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Definition The centre of mass of a body is
defined as the point at which the entire mass of the body may be
assumed to be concentrated in order to study the motion of the body
under the influence of an external force.
Centre of Mass of an 'n' Particle System
The co-ordinates (x, y, z) of the centre of mass of an n particle
system are given by the following relationships.
yi and zi are the x, y and z
co-ordinates, respectively, of the i th
and mi is the
mass of the i th particle.
i = 1
The position vector , of the centre of mass is given
is the total mass of the system
and i and mi are the
position vector and
the mass, respectively, of the ith particle.
Definition A body is said to be rigid if the
distance between any two particles of the body remains constant,
whatever be the applied force. The position vector of the centre of
mass of a rigid body is given by the relation,
the position vector of an
element of the body of mass dm.
of Inertia and it's Physical Significance
In rotational motion, the moment of inertia is the
measure of the rotational inertia. Moment of inertia plays the same
role in rotational motion as that of mass in linear motion.
of Inertia (I)
Definition The moment of inertia (I) of a body
is defined as the sum of the products of the masses of the particles
of the body and the squares of their respective distances from the
axis of rotation.
i.e. I =
where mi is the mass
of the i th particle
is the distance of the ith particle
from the axis of rotation.
Units of I :Kilogram-metre2 (kg-m2)
Dimensions of I:[M1L2T0]
Radius of Gyration (K)
The radius of gyration (K) of a body rotating about any
axis is the distance between the axis of rotation and the point at
which the entire mass of the body can be assumed to be concentrated
so as to give the same moment of inertia as that of the body about
the given axis.[ \ I = MK2 ]
Radius of gyration, K =
where I is the moment of inertia of the body about the
given axis and M is the mass of the body.
Physical Significance: Radius of
gyration of a body about a given axis of rotation is defined as the
distance from the axis of rotation to the point at which whole mass
of the body is supposed to be concentrated so as to produce the same
moment of inertia as that of the body.
If M is the total mass of a body, I is the M.I. about a given axis
and K is the radius of gyration then
I = MK2 =
\ K =
Energy of a Rotating Body
= E =
m2v22 + ….
m2r2w2 + ….
(... V = rw)
\ E =
( ... I =
\ K. E. of rotation of a body = E
where I is the moment of inertia of the
body about a given axis.