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Centre of Mass

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.

x =

n

m_{i}x_{i}

å

i = 1

m_{i}

where x_{i}, y_{i} and z_{i} are the x, y and z co-ordinates, respectively, of the i ^{th} particle and m_{i} is the mass of the i ^{th} particle.

y =

m_{i}y_{i}

z =

m_{i}z_{i}

The position vector , of the centre of mass is given by

m_{i}_{i}

M

=

where M is the total mass of the system and i and m_{i} are the position vector and the mass, respectively, of the i^{th} particle.

Rigid Body

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,

1

ò dm

where is the position vector of an element of the body of mass dm.

Moment 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.

Moment 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 =

m_{i}r^{2}_{i}

where m_{i} is the mass of the i th particle and r_{i} is the distance of the i^{th} particle from the axis of rotation.

SI Units of I :Kilogram-metre^{2} (kg-m^{2}) Dimensions of I:[M^{1}L^{2}T^{0}]

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 = MK^{2} ]

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.

Unit: Metre Dimensions [M^{0}L^{1}T^{0}]

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 = MK^{2} =

N

\ K =

Kinetic Energy of a Rotating Body

= E =

2

m_{1}v^{2}_{1} +

m_{2}v^{2}_{2} + ….

m_{n}v^{2}_{n}

E =

m_{1}r^{2}_{1} w^{2}+

m_{2}r^{2}w^{2} + ….

m_{n}r^{2}w^{2}

(^{.}.^{.} V = rw)

\ E =

w^{2}

Iw^{2}

^{( .}.^{.} I =

m_{i}r^{2}_{i})

\ K. E. of rotation of a body = E =

where I is the moment of inertia of the body about a given axis.