“A particle is a hypothetical object having mass but no size and internal structure”. Such several independent particles moving with different velocities and changing their relative positions from time to time can be regarded as a “system of particles”.
We start our study with the simplest of such system of particles namely a “rigid body” which may have translational and rotational motions at the same time. A rigid body is a system of particles having a continuous distribution of matter, the relative distances between the particles remaining unchanged even under the action of external forces.
What kind of motion can a rigid body have?
Let us try to explore this question by taking few examples of motion of the bodies.
Example - I: Let us consider a rectangular body (block) sliding down an inclined plane without any sidewise movement. The block is a rigid body. Its motion down the plane is such that all the particles of the body are moving together (i.e. same velocity at any instant of time).
In this case the motion of a body is pure translational motion.
''In pure translational motion at any instant of time all particles of the body have the same velocity.''
Example - II: Consider a rigid body, namely cylinder, shifts from top to bottom of the inclined plane, then it has translational motion, but all particles are not moving with the same velocity at any instant.
The body therefore, is not in pure translation plus rotation. In rotation of a rigid body about a fixed axis, every particle of the body moves in circle, which lies in a plane perpendicular to the axis and has its centre on the axis.
The line along which the body is fixed is turned as its axis of rotation. Example of rotation about an axis, a ceiling fan, a potter’s wheel, a giant wheel, a merry-go-round and so on.
In few cases like spinning top, the axis of rotation may not be fixed. The axis moves around the vertical through its point of contact with the ground, sweeps out a cone.
This movement of axis of rotation is called “Precession”. In the present chapter consider rotational motion about a fixed axis only.
Centre of Mass:
The motion of rigid bodies can be studied without considering the interaction between the particles by the concept of centre of mass.
For example consider a wheel, rolling on a horizontal surface without slipping.
Let us choose three particles A, B and C as shown in figure.
The particle ‘A’ is on the rim of the wheel, which follows a complicated cycloid path. Similarly another particle ‘B’ also follows a complicated path. But the particle ‘C’ moves along a straight line. This point ‘C’ represents “Centre of mass” of the wheel. This point ‘C’ moves as if the total external force acting on it.
From the above example, we can say that different particles in a body may have different complicated paths, but certain point of the body has simple pure translational motion. The motion of this single point represents the motion of the entire body. Thus ''Centre of mass of a body or a system is a point where entire mass of the body or the system is supposed to be concentrated to describe its translatory motion''. It lies within the boundary of the body.
Position of centre of mass of some symmetrical bodies with uniform mass distribution.
Coordinates of Centre of mass: [for two particle system along a line (one dimension)]
Consider two particles of masses m1, m2 situated at distances x1, x2 from origin on X - axis. ‘X’ is the distance of centre of mass from origin. Let d1, d2 be distances of first (m1) and second (m2) particles from the centre of mass. Centre of mass of two particle system will be nearer to heavier mass. so, the distance of centre of mass from any particle is inversely proportional to the mass of particle.
But d1 = X - x1, d2 = x2 - X
(X - x1)m1 = (x2 - X)m2
m1 X - m1x1 = m2x2 - m2X
⇒ m1X + m2X = m1 x1 + m2x2
⇒ X(m1+ m2) = m1x1 + m2x2
i.e., the centre of mass of the system of particles depends on masses of the particles and their positions.
If m1 = m2 = m, then X cm = 
Thus, for two particles of equal masses the centre of mass lies exactly midway between them.
If there are n-particles of masses m1, m2, ...., mn situated along X - axis having coordinates x1, x2,... xn respectively.
Then position of centre of mass of system of particles is
'M' is total mass of the system.
Suppose that we have three particles, not lying in a straight line. The position of three particles can be represented by co-ordinates (x1, y1), (x2, y2), (x3, y3). The centre of mass of system of three particles is defined and located by coordinates (X cm, Y cm).
Thus for three particles of equal mass, the co-ordinates of centre of mass coincides with the centroid of the triangle.
If n-particles are distributed in space, then position of co-ordinates of centre of mass (Xcm, Ycm, Zcm) are expressed as.
The distance of centre of mass from origin in space is
Where M is total mass of the body, the co - ordinates of centre of mass are (for continuous mass distribution)
If the centre of mass as the origin of our co-ordinate system, then
Motion of centre of mass:
Consider a system of ‘n’- particles of masses m1, m2, ..., mn and their position vectors are .
differentiate on both sides with respect to time
are the velocities of individual particles of the system.
For two particle system, 
If two particles move in opposite direction then
If they move perpendicular to each other, then
Momentum of centre of mass:
From equation (1)
i.e., momentum of centre of mass is equal to the sum of the momentum of the particles in the system.
Acceleration of centre of mass:
From equation 
Differentiate on both sides with respect to ‘t’
From the above relation we can write
Let Σ F internal = 0, as all internal forces cancel each other because they are action and reaction pairs.
Thus, the centre of mass of a system of particle moves, as if it was a particle of mass equal to that of whole system with all the external forces acting directly on it.
vcm = Constant and from newton’s second Law
Thus, when total external force acting on the system of particle is zero, the total linear momentum of the system is constant. This is ''The Law of Conservation of linear momentum''.
Yet, if the total external force acting on the system is zero, the centre of mass moves with a constant velocity i.e., it moves uniformly in a straight line like a free particle.
Example-1: In solar system, moon moves around the earth in circular orbit and earth moves around the sun in an elliptical orbit. But earth-moon system rotates about common centre of mass. The centre of mass of the system is located very nearer to centre of mass of earth (inside the earth), because earth is heavier than moon. The interaction of earth and moon does not affect the motion of the centre of earth-moon system.
Gravitational force of sun is acts as external force on the system. Thus the centre of mass of the system moves in an elliptical path round the sun.
Example-2: Let us consider a cracker which explodes in air. Before explosion, the cracker moves along the parabolic path after explosion, under the action of gravity (external force). When, cracker explodes in the mid air under the action of internal forces.
These internal forces cannot change the motion of centre of mass. Thus the centre of mass follows the same parabolic path.
Centre of Gravity:
Let us take an irregular shaped cardboard and a narrow tipped object like a nail or a pencil. You can locate the point ‘G’, where it can be balanced on the tip by trial and error basis. Similarly many of us try to balance the notebook on the tip of a finger. In the above two cases the point of balance is called “centre of gravity”.
The tip of pencil or nail provides a vertically upward force due to which cardboard is in mechanical equilibrium. The reaction of tip is equal and opposite to ‘mg’, the total weight of cardboard. Hence the cardboard is in translational equilibrium as well as in rotational equilibrium.
Torques are developed on cardboard due to the force of gravity like m1g, m2g, ... etc acting on individual particles that balance the cardboard. At centre of gravity the total torque on it due to individual forces. m1g, m2g ... etc is zero. 
Thus we can define centre of gravity a body is a point where total gravitational torque is zero. The centre of gravity describes the stability of bodies. For smaller bodies centre of mass and centre of gravity coincide with each other.
However, for a body of such large dimensions, the value of ‘g’ is different for its different parts, the centre of mass may not coincide with centre of gravity. At a gravity free place, the centre of gravity makes no sense, where as centre of mass has still a meaning.
Rotational Motion:
If a particle in a rigid body is fixed and other particles are in motion, the particles should go round a fixed point/ particle, describing circles still retaining their relative positions. Such motion is called "Circular motion".
e.g.: Motion of a small body which is tied to a string and whirled round a fixed point.
" A body is said to be in rotatory motion if every particle moves in a circular path about a fixed point on the axis of rotation".
The locus of the centres of circular paths of the particles in a rotating body is called axis of rotation. It is an imaginary line, always perpendicular to the plane.
e.g.: 1) Motion of earth around the Sun.
2) Motion of planets around the Sun.
3) Motion of the fly wheel.
Radius vector: When a particle is moving on circumference of a circle, the line joining the position of particle at any instant of time and centre of circle is called radius vector.
Angular displacement (θ):
When a particle is moving along the circumference of a circle, the radius vector rotates. The angle described by radius vector in a given interval of time is called 'angular displacement' (θ).
In figure 'O' represents centre of circle.
P0 be the position of particle at t = 0, P be the position of particle at an instant of time 't'. 'θ' be the angular displacement during the time interval between 0 to t.
Unit of 'θ' is 'radian'. It has no dimensions.
1 radian = 
Angular velocity (ω):
The rate of change of angular displacement of a particle is called angular velocity (ω).
For one complete rotation θ = 2π
'n' be the no. of rotations made in 1 sec.
In the time t1, the radius vector describes an angle θ1, in the time t2, the angle described by radius vector is θ2, then average angular velocity
The angular velocity of a particle at a particular instant of time is called "Instantaneous angular velocity".
Unit of angular velocity is rad/ sec.
Dimensional formula is [T−1].
It is pseudo vector. The direction of angular velocity will be along the axis of rotation.
* Angular velocity of seconds hand = 
* Angular velocity of minutes hand = 
* Angular velocity of hours hand = 
* Angular velocity of self rotation of earth = 
Example - I
Find the average angular velocity of spinning motion of the earth.
Sol: Earth completes one rotation in 24 hours i.e., n = 1
Angular velocity ω = 
Example - II
When a motor cyclist takes a U-turn in 4 seconds. What is the average angular velocity of the motor cyclist?
Sol: When cyclist takes U-turn, angular displacement θ = π rad.
Given t = 4 sec.
Average angular velocity ω = 
= 0.7855 rad/s
Example - III
The angular displacement of a particle is θ = 3t3 + 9t2 + 6t + 4.
θ is in radian and 't' is in seconds. Find its angular velocity at t = 1 sec.
Sol: 
= 3 × 3t2 + 18t + 6
= 9 + 18 + 6 = 33 rad − sec−1
Relation between linear velocity (v) and angular velocity (ω):
In rotational motion of a rigid body about a fixed axis, 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.
Consider a particle moving in a circle of radius 'r' (which is the perpendicular distance between point 'P' and axis of rotation). Let 
be the linear velocity of the particle at point 'P', its direction is along the tangent drawn at 'P' to the circle. 'P' be the position of particle after an interval of time ∆t. The angle PCP' describes angular displacement dθ or ∆θ in the time interval ∆t.
In pure rotation by all parts of the body having same angular velocity at any instant of time.
In the vector notation 
= 
× 
, the product between and is "Vector Product" (or) "Cross Product".
Vector Product of Two Vectors:
In the study of rotational motion, two important quantities namely moment of force and angular momentum are defined as vector products.
Definition of Vector Product:
A vector product of two vectors a and b is a vector. 
"The vector product of two vectors is defined as a vector having magnitude equal to the product of their magnitudes and sine of the angle between them and direction perpendicular to the plane containing the two vectors."
= ab sin θ 
where 
, 
are two vectors, a, b are their corresponding magnitudes. is unit normal vector.
is perpendicular to the plane containing both and in accordance with right hand thumb rule (or) right hand screw rule.
Right hand thumb rule:
The two vectors and are drawn such that their tails coincide. The right hand palm is stretched and placed perpendicular to the plane containing and . Curl the fingers pointing from a to b, stretched thumb points in the direction of .
Right hand screw rule:
A right hand screw is set with its axis perpendicular to the plane containing vectors and . If screw is rotated in a direction from and through small angle between them, the direction in which the screw advances gives the direction of . If rotation of screw is reversed, its axis advances in opposite direction to .
Properties of Vector Product:
* The cross product does not obey commutative law.
Example: Find the scalar and vector product of two vectors
Sol: Scalar product
Relation between linear acceleration and angular acceleration:
Relation between linear velocity and angular velocity is v = rω
In vector rotation 
radius vector is normal to angular acceleration.
Equations of rotational kinematics:
There exists the analogy between rotational motion and translational motion (which corresponds to linear motion). The kinematical quantities in rotational motion, angular displacement (θ), angular velocity (ω), angular acceleration (α) corresponds to kinematic quantities in linear motion are displacement (x), velocity (v) and acceleration (a). Equations of linear motion with uniform acceleration are
x0 = initial displacement, v0 = initial velocity.
Kinematic equations for rotational motion with uniform acceleration are,
θ0 = initial angular displacement, ω0 = initial angular velocity.
Torque and Angular momentum:
Motion of a rigid body is a combination of rotation and translation. If the body is fixed at one edge, it has only rotational motion. To study about such motion, consider the example of opening (or) closing of a door. Try to rotate the door by applying force at various points on the door. If force is applied exactly, on the hinges, the door cannot move. Here the point of application of force is nearer to axis of rotation. Thus it is difficult to rotate the door. This example suggests that rotating affect is decided not only by the magnitude of applied force but also by the way it is applied. The rotational analogue of force is "moment of force". It is also referred as Torque (or) Couple.
Moment of force:
Turning effect of a force about the axis of rotation or fulcrum is called torque or moment of force is measured as the product of force and its perpendicular distance from the centre of rotation (r).
If a force acts on a single particle at a point P whose position with respect to origin 'O' is given by the position vector 
. The moment of force acting on the particle with respect to the origin 'O' is defined as the vector product 
In figure, r sin θ = OP, θ is angle between r and F.
Dimensions of moment of force and work are same [ML2T−2]. But both are different. S.I. unit of moment of force is N-m. It is a vector and direction is perpendicular to the plane of rotation produced in the body obeying right hand screw rule. It tends to produce rotation in anticlock wise direction torque taken as positive where as it tends to produce rotation in clockwise torque direction is taken as negative.
Couple: A pair of equal and unlike parallel forces acting at different points of a rigid body is called couple.
e.g.: 1) When we use a screw driver, we apply a couple on its head.
2) When we turn a tap, we apply a couple on its head.
3) To spin a ball, a couple required.
Moment of Inertia:
In translatory motion mass is the measure of inertia. In rotational motion the property by virtue of which the body in rotatory motion opposes to change its state is called moment of inertia. If a particle of mass 'm' rotating about an axis at a distance r, the moment of inertia of the particle is given by I = mr2.
Consider a rigid body is imagined to be divided into several small particles of masses m1, m2, m3, ....... mn and located at perpendicular distances r1, r2, ....... rn respectively from axis of rotation. The sum of products of the mass of a particle and square of perpendicular distance about centre of rotation is called Inertia (I).
Unit of I is Kg-m2 in S.I., gm−cm2 in C.G.S. system.
I depends on (1) mass of the body (2) the distribution of mass in the body about axis of rotation (3) shape of the body.
Radius of gyration:
This is an imaginary quantity and is devised to simplify the method of calculating M.I. of the body.
The distance of a point from the axis of rotation at which the entire mass of the body is supposed to be concentrated is called "radius of gyration" of the body about given axis of rotation.
S.I. unit of k is m, cm is in C.G.S. system.
It depends on position of axis of rotation, distribution of mass about axis of rotation.
Moment of Inertia of some regular shaped bodies: Theorem of perpendicular axes:
This theorem is used to find moment of inertia of some regular shaped bodies which are planar. This means theorem applies to flat bodies whose thickness is very small compared to their other dimensions (e.g.: length, breadth or radius).
"The moment of inertia of a "Plane lamina about an axis perpendicular to its plane is equal to the sum of moment of inertia of the plane lamina about an axes perpendicular to each other in its own plane and intersecting each other at a point, where the axis perpendicular to the plane passes".
Consider a planar body, an axis perpendicular to the body through a point 'O' is taken as Z-axis. Two mutually perpendicular axes (X, Y) lying in the plane of the body and concurrent with
Z -axis and passing through 'O'. Ix and Iy are moment of inertia of plane lamina about Ox and Oy. Mathematically Iz = Ix + Iy
Ix = Σ my2, Iy = Σ mx2, Iz = Σ m(OP)2,
Iz = Σ m(x2 + y2)
Iz = Σ m(OP)2 = Σ m(x2 + y2)
= Σ mx2 + Σ my2
Theorem of parallel axes:
This theorem is applicable for a body of any shape. It is used to find moment of inertia of the body about parallel axes through the centre of mass of the body. The moment of inertia of a body about any axis is equal to the sum of moment of inertia of the body about a parallel axis passing through its centre of mass and product of its mass and square of the distance between two parallel axes.
I = I0 + mr2
Consider a rigid body of mass m, 'I' is the moment of inertia of it about any axis, I0 is the moment of inertia of the body about a parallel axis through the centre of mass of the body, r is the distance between two parallel axes.
Moment of inertia of the rigid body about the axis AB is I = Σ m(AP)2
Moment of inertia of the rigid body about the axis 'CD' is I0 = Σ m(CP)2
Extend AC and draw a normal to it. So that it passes through P.
In ∆APQ, (AP)2 = (AQ)2 + (PQ)2
(AP)2 = (AC + CQ)2 + (PQ)2
= (AC)2 + (CQ)2+ 2(AC)(CQ) + (PQ)2
= (AC)2 + [(CQ)2 + (PQ)2] + 2(AC)(CQ)
(AP)2 = (AC)2 + (PC)2 + 2(AC)(CQ)
I = Σ m(AP)2
= Σ m[(AC)2 + (CP)2 + 2(AC)(CQ)]
= Σ m(AC)2 + Σ m(CP)2 + 2 Σm(AC)(CQ)
I = mr2 + I0 + 2 Σm(AC)(CQ) [... mr(CQ) = 0]
i.e., sum of moments of mass about its centre of mass is zero.
I = mr2 + I0
Dynamics of rotational motion:
In rotational motion, moment of inertia and torque play the same role as mass and force in linear motion. In linear motion workdone is given by F dx, in rotational motion about a fixed axis it should be τ dθ. Where dx analogous to dθ, F analogous to τ.
Comparison of Translational and Rotational Motion:
Angular momentum:
Angular momentum is the rotational analogue of linear momentum. It could also be referred to as moment of momentum i.e., the product of momentum of the particle and perpendicular distance of direction of linear momentum from the axis of rotation gives the angular momentum.
Consider a body rotating about an axis XY passing through 'O' with uniform velocity ω. Let 'v' be the linear velocity of particle of mass 'm'.
Moment of momentum (l) = Momentum × Perpendicular distance = mv (r sin θ)
In circular motion, angle between 
and 
is always 90°. Hence the angular momentum (l) = mvr
= mr2ω
In case of rigid body,
It is a vector quantity.
Units: In C.G.S. system, gm − cm2 − s−1
In S.I. system, Kg − m2 − s−1 or Js
Dimensional formula: [ML2T−1]
Relation between Angular Momentum and Torque:
* A diver jumps from a diving board with both legs and hands kept far off from the body. Then inertia of the body is more and angular velocity is less. Afterwards he fold his hands and legs closer to the body. Then moment of inertia decreases and angular velocity increases. Due to this reason diver makes somersaults in air.
* If radius of earth suddenly changes without changing its mass then according to law of conservation angular moment Iω = constant; If radius contracts, inertia decreases and angular velocity increases. As a result duration of the day decreases.
Rolling motion:
The most common motion observed in daily life is rolling motion. It is a combination of rotation and translation. The translational motion of system of particles is the motion of its centre of mass.
Consider a wheel moving along a straight track, the centre of wheel moves forward in pure translation. A point on the rim of wheel, traces a complex curve called cycloid.
In one complete rotation of the centre of mass displaces s = 2ΠR
If Vcm = Rω, wheel is in pure rolling. Velocity of point of contact is zero.
If Vcm > Rω, wheel slips forward, wheel moves greater than 2ΠR in one full rotation.
If Vcm < Rω, wheel slips backward. It moves through a distance less than 2ΠR in one full rotation.
Velocities at different points on rolling body:
Kinetic Energy of Rolling Motion:
Kinetic energy of a rolling body can be separated into kinetic energy of translation and kinetic energy of rotation.
β is a dimensionless quantity, which varies from body to body and axis to axis.
K.E. of rolling body without slipping can be written as
Where IP is moment of inertia of the wheel about the point of contact.
Rolling a body on an inclined plane:
Rolling of a body does not take place on smooth inclined plane, but it take place along rough inclined plane. Consider a rigid round body of mass 'm' and radius 'R' is placed on rough inclined plane making an angle θ with horizontal.
There is no loss of energy due to friction etc. The PE lost by the body during rolling down the inclined plane is equal to KE gained.
Smaller the β, smaller the time taken by the rolling body to reach he bottom of the inclined plane.
βsolid sphere< βdisc< βhallow sphere < βring
If bodies are sliding (β = 0), all bodies reach the bottom of the inclined plane at the same time.
