results
from the transmission of vibrations of water particles.
accomplished almost single-handedly by Isaac
Newton, one of the greatest scientists of all times.
Newton built on Galileo’s ideas and laid the
foundation of mechanics in terms of three laws
of  motion that go by his name.  Galileo’s law of
inertia was his starting point which he
formulated as the first law of motion:
Every body continues to be in its state
of rest or of uniform motion in a straight
line unless compelled by some external
force to act otherwise.
The state of rest or uniform linear motion both
imply zero acceleration. The first law of motion  can,
therefore, be simply expressed as:
If the net external force on a body is zero, its
acceleration is zero.  Acceleration can be non
zero only if there is a net external force on
the body.
Two kinds of situations are encountered in the
application of this law in practice. In some
examples, we know that the net external force
on the object is zero. In that case we can
conclude that the acceleration of the object is
zero.  For example, a spaceship out in
interstellar space, far from all other objects and
with all its rockets turned off, has no net
external force acting on it.  Its acceleration,
according to the first law, must be zero.  If it is
in motion, it must continue to move with a
uniform velocity.
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PHYSICS 92
More often, however, we do not know all the
forces to begin with.  In that case, if we know
that an object is unaccelerated (i.e. it is either
at rest or in uniform linear motion), we can infer
from the first law that the net external force on
the object must be zero. Gravity is everywhere.
For terrestrial phenomena, in particular, every
object experiences gravitational force due to the
earth.  Also objects in motion generally experience
friction, viscous drag, etc. If then, on earth, an
object is at rest or in uniform linear motion, it is
not because there are no forces acting on it, but
because the various external forces cancel out
i.e. add up to zero net external force.
Consider a book at rest on a horizontal surface
Fig. (5.2(a)).  It is subject to two external forces :
the force due to gravity (i.e. its weight W) acting
downward and the upward force on the book by
the table, the normal force R . R is a self-adjusting
force. This is an example of the kind of situation
mentioned above. The forces are not quite known
fully but the state of motion is known. We observe
the book to be at rest.  Therefore, we conclude
from the first law that the magnitude of R equals
that of W. A statement often encountered is :
“Since W = R, forces cancel and, therefore, the book
is at rest”. This is incorrect reasoning. The correct
statement is : “Since the book is observed to be at
rest, the net external force on it must be zero,
according to the first law. This implies that the
Galileo Galilei, born in Pisa, Italy in 1564 was a key figure in the scientific revolution
in Europe about four centuries ago.  Galileo proposed the concept of acceleration.
From experiments on motion of bodies on inclined planes or falling freely, he
contradicted the Aristotelian notion that a force was required to keep a body in
motion, and that heavier bodies fall faster than lighter bodies under gravity.  He
thus arrived at the law of inertia that was the starting point of the subsequent
epochal work of Isaac Newton.
Galileo’s discoveries in astronomy were equally revolutionary.  In 1609, he designed
his own telescope (invented earlier in Holland) and used it to make a number of
startling  observations :  mountains  and depressions on  the  surface of the moon;
dark spots on the sun; the moons of Jupiter and the phases of Venus.  He concluded
that the Milky Way derived its luminosity because of a large number of stars not visible to the naked eye.
In his masterpiece of scientific reasoning : Dialogue on the Two Chief World Systems, Galileo advocated
the heliocentric theory of the solar system proposed by Copernicus, which eventually got universal
acceptance.
With Galileo came a turning point in the very method of scientific inquiry. Science was no longer
merely observations of nature and inferences from them. Science meant devising and doing experiments
to verify or refute theories. Science meant measurement of quantities and a search for mathematical
relations between them. Not undeservedly, many regard Galileo as the father of modern science.
normal force R  must be equal and opposite to the
weight W ”.
Fig. 5.2 (a) a book at rest on the table, and (b) a car
moving with uniform velocity. The net force
is zero in each case.
Consider the motion of a car starting from
rest, picking up speed and then moving on a
smooth straight road with uniform speed (Fig.
(5.2(b)).  When the car is stationary, there is no
net force acting on it. During pick-up, it
accelerates. This must happen due to a net
external force. Note, it has to be an external force.
The acceleration of the car cannot be accounted
for by any internal force.  This might sound
surprising, but it is true.  The only conceivable
external force along the road is the force of
friction.  It is the frictional force that accelerates
the car as a whole.  (You will learn about friction
in section 5.9).  When the car moves with
constant velocity, there is no net external force.
Galileo Galilei (1564 - 1642)
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LAWS OF MOTION 93
t
It relates the net external force to the
acceleration of the body.
Momentum
Momentum of a body is defined to be the product
of its mass m and velocity v, and is denoted
by p:
p = m v            (5.1)
Momentum is clearly a vector quantity.  The
following common experiences indicate the
importance of this quantity for considering the
effect of force on motion.
• Suppose a light-weight vehicle (say a small
car) and a heavy weight vehicle (say a loaded
truck) are parked on a horizontal road. We all
know that a much greater force is needed to
push the truck than the car to bring them to
the same speed in same time.  Similarly, a
greater opposing force is needed to stop a
heavy body than a light body in the same time,
if they are moving with the same speed.
• If two stones, one light and the other heavy,
are dropped from the top of a building, a
person on the ground will find it easier to catch
the light stone than the heavy stone.  The
mass of a body is thus an important
parameter that determines the effect of force
on its motion.
• Speed is another important parameter to
consider. A bullet fired by a gun can easily
pierce human tissue before it stops, resulting
in casualty.  The same bullet fired with
moderate speed will not cause much damage.
Thus for a given mass, the greater the speed,
the greater is the opposing force needed to stop
the body in a certain time.  Taken together,
the product of mass and velocity, that is
momentum, is evidently a relevant variable
of motion. The greater the change in the
momentum in a given time, the greater is the
force that needs to be applied.
• A seasoned cricketer catches a cricket ball
coming in with great speed far more easily
than a novice, who can hurt his hands in the
act.  One reason is that the cricketer allows a
longer time for his hands to stop the ball.  As
you may have noticed, he draws in the hands
backward in the act of catching the ball (Fig.
5.3).  The novice, on the other hand, keeps
his hands fixed and tries to catch the ball
almost instantly. He needs to provide a much
greater force to stop the ball instantly, and
The property of inertia contained in the First
law is evident in many situations.  Suppose we
are standing in a stationary  bus and the driver
starts the bus suddenly. We get thrown
backward with a jerk. Why ? Our feet are in touch
with the floor. If there were no friction, we would
remain where we were, while the floor of the bus
would simply slip forward under our feet and the
back of the bus would hit us.  However,
fortunately, there is some friction between the
feet and the floor.  If the start is not too sudden,
i.e. if the acceleration is moderate, the frictional
force would be enough to accelerate our feet
along with the bus.  But our body is not strictly
a rigid body. It is deformable, i.e. it allows some
relative displacement between different parts.
What this means is that while our feet go with
the bus, the rest of the body remains where it is
due to inertia.  Relative to the bus, therefore, we
are thrown backward.  As soon as that happens,
however, the muscular forces on the rest of the
body (by the feet) come into play to move the body
along with the bus. A similar thing happens
when the bus suddenly stops.  Our feet stop due
to the friction which does not allow relative
motion between the feet and the floor of the bus.
But the rest of the body continues to move
forward due to inertia.  We are thrown forward.
The restoring muscular forces again come into
play and bring the body to rest.
Example 5.1  An astronaut accidentally
gets separated out of his small spaceship
accelerating in inter stellar space at a
constant rate of 100 m s
–2
.  What is the
acceleration of the astronaut the instant after
he is outside the spaceship ? (Assume that
there are no nearby stars to exert
gravitational force on him.)
Answer  Since there are no nearby stars to exert
gravitational force on him and the small
spaceship exerts negligible gravitational
attraction on him, the net force acting on the
astronaut, once he is out of the spaceship, is
zero. By the first law of motion the acceleration
of the astronaut is zero.             t
5.5  NEWTON’S SECOND LAW OF MOTION
The first law refers to the simple case when the
net external force on a body is zero.  The second
law of motion refers to the general situation when
there is a net external force acting on the body.
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