NCERT Textbook - Systems of Particles and Rotational Motion Class 11 Notes | EduRev

 Page 1


CHAPTER SEVEN
SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
7.1 INTRODUCTION
In the earlier chapters we primarily considered the motion
of a single particle. (A particle is ideally represented as a
point mass having no size.)  We applied the results of our
study even to the motion of bodies of finite size, assuming
that motion of such bodies can be described in terms of the
motion of a particle.
Any real body which we encounter in daily life has a
finite size.  In dealing with the motion of extended bodies
(bodies of finite size) often the idealised model of a particle is
inadequate.  In this chapter we shall try to go beyond this
inadequacy.  We shall attempt to build an understanding of
the motion of extended bodies.  An extended body, in the
first place, is a system of particles.  We shall begin with the
consideration of motion of the system as a whole.  The centre
of mass of a system of particles will be a key concept here.
We shall discuss the motion of the centre of mass of a system
of particles and usefulness of this concept in understanding
the motion of extended bodies.
A large class of problems with extended bodies can be
solved by considering them to be rigid bodies.  Ideally a
rigid body is a body with a perfectly definite and
unchanging shape.  The distances between all pairs of
particles of such a body do not change. It is evident from
this definition of a rigid body that no real body is truly rigid,
since real bodies deform under the influence of forces. But in
many situations the deformations are negligible.  In a number
of situations involving bodies such as wheels, tops, steel
beams, molecules and planets on the other hand, we can ignore
that they warp (twist out of shape), bend or vibrate and treat
them as rigid.
7.1.1 What kind of motion can a rigid body have?
Let us try to explore this question by taking some examples
of the motion of rigid bodies.  Let us begin with a rectangular
7.1 Introduction
7.2 Centre of mass
7.3 Motion of centre of mass
7.4 Linear momentum of a
system of particles
7.5 Vector product of two
vectors
7.6 Angular velocity and its
relation with linear velocity
7.7 Torque and angular
momentum
7.8 Equilibrium of a rigid body
7.9 Moment of inertia
7.10 Theorems of perpendicular
and parallel axes
7.11 Kinematics of rotational
motion about a fixed axis
7.12 Dynamics of rotational
motion about a fixed axis
7.13 Angular momentum in case
of rotation about a fixed axis
7.14 Rolling motion
Summary
Points to Ponder
Exercises
Additional exercises
2020-21
Page 2


CHAPTER SEVEN
SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
7.1 INTRODUCTION
In the earlier chapters we primarily considered the motion
of a single particle. (A particle is ideally represented as a
point mass having no size.)  We applied the results of our
study even to the motion of bodies of finite size, assuming
that motion of such bodies can be described in terms of the
motion of a particle.
Any real body which we encounter in daily life has a
finite size.  In dealing with the motion of extended bodies
(bodies of finite size) often the idealised model of a particle is
inadequate.  In this chapter we shall try to go beyond this
inadequacy.  We shall attempt to build an understanding of
the motion of extended bodies.  An extended body, in the
first place, is a system of particles.  We shall begin with the
consideration of motion of the system as a whole.  The centre
of mass of a system of particles will be a key concept here.
We shall discuss the motion of the centre of mass of a system
of particles and usefulness of this concept in understanding
the motion of extended bodies.
A large class of problems with extended bodies can be
solved by considering them to be rigid bodies.  Ideally a
rigid body is a body with a perfectly definite and
unchanging shape.  The distances between all pairs of
particles of such a body do not change. It is evident from
this definition of a rigid body that no real body is truly rigid,
since real bodies deform under the influence of forces. But in
many situations the deformations are negligible.  In a number
of situations involving bodies such as wheels, tops, steel
beams, molecules and planets on the other hand, we can ignore
that they warp (twist out of shape), bend or vibrate and treat
them as rigid.
7.1.1 What kind of motion can a rigid body have?
Let us try to explore this question by taking some examples
of the motion of rigid bodies.  Let us begin with a rectangular
7.1 Introduction
7.2 Centre of mass
7.3 Motion of centre of mass
7.4 Linear momentum of a
system of particles
7.5 Vector product of two
vectors
7.6 Angular velocity and its
relation with linear velocity
7.7 Torque and angular
momentum
7.8 Equilibrium of a rigid body
7.9 Moment of inertia
7.10 Theorems of perpendicular
and parallel axes
7.11 Kinematics of rotational
motion about a fixed axis
7.12 Dynamics of rotational
motion about a fixed axis
7.13 Angular momentum in case
of rotation about a fixed axis
7.14 Rolling motion
Summary
Points to Ponder
Exercises
Additional exercises
2020-21
142 PHYSICS
block sliding down an inclined plane without any
sidewise movement.  The block is taken as a rigid
body.  Its motion down the plane is such that all
the particles of the body are moving together,
i.e. they have the same velocity at any instant
of time. The rigid body here is in pure
translational motion (Fig. 7.1).
In pure translational motion at any
instant of time, all particles of the body have
the same velocity.
Consider now the rolling motion of a solid
metallic or wooden cylinder down the same
inclined plane (Fig. 7.2). The rigid body in this
problem, namely the cylinder, shifts from the
top to the bottom of the inclined plane, and thus,
seems to have translational motion.  But as Fig.
7.2 shows, all its particles are not moving with
the same velocity at any instant. The body,
therefore, is not in pure translational motion.
Its motion is translational plus ‘something else.’
In order to understand what this ‘something
else’ is, let us take a rigid body so constrained
that it cannot have translational motion.  The
most common way to constrain a rigid body so
that it does not have translational motion is to
fix it along a straight line. The only possible
motion of such a rigid body is rotation. The line
or fixed axis about which the body is rotating is
its axis of rotation. If you look around, you will
come across many examples of rotation about
an axis, a ceiling fan, a potter’s wheel, a giant
wheel in a fair, a merry-go-round and so on (Fig
7.3(a) and (b)).
(a)
(b)
Fig. 7.3 Rotation about a fixed axis
(a) A ceiling fan
(b) A potter’s wheel.
Let us try to understand what rotation is,
what characterises rotation.  You may notice that
in rotation of a rigid body about a fixed axis,
Fig 7.1 Translational (sliding) motion of a block down
an inclined plane.
(Any point like P
1
 or P
2 
of the block moves
with the same velocity at any instant of time.)
Fig. 7.2 Rolling motion of a cylinder. It is not pure
translational motion. Points P
1
, P
2
,
 
P
3
 and P
4
have different velocities (shown by arrows)
at any instant of time. In fact, the velocity of
the point of contact P
3 
is zero at any instant,
if the cylinder rolls without slipping.
2020-21
Page 3


CHAPTER SEVEN
SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
7.1 INTRODUCTION
In the earlier chapters we primarily considered the motion
of a single particle. (A particle is ideally represented as a
point mass having no size.)  We applied the results of our
study even to the motion of bodies of finite size, assuming
that motion of such bodies can be described in terms of the
motion of a particle.
Any real body which we encounter in daily life has a
finite size.  In dealing with the motion of extended bodies
(bodies of finite size) often the idealised model of a particle is
inadequate.  In this chapter we shall try to go beyond this
inadequacy.  We shall attempt to build an understanding of
the motion of extended bodies.  An extended body, in the
first place, is a system of particles.  We shall begin with the
consideration of motion of the system as a whole.  The centre
of mass of a system of particles will be a key concept here.
We shall discuss the motion of the centre of mass of a system
of particles and usefulness of this concept in understanding
the motion of extended bodies.
A large class of problems with extended bodies can be
solved by considering them to be rigid bodies.  Ideally a
rigid body is a body with a perfectly definite and
unchanging shape.  The distances between all pairs of
particles of such a body do not change. It is evident from
this definition of a rigid body that no real body is truly rigid,
since real bodies deform under the influence of forces. But in
many situations the deformations are negligible.  In a number
of situations involving bodies such as wheels, tops, steel
beams, molecules and planets on the other hand, we can ignore
that they warp (twist out of shape), bend or vibrate and treat
them as rigid.
7.1.1 What kind of motion can a rigid body have?
Let us try to explore this question by taking some examples
of the motion of rigid bodies.  Let us begin with a rectangular
7.1 Introduction
7.2 Centre of mass
7.3 Motion of centre of mass
7.4 Linear momentum of a
system of particles
7.5 Vector product of two
vectors
7.6 Angular velocity and its
relation with linear velocity
7.7 Torque and angular
momentum
7.8 Equilibrium of a rigid body
7.9 Moment of inertia
7.10 Theorems of perpendicular
and parallel axes
7.11 Kinematics of rotational
motion about a fixed axis
7.12 Dynamics of rotational
motion about a fixed axis
7.13 Angular momentum in case
of rotation about a fixed axis
7.14 Rolling motion
Summary
Points to Ponder
Exercises
Additional exercises
2020-21
142 PHYSICS
block sliding down an inclined plane without any
sidewise movement.  The block is taken as a rigid
body.  Its motion down the plane is such that all
the particles of the body are moving together,
i.e. they have the same velocity at any instant
of time. The rigid body here is in pure
translational motion (Fig. 7.1).
In pure translational motion at any
instant of time, all particles of the body have
the same velocity.
Consider now the rolling motion of a solid
metallic or wooden cylinder down the same
inclined plane (Fig. 7.2). The rigid body in this
problem, namely the cylinder, shifts from the
top to the bottom of the inclined plane, and thus,
seems to have translational motion.  But as Fig.
7.2 shows, all its particles are not moving with
the same velocity at any instant. The body,
therefore, is not in pure translational motion.
Its motion is translational plus ‘something else.’
In order to understand what this ‘something
else’ is, let us take a rigid body so constrained
that it cannot have translational motion.  The
most common way to constrain a rigid body so
that it does not have translational motion is to
fix it along a straight line. The only possible
motion of such a rigid body is rotation. The line
or fixed axis about which the body is rotating is
its axis of rotation. If you look around, you will
come across many examples of rotation about
an axis, a ceiling fan, a potter’s wheel, a giant
wheel in a fair, a merry-go-round and so on (Fig
7.3(a) and (b)).
(a)
(b)
Fig. 7.3 Rotation about a fixed axis
(a) A ceiling fan
(b) A potter’s wheel.
Let us try to understand what rotation is,
what characterises rotation.  You may notice that
in rotation of a rigid body about a fixed axis,
Fig 7.1 Translational (sliding) motion of a block down
an inclined plane.
(Any point like P
1
 or P
2 
of the block moves
with the same velocity at any instant of time.)
Fig. 7.2 Rolling motion of a cylinder. It is not pure
translational motion. Points P
1
, P
2
,
 
P
3
 and P
4
have different velocities (shown by arrows)
at any instant of time. In fact, the velocity of
the point of contact P
3 
is zero at any instant,
if the cylinder rolls without slipping.
2020-21
SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 143
every particle of the body moves in a circle,
which lies in a plane perpendicular to the axis
and has its centre on the axis.  Fig. 7.4 shows
the rotational motion of a rigid body about a fixed
axis (the z-axis of the frame of reference).  Let P
1
be a particle of the rigid body, arbitrarily chosen
and at a distance r
1
 from fixed axis.  The particle
P
1
 describes a circle of radius  r
1 
with its centre
C
1
 on the fixed axis.  The circle lies in a plane
perpendicular to the axis. The figure also shows
another particle P
2
 of the rigid body, P
2 
is at a
distance r
2 
from the fixed axis. The particle P
2
moves in a circle of radius r
2 
and with centre C
2
on the axis. This circle, too, lies in a plane
perpendicular to the axis.  Note that the circles
described by P
1
 and P
2
 may lie in different planes;
both these planes, however, are perpendicular
to the fixed axis.  For any particle on the axis
like P
3
, r = 0. Any such particle remains
stationary while the body rotates.  This is
expected since the axis of rotation is fixed.
Fig. 7.5 (a) A spinning top
(The point of contact of the top with the
ground, its tip O, is fixed.)
Fig. 7.5 (b) An oscillating table fan with rotating
blades. The pivot of the fan, point O, is
fixed. The blades of the fan are under
rotational motion, whereas, the axis of
rotation of  the fan blades is oscillating.
Fig. 7.4 A rigid body rotation about the z-axis (Each
point of the body such as P
1
 or
P
2
 describes a circle with its centre (C
1
or C
2
) on the axis of rotation.  The radius of
the circle (r
1
or r
2
) is the perpendicular
distance of the point (P
1
 or P
2
) from the
axis. A point on the axis like P
3 
remains
stationary).
Axis of oscillation
Axis of
rotation
from blades
In some examples of rotation, however, the
axis may not be fixed.  A prominent example of
this kind of rotation is a top spinning in place
[Fig. 7.5(a)].  (We assume that the top does not
slip from place to place and so does not have
translational motion.)  We know from experience
that the axis of such a spinning top moves
around the vertical through its point of contact
with the ground, sweeping out a cone as shown
in Fig. 7.5(a).  (This movement of the axis of the
top around the vertical is termed precession.)
Note, the point of contact of the top with
ground is fixed. The axis of rotation of the top
at any instant passes through the point of
contact. Another simple example of this kind of
rotation is the oscillating table fan or a pedestal
fan [Fig.7.5(b)]. You may have observed that the
2020-21
Page 4


CHAPTER SEVEN
SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
7.1 INTRODUCTION
In the earlier chapters we primarily considered the motion
of a single particle. (A particle is ideally represented as a
point mass having no size.)  We applied the results of our
study even to the motion of bodies of finite size, assuming
that motion of such bodies can be described in terms of the
motion of a particle.
Any real body which we encounter in daily life has a
finite size.  In dealing with the motion of extended bodies
(bodies of finite size) often the idealised model of a particle is
inadequate.  In this chapter we shall try to go beyond this
inadequacy.  We shall attempt to build an understanding of
the motion of extended bodies.  An extended body, in the
first place, is a system of particles.  We shall begin with the
consideration of motion of the system as a whole.  The centre
of mass of a system of particles will be a key concept here.
We shall discuss the motion of the centre of mass of a system
of particles and usefulness of this concept in understanding
the motion of extended bodies.
A large class of problems with extended bodies can be
solved by considering them to be rigid bodies.  Ideally a
rigid body is a body with a perfectly definite and
unchanging shape.  The distances between all pairs of
particles of such a body do not change. It is evident from
this definition of a rigid body that no real body is truly rigid,
since real bodies deform under the influence of forces. But in
many situations the deformations are negligible.  In a number
of situations involving bodies such as wheels, tops, steel
beams, molecules and planets on the other hand, we can ignore
that they warp (twist out of shape), bend or vibrate and treat
them as rigid.
7.1.1 What kind of motion can a rigid body have?
Let us try to explore this question by taking some examples
of the motion of rigid bodies.  Let us begin with a rectangular
7.1 Introduction
7.2 Centre of mass
7.3 Motion of centre of mass
7.4 Linear momentum of a
system of particles
7.5 Vector product of two
vectors
7.6 Angular velocity and its
relation with linear velocity
7.7 Torque and angular
momentum
7.8 Equilibrium of a rigid body
7.9 Moment of inertia
7.10 Theorems of perpendicular
and parallel axes
7.11 Kinematics of rotational
motion about a fixed axis
7.12 Dynamics of rotational
motion about a fixed axis
7.13 Angular momentum in case
of rotation about a fixed axis
7.14 Rolling motion
Summary
Points to Ponder
Exercises
Additional exercises
2020-21
142 PHYSICS
block sliding down an inclined plane without any
sidewise movement.  The block is taken as a rigid
body.  Its motion down the plane is such that all
the particles of the body are moving together,
i.e. they have the same velocity at any instant
of time. The rigid body here is in pure
translational motion (Fig. 7.1).
In pure translational motion at any
instant of time, all particles of the body have
the same velocity.
Consider now the rolling motion of a solid
metallic or wooden cylinder down the same
inclined plane (Fig. 7.2). The rigid body in this
problem, namely the cylinder, shifts from the
top to the bottom of the inclined plane, and thus,
seems to have translational motion.  But as Fig.
7.2 shows, all its particles are not moving with
the same velocity at any instant. The body,
therefore, is not in pure translational motion.
Its motion is translational plus ‘something else.’
In order to understand what this ‘something
else’ is, let us take a rigid body so constrained
that it cannot have translational motion.  The
most common way to constrain a rigid body so
that it does not have translational motion is to
fix it along a straight line. The only possible
motion of such a rigid body is rotation. The line
or fixed axis about which the body is rotating is
its axis of rotation. If you look around, you will
come across many examples of rotation about
an axis, a ceiling fan, a potter’s wheel, a giant
wheel in a fair, a merry-go-round and so on (Fig
7.3(a) and (b)).
(a)
(b)
Fig. 7.3 Rotation about a fixed axis
(a) A ceiling fan
(b) A potter’s wheel.
Let us try to understand what rotation is,
what characterises rotation.  You may notice that
in rotation of a rigid body about a fixed axis,
Fig 7.1 Translational (sliding) motion of a block down
an inclined plane.
(Any point like P
1
 or P
2 
of the block moves
with the same velocity at any instant of time.)
Fig. 7.2 Rolling motion of a cylinder. It is not pure
translational motion. Points P
1
, P
2
,
 
P
3
 and P
4
have different velocities (shown by arrows)
at any instant of time. In fact, the velocity of
the point of contact P
3 
is zero at any instant,
if the cylinder rolls without slipping.
2020-21
SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 143
every particle of the body moves in a circle,
which lies in a plane perpendicular to the axis
and has its centre on the axis.  Fig. 7.4 shows
the rotational motion of a rigid body about a fixed
axis (the z-axis of the frame of reference).  Let P
1
be a particle of the rigid body, arbitrarily chosen
and at a distance r
1
 from fixed axis.  The particle
P
1
 describes a circle of radius  r
1 
with its centre
C
1
 on the fixed axis.  The circle lies in a plane
perpendicular to the axis. The figure also shows
another particle P
2
 of the rigid body, P
2 
is at a
distance r
2 
from the fixed axis. The particle P
2
moves in a circle of radius r
2 
and with centre C
2
on the axis. This circle, too, lies in a plane
perpendicular to the axis.  Note that the circles
described by P
1
 and P
2
 may lie in different planes;
both these planes, however, are perpendicular
to the fixed axis.  For any particle on the axis
like P
3
, r = 0. Any such particle remains
stationary while the body rotates.  This is
expected since the axis of rotation is fixed.
Fig. 7.5 (a) A spinning top
(The point of contact of the top with the
ground, its tip O, is fixed.)
Fig. 7.5 (b) An oscillating table fan with rotating
blades. The pivot of the fan, point O, is
fixed. The blades of the fan are under
rotational motion, whereas, the axis of
rotation of  the fan blades is oscillating.
Fig. 7.4 A rigid body rotation about the z-axis (Each
point of the body such as P
1
 or
P
2
 describes a circle with its centre (C
1
or C
2
) on the axis of rotation.  The radius of
the circle (r
1
or r
2
) is the perpendicular
distance of the point (P
1
 or P
2
) from the
axis. A point on the axis like P
3 
remains
stationary).
Axis of oscillation
Axis of
rotation
from blades
In some examples of rotation, however, the
axis may not be fixed.  A prominent example of
this kind of rotation is a top spinning in place
[Fig. 7.5(a)].  (We assume that the top does not
slip from place to place and so does not have
translational motion.)  We know from experience
that the axis of such a spinning top moves
around the vertical through its point of contact
with the ground, sweeping out a cone as shown
in Fig. 7.5(a).  (This movement of the axis of the
top around the vertical is termed precession.)
Note, the point of contact of the top with
ground is fixed. The axis of rotation of the top
at any instant passes through the point of
contact. Another simple example of this kind of
rotation is the oscillating table fan or a pedestal
fan [Fig.7.5(b)]. You may have observed that the
2020-21
144 PHYSICS
axis of rotation of such a fan has an oscillating
(sidewise) movement in a horizontal plane about
the vertical through the point at which the axis
is pivoted (point O in Fig. 7.5(b)).
While the fan rotates and its axis moves
sidewise, this point is fixed.  Thus, in more
general cases of rotation, such as the rotation
of a top or a pedestal fan, one point and not
one line, of the rigid body is fixed. In this case
the axis is not fixed, though it always passes
through the fixed point. In our study, however,
we mostly deal with the simpler and special case
of rotation in which one line (i.e. the axis) is fixed.
Thus, for us rotation will be about a fixed axis
only unless stated otherwise.
The rolling motion of a cylinder down an
inclined plane is a combination of rotation about
a fixed axis and translation.  Thus, the
‘something else’ in the case of rolling motion
which we referred to earlier is rotational motion.
You will find Fig. 7.6(a) and (b) instructive from
this point of view. Both these figures show
motion of the same body along identical
translational trajectory. In one case, Fig. 7.6(a),
the motion is a pure translation; in the other
case [Fig. 7.6(b)] it is a combination of
translation and rotation. (You may try to
reproduce the two types of motion shown, using
a rigid object like a heavy book.)
We now recapitulate the most important
observations of the present section: The motion
of a rigid body which is not pivoted or fixed in
some way is either a pure translation or a
combination of translation and rotation. The
motion of a rigid body which is pivoted or fixed
in some way is rotation.  The rotation may be
about an axis that is fixed (e.g. a ceiling fan) or
moving (e.g. an oscillating table fan [Fig.7.5(b)]).
We shall, in the present chapter, consider
rotational motion about a fixed axis only.
7.2  CENTRE OF MASS
We shall first see what the centre of mass of a
system of particles is and then discuss its
significance. For simplicity we shall start with
a two particle system. We shall take the line
joining the two particles to be the x- axis.
Fig. 7.7
Let the distances of the two particles be x
1
and x
2
 respectively from some origin O. Let m
1
and m
2
 be respectively the masses of the two
Fig. 7.6(a) Motion of a rigid body which is pure
translation.
Fig. 7.6(b) Motion of a rigid body which is a
combination of translation and
rotation.
Fig 7.6 (a) and 7.6 (b) illustrate different motions of
the same body. Note P is an arbitrary point of the
body; O is the centre of mass of the body, which is
defined in the next section. Suffice to say here that
the trajectories of O are the translational trajectories
Tr
1
 and Tr
2 
of the body. The positions O and P at
three different instants of time are shown by O
1
, O
2
,
and O
3
, and P
1
, P
2
 and P
3
, respectively, in both
Figs. 7.6 (a) and (b) . As seen from Fig. 7.6(a), at any
instant the velocities of any particles like O and P of
the body are the same in pure translation. Notice, in
this case the orientation of OP, i.e. the angle OP makes
with a fixed direction, say the horizontal, remains
the same, i.e. a
1 
= a
2
 = a
3
. Fig. 7.6 (b) illustrates a
case of combination of translation and rotation. In
this case, at any instants the velocities of O and P
differ. Also, a
1
, a
2
 and a
3
 may all be different.
2020-21
Page 5


CHAPTER SEVEN
SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
7.1 INTRODUCTION
In the earlier chapters we primarily considered the motion
of a single particle. (A particle is ideally represented as a
point mass having no size.)  We applied the results of our
study even to the motion of bodies of finite size, assuming
that motion of such bodies can be described in terms of the
motion of a particle.
Any real body which we encounter in daily life has a
finite size.  In dealing with the motion of extended bodies
(bodies of finite size) often the idealised model of a particle is
inadequate.  In this chapter we shall try to go beyond this
inadequacy.  We shall attempt to build an understanding of
the motion of extended bodies.  An extended body, in the
first place, is a system of particles.  We shall begin with the
consideration of motion of the system as a whole.  The centre
of mass of a system of particles will be a key concept here.
We shall discuss the motion of the centre of mass of a system
of particles and usefulness of this concept in understanding
the motion of extended bodies.
A large class of problems with extended bodies can be
solved by considering them to be rigid bodies.  Ideally a
rigid body is a body with a perfectly definite and
unchanging shape.  The distances between all pairs of
particles of such a body do not change. It is evident from
this definition of a rigid body that no real body is truly rigid,
since real bodies deform under the influence of forces. But in
many situations the deformations are negligible.  In a number
of situations involving bodies such as wheels, tops, steel
beams, molecules and planets on the other hand, we can ignore
that they warp (twist out of shape), bend or vibrate and treat
them as rigid.
7.1.1 What kind of motion can a rigid body have?
Let us try to explore this question by taking some examples
of the motion of rigid bodies.  Let us begin with a rectangular
7.1 Introduction
7.2 Centre of mass
7.3 Motion of centre of mass
7.4 Linear momentum of a
system of particles
7.5 Vector product of two
vectors
7.6 Angular velocity and its
relation with linear velocity
7.7 Torque and angular
momentum
7.8 Equilibrium of a rigid body
7.9 Moment of inertia
7.10 Theorems of perpendicular
and parallel axes
7.11 Kinematics of rotational
motion about a fixed axis
7.12 Dynamics of rotational
motion about a fixed axis
7.13 Angular momentum in case
of rotation about a fixed axis
7.14 Rolling motion
Summary
Points to Ponder
Exercises
Additional exercises
2020-21
142 PHYSICS
block sliding down an inclined plane without any
sidewise movement.  The block is taken as a rigid
body.  Its motion down the plane is such that all
the particles of the body are moving together,
i.e. they have the same velocity at any instant
of time. The rigid body here is in pure
translational motion (Fig. 7.1).
In pure translational motion at any
instant of time, all particles of the body have
the same velocity.
Consider now the rolling motion of a solid
metallic or wooden cylinder down the same
inclined plane (Fig. 7.2). The rigid body in this
problem, namely the cylinder, shifts from the
top to the bottom of the inclined plane, and thus,
seems to have translational motion.  But as Fig.
7.2 shows, all its particles are not moving with
the same velocity at any instant. The body,
therefore, is not in pure translational motion.
Its motion is translational plus ‘something else.’
In order to understand what this ‘something
else’ is, let us take a rigid body so constrained
that it cannot have translational motion.  The
most common way to constrain a rigid body so
that it does not have translational motion is to
fix it along a straight line. The only possible
motion of such a rigid body is rotation. The line
or fixed axis about which the body is rotating is
its axis of rotation. If you look around, you will
come across many examples of rotation about
an axis, a ceiling fan, a potter’s wheel, a giant
wheel in a fair, a merry-go-round and so on (Fig
7.3(a) and (b)).
(a)
(b)
Fig. 7.3 Rotation about a fixed axis
(a) A ceiling fan
(b) A potter’s wheel.
Let us try to understand what rotation is,
what characterises rotation.  You may notice that
in rotation of a rigid body about a fixed axis,
Fig 7.1 Translational (sliding) motion of a block down
an inclined plane.
(Any point like P
1
 or P
2 
of the block moves
with the same velocity at any instant of time.)
Fig. 7.2 Rolling motion of a cylinder. It is not pure
translational motion. Points P
1
, P
2
,
 
P
3
 and P
4
have different velocities (shown by arrows)
at any instant of time. In fact, the velocity of
the point of contact P
3 
is zero at any instant,
if the cylinder rolls without slipping.
2020-21
SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 143
every particle of the body moves in a circle,
which lies in a plane perpendicular to the axis
and has its centre on the axis.  Fig. 7.4 shows
the rotational motion of a rigid body about a fixed
axis (the z-axis of the frame of reference).  Let P
1
be a particle of the rigid body, arbitrarily chosen
and at a distance r
1
 from fixed axis.  The particle
P
1
 describes a circle of radius  r
1 
with its centre
C
1
 on the fixed axis.  The circle lies in a plane
perpendicular to the axis. The figure also shows
another particle P
2
 of the rigid body, P
2 
is at a
distance r
2 
from the fixed axis. The particle P
2
moves in a circle of radius r
2 
and with centre C
2
on the axis. This circle, too, lies in a plane
perpendicular to the axis.  Note that the circles
described by P
1
 and P
2
 may lie in different planes;
both these planes, however, are perpendicular
to the fixed axis.  For any particle on the axis
like P
3
, r = 0. Any such particle remains
stationary while the body rotates.  This is
expected since the axis of rotation is fixed.
Fig. 7.5 (a) A spinning top
(The point of contact of the top with the
ground, its tip O, is fixed.)
Fig. 7.5 (b) An oscillating table fan with rotating
blades. The pivot of the fan, point O, is
fixed. The blades of the fan are under
rotational motion, whereas, the axis of
rotation of  the fan blades is oscillating.
Fig. 7.4 A rigid body rotation about the z-axis (Each
point of the body such as P
1
 or
P
2
 describes a circle with its centre (C
1
or C
2
) on the axis of rotation.  The radius of
the circle (r
1
or r
2
) is the perpendicular
distance of the point (P
1
 or P
2
) from the
axis. A point on the axis like P
3 
remains
stationary).
Axis of oscillation
Axis of
rotation
from blades
In some examples of rotation, however, the
axis may not be fixed.  A prominent example of
this kind of rotation is a top spinning in place
[Fig. 7.5(a)].  (We assume that the top does not
slip from place to place and so does not have
translational motion.)  We know from experience
that the axis of such a spinning top moves
around the vertical through its point of contact
with the ground, sweeping out a cone as shown
in Fig. 7.5(a).  (This movement of the axis of the
top around the vertical is termed precession.)
Note, the point of contact of the top with
ground is fixed. The axis of rotation of the top
at any instant passes through the point of
contact. Another simple example of this kind of
rotation is the oscillating table fan or a pedestal
fan [Fig.7.5(b)]. You may have observed that the
2020-21
144 PHYSICS
axis of rotation of such a fan has an oscillating
(sidewise) movement in a horizontal plane about
the vertical through the point at which the axis
is pivoted (point O in Fig. 7.5(b)).
While the fan rotates and its axis moves
sidewise, this point is fixed.  Thus, in more
general cases of rotation, such as the rotation
of a top or a pedestal fan, one point and not
one line, of the rigid body is fixed. In this case
the axis is not fixed, though it always passes
through the fixed point. In our study, however,
we mostly deal with the simpler and special case
of rotation in which one line (i.e. the axis) is fixed.
Thus, for us rotation will be about a fixed axis
only unless stated otherwise.
The rolling motion of a cylinder down an
inclined plane is a combination of rotation about
a fixed axis and translation.  Thus, the
‘something else’ in the case of rolling motion
which we referred to earlier is rotational motion.
You will find Fig. 7.6(a) and (b) instructive from
this point of view. Both these figures show
motion of the same body along identical
translational trajectory. In one case, Fig. 7.6(a),
the motion is a pure translation; in the other
case [Fig. 7.6(b)] it is a combination of
translation and rotation. (You may try to
reproduce the two types of motion shown, using
a rigid object like a heavy book.)
We now recapitulate the most important
observations of the present section: The motion
of a rigid body which is not pivoted or fixed in
some way is either a pure translation or a
combination of translation and rotation. The
motion of a rigid body which is pivoted or fixed
in some way is rotation.  The rotation may be
about an axis that is fixed (e.g. a ceiling fan) or
moving (e.g. an oscillating table fan [Fig.7.5(b)]).
We shall, in the present chapter, consider
rotational motion about a fixed axis only.
7.2  CENTRE OF MASS
We shall first see what the centre of mass of a
system of particles is and then discuss its
significance. For simplicity we shall start with
a two particle system. We shall take the line
joining the two particles to be the x- axis.
Fig. 7.7
Let the distances of the two particles be x
1
and x
2
 respectively from some origin O. Let m
1
and m
2
 be respectively the masses of the two
Fig. 7.6(a) Motion of a rigid body which is pure
translation.
Fig. 7.6(b) Motion of a rigid body which is a
combination of translation and
rotation.
Fig 7.6 (a) and 7.6 (b) illustrate different motions of
the same body. Note P is an arbitrary point of the
body; O is the centre of mass of the body, which is
defined in the next section. Suffice to say here that
the trajectories of O are the translational trajectories
Tr
1
 and Tr
2 
of the body. The positions O and P at
three different instants of time are shown by O
1
, O
2
,
and O
3
, and P
1
, P
2
 and P
3
, respectively, in both
Figs. 7.6 (a) and (b) . As seen from Fig. 7.6(a), at any
instant the velocities of any particles like O and P of
the body are the same in pure translation. Notice, in
this case the orientation of OP, i.e. the angle OP makes
with a fixed direction, say the horizontal, remains
the same, i.e. a
1 
= a
2
 = a
3
. Fig. 7.6 (b) illustrates a
case of combination of translation and rotation. In
this case, at any instants the velocities of O and P
differ. Also, a
1
, a
2
 and a
3
 may all be different.
2020-21
SYSTEMS OF PARTICLES AND ROTATIONAL MOTION 145
particles. The centre of mass of the system is
that   point C which is at a distance X from O,
where X is given by
1 1 2 2
1 2
m x m x
X
m m
+
=
+
(7.1)
In Eq. (7.1), X can be regarded as the mass-
weighted mean of  x
1 
and x
2
. If the two particles
have the same mass m
1
 = m
2
 = m
, 
then
1 2 1 2
2 2
mx mx x x
X
m
+ +
= =
Thus, for two particles of equal mass the
centre of mass lies exactly midway between
them.
       If we have n particles of masses m
1
, m
2
,
...m
n
 respectively, along a straight line taken as
the x- axis, then by definition the position of the
centre of the mass of the system of particles is
given by.
X
m x m x m x
m m m
m x
m
m x
m
n n
n
i i
i
n
i
i
n
i i
i
= = =
=
=
?
?
?
?
1 1 2 2
1 2
1
1
+ + ... +
+ +... +
  (7.2)
where  x
1
, x
2
,...x
n
 are the distances of the
particles from the origin; X is also measured from
the same origin. The symbol 
?
(the Greek letter
sigma) denotes summation, in this case over n
particles. The sum
i
m M =
?
is the total mass of the system.
Suppose that we have three particles, not
lying in a straight line. We may define x– and y–
axes in the plane in which the particles lie and
represent the positions of the three particles by
coordinates (x
1
,y
1
), (x
2
,y
2
) and (x
3
,y
3
) respectively.
Let the masses of the three particles be m
1
, m
2
and m
3 
respectively. The centre of mass C of
the system of the three particles is defined and
located by the coordinates (X, Y) given by
1 1 2 2 3 3
1 2 3
m x m x m x
X
m m m
+ +
=
+ +
(7.3a)
1 1 2 2 3 3
1 2 3
m y m y m y
Y
m m m
+ +
=
+ +
(7.3b)
For the particles of equal mass m = m
1
 = m
2
= m
3
,
1 2 3 1 2 3
( )
3 3
m x x x x x x
X
m
+ + + +
= =
1 2 3 1 2 3
( )
3 3
m y y y y y y
Y
m
+ + + +
= =
Thus, for three particles of equal mass, the
centre of mass coincides with the centroid of the
triangle formed by the particles.
Results of Eqs. (7.3a) and (7.3b) are
generalised easily to a system of n particles, not
necessarily lying in a plane, but distributed in
space. The centre of mass of such a system is
at (X, Y, Z ), where
i i
m x
X
M
=
?
(7.4a)
i i
m y
Y
M
=
?
(7.4b)
and  
i i
m z
Z
M
=
?
(7.4c)
Here M = 
i
m
?
is the total mass of the
system. The index i runs from 1 to n; m
i
 is the
mass of the i
th
 particle and the position of the
i
th
 particle is given by (x
i
, y
i
, z
i
).
Eqs. (7.4a), (7.4b) and (7.4c) can be
combined into one equation using the notation
of position vectors. Let  
i
r be the position vector
of the i
th
 particle and R be the position vector of
the centre of mass:
 


i i i i
x y z = + + r i j k
 
and 


X Y Z = + + R i j k
 
Then  
i i
m
M
=
?
r
R (7.4d)
The sum on the right hand side is a vector
sum.
Note the economy of expressions we achieve
by use of vectors. If the origin of the frame of
reference (the coordinate system) is chosen to
be the centre of mass then  0
i i
m =
?
r for the
given system of particles.
A rigid body, such as a metre stick or a
flywheel, is a system of closely packed particles;
Eqs. (7.4a), (7.4b), (7.4c) and (7.4d) are
therefore, applicable to a rigid body. The number
of particles (atoms or molecules) in such a body
is so large that it is impossible to carry out the
summations over individual particles in these
equations. Since the spacing of the particles is
2020-21