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MHT-CET : Physics Entrance Exam

MHT - CET : Physics - Rotational motion Page 1

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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

mixi

i = 1

n

mi

i = 1

where xi,
yi and zi are the x, y and z
co
-ordinates, respectively, of the i th particle
and
mi is the mass of the i th particle.

y =

n

miyi

i = 1

n

mi

i = 1

z =

n

mizi

i = 1

n

mi

i = 1

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

 

n

mii

i = 1

M

 

 

=

where M is the total mass of the system
and
i and mi are the position vector and
the mass, respectively, of the
ith 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

M

  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 =

n

mir2i

i = 1


where
mi is the mass of the i th particle and ri is the distance of the ith particle from the axis of rotation.

 

 

SI 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.

 

 

Unit: Metre
Dimensions [M0L1T0]

 

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 =

N

mir2i

i = 1

 

\ K =

 

 

Kinetic Energy of a Rotating Body

 

  • Rotational K.E. of the body = Sum of the K.E. of individual particles.
  • Rotational K.E. of the body

= E =

1

2

m1v21 +

1

2

m2v22 + .

1

2

mnv2n

 

E =

1

2

m1r21 w2+

1

2

m2r2w2 + .

1

2

mnr2w2

(... V = rw)

 

\ E =

1

2

n

mir2i

i = 1

w2

=

1

2

Iw2

( ... I =

n

mir2i)

i = 1

 

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

1

2

Iw2

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

 

 

 

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