System of Particles & Rotational Motion Class 11 Notes Physics Chapter 6

 Page 1


Physics Class XI
110
5.1 Introduction
Rigid body : A rigid body is a body that can rotate with all the parts locked  
 together and without any change in its shape.
5.2 Centre of Mass
Centre of mass of a system is a point that moves as though all the mass were  
concentrated there and all external forces were applied there.
(1) Position vector of centre of mass for n particle system : If a system 
consists of n particles of masses m
1
, m
2
, m
3
 ........ m
n
, whose positions 
vectors are  respectively then position vector of centre 
of mass
If two masses are equal i.e., m
1
 = m
2
, then position vector of centre of  
 mass 
(2) Important points about centre of mass
(i) The position of centre of mass is independent of the co-ordinate 
system chosen.
(ii) The position of centre of mass depends upon the shape of the body 
and distribution of mass.
(iii) In symmetrical bodies in which the distribution of mass is 
homogenous, the centre of mass coincides with the geometrical 
centre or centre of symmetry of the body. Centre of mass of cone 
Page 2


Physics Class XI
110
5.1 Introduction
Rigid body : A rigid body is a body that can rotate with all the parts locked  
 together and without any change in its shape.
5.2 Centre of Mass
Centre of mass of a system is a point that moves as though all the mass were  
concentrated there and all external forces were applied there.
(1) Position vector of centre of mass for n particle system : If a system 
consists of n particles of masses m
1
, m
2
, m
3
 ........ m
n
, whose positions 
vectors are  respectively then position vector of centre 
of mass
If two masses are equal i.e., m
1
 = m
2
, then position vector of centre of  
 mass 
(2) Important points about centre of mass
(i) The position of centre of mass is independent of the co-ordinate 
system chosen.
(ii) The position of centre of mass depends upon the shape of the body 
and distribution of mass.
(iii) In symmetrical bodies in which the distribution of mass is 
homogenous, the centre of mass coincides with the geometrical 
centre or centre of symmetry of the body. Centre of mass of cone 
or pyramid lies on the axis of the cone at point distance  from 
the vertex where h is the height of cone.
(iv) The centre of mass changes its position only under the translatory 
motion. There is no effect of rotatory motion on centre of mass of 
the body.
(v) If the origin is at the centre of mass, then the sum of the moments 
of the masses of the system about the centre of mass is zero i.e., 
.
(vi) If a system of particles of masses m
1
, m
2
, m
3
, ...... move with velocity  
v
1
, v
2
, v
3
, ......... then the velocity of centre of mass
(vii)  If a system of particles of masses m
1
, m
2
, m
3
, ...... move with 
accelerations a
1
, a
2
, a
3
, .......... then the acceleration of centre of 
mass
(viii)  I f  is a position vector of centre of mass of a system then velocity 
of centre of mass  = .
(ix) Acceleration of centre of mass  = .
(x) Force on a rigid body .
(xi) For an isolated system external force on the body is zero 
=
 ?  = constant.
  i.e., centre of mass of an isolated system moves with uniform velocity 
along a straight-line path.
Page 3


Physics Class XI
110
5.1 Introduction
Rigid body : A rigid body is a body that can rotate with all the parts locked  
 together and without any change in its shape.
5.2 Centre of Mass
Centre of mass of a system is a point that moves as though all the mass were  
concentrated there and all external forces were applied there.
(1) Position vector of centre of mass for n particle system : If a system 
consists of n particles of masses m
1
, m
2
, m
3
 ........ m
n
, whose positions 
vectors are  respectively then position vector of centre 
of mass
If two masses are equal i.e., m
1
 = m
2
, then position vector of centre of  
 mass 
(2) Important points about centre of mass
(i) The position of centre of mass is independent of the co-ordinate 
system chosen.
(ii) The position of centre of mass depends upon the shape of the body 
and distribution of mass.
(iii) In symmetrical bodies in which the distribution of mass is 
homogenous, the centre of mass coincides with the geometrical 
centre or centre of symmetry of the body. Centre of mass of cone 
or pyramid lies on the axis of the cone at point distance  from 
the vertex where h is the height of cone.
(iv) The centre of mass changes its position only under the translatory 
motion. There is no effect of rotatory motion on centre of mass of 
the body.
(v) If the origin is at the centre of mass, then the sum of the moments 
of the masses of the system about the centre of mass is zero i.e., 
.
(vi) If a system of particles of masses m
1
, m
2
, m
3
, ...... move with velocity  
v
1
, v
2
, v
3
, ......... then the velocity of centre of mass
(vii)  If a system of particles of masses m
1
, m
2
, m
3
, ...... move with 
accelerations a
1
, a
2
, a
3
, .......... then the acceleration of centre of 
mass
(viii)  I f  is a position vector of centre of mass of a system then velocity 
of centre of mass  = .
(ix) Acceleration of centre of mass  = .
(x) Force on a rigid body .
(xi) For an isolated system external force on the body is zero 
=
 ?  = constant.
  i.e., centre of mass of an isolated system moves with uniform velocity 
along a straight-line path.
5.6 Equations of Linear Motion and Rotational Motion
Rotational Motion
If angular acceleration is 0, ? = constant and ? = ?t
If angular acceleration a = constant then
(i) 
(ii) 
(iii) ?
2
 = ?
1
 + at
(iv) 
(v) 
(vi) 
If acceleration is not constant, the above equation will not be applicable. In this case
(i) 
(ii) 
(iii) ?d ? = ad?
5.7 Moment of Inertia
Moment of inertia plays the same role in rotational motion as mass plays in  
linear motion. It is the property of a body due to which it opposes any change 
in its state of rest or of uniform rotation.
(1) Moment of inertia of a particle I = mr
2
; where r is the perpendicular 
distance of particle from rotational axis.
(2) Moment of inertia of a body made up of number of particles (discrete 
distribution)
I = 
(3) Moment of inertia of a continuous distribution of mass, dI = dmr
2
 i.e.,
I = 
(4) Dimension : [ML
2
T
0
]
Page 4


Physics Class XI
110
5.1 Introduction
Rigid body : A rigid body is a body that can rotate with all the parts locked  
 together and without any change in its shape.
5.2 Centre of Mass
Centre of mass of a system is a point that moves as though all the mass were  
concentrated there and all external forces were applied there.
(1) Position vector of centre of mass for n particle system : If a system 
consists of n particles of masses m
1
, m
2
, m
3
 ........ m
n
, whose positions 
vectors are  respectively then position vector of centre 
of mass
If two masses are equal i.e., m
1
 = m
2
, then position vector of centre of  
 mass 
(2) Important points about centre of mass
(i) The position of centre of mass is independent of the co-ordinate 
system chosen.
(ii) The position of centre of mass depends upon the shape of the body 
and distribution of mass.
(iii) In symmetrical bodies in which the distribution of mass is 
homogenous, the centre of mass coincides with the geometrical 
centre or centre of symmetry of the body. Centre of mass of cone 
or pyramid lies on the axis of the cone at point distance  from 
the vertex where h is the height of cone.
(iv) The centre of mass changes its position only under the translatory 
motion. There is no effect of rotatory motion on centre of mass of 
the body.
(v) If the origin is at the centre of mass, then the sum of the moments 
of the masses of the system about the centre of mass is zero i.e., 
.
(vi) If a system of particles of masses m
1
, m
2
, m
3
, ...... move with velocity  
v
1
, v
2
, v
3
, ......... then the velocity of centre of mass
(vii)  If a system of particles of masses m
1
, m
2
, m
3
, ...... move with 
accelerations a
1
, a
2
, a
3
, .......... then the acceleration of centre of 
mass
(viii)  I f  is a position vector of centre of mass of a system then velocity 
of centre of mass  = .
(ix) Acceleration of centre of mass  = .
(x) Force on a rigid body .
(xi) For an isolated system external force on the body is zero 
=
 ?  = constant.
  i.e., centre of mass of an isolated system moves with uniform velocity 
along a straight-line path.
5.6 Equations of Linear Motion and Rotational Motion
Rotational Motion
If angular acceleration is 0, ? = constant and ? = ?t
If angular acceleration a = constant then
(i) 
(ii) 
(iii) ?
2
 = ?
1
 + at
(iv) 
(v) 
(vi) 
If acceleration is not constant, the above equation will not be applicable. In this case
(i) 
(ii) 
(iii) ?d ? = ad?
5.7 Moment of Inertia
Moment of inertia plays the same role in rotational motion as mass plays in  
linear motion. It is the property of a body due to which it opposes any change 
in its state of rest or of uniform rotation.
(1) Moment of inertia of a particle I = mr
2
; where r is the perpendicular 
distance of particle from rotational axis.
(2) Moment of inertia of a body made up of number of particles (discrete 
distribution)
I = 
(3) Moment of inertia of a continuous distribution of mass, dI = dmr
2
 i.e.,
I = 
(4) Dimension : [ML
2
T
0
]
(5) S.I. unit : kgm
2
.
(6) Moment of inertia depends on mass, distribution of mass and on the 
position of axis of rotation.
(7) Moment of inertia is a tensor quantity.
5.8 Radius of Gyration
Radius of gyration of a body about a given axis is the perpendicular distance 
of a point from the axis, where if whole mass of the body were concentrated, 
the body shall have the same moment of inertia as it has with the actual 
distribution of mass.
When square of radius of gyration is multiplied with the mass of the body 
gives the moment of inertia of the body about the given axis.
I = Mk
2
 or k = 
Here k is called radius of gyration.
k = 
Note :
• For a given body inertia is constant whereas moment of inertia is variable.
5.9 Theorem of Parallel Axes
Moment of inertia of a body about a given axis 
I is equal to the sum of moment of inertia of the 
body about an axis parallel to given axis and 
passing through centre of mass of the body I
g
 
and Ma
2
 where M is the mass of the body and 
a is the perpendicular distance between the 
two axes.
I = I
g
 + Ma
2
5.10 Theorem of Perpendicular Axes
According to this theorem the sum of moment 
of inertia of a plane lamina about two mutually 
perpendicular axes lying in its plane is equal to 
its moment of inertia about an axis perpendicular 
to the plane of lamina and passing through the 
point of intersection of first two axis.
I
z
 = I
x
 + I
y
Ig
Page 5


Physics Class XI
110
5.1 Introduction
Rigid body : A rigid body is a body that can rotate with all the parts locked  
 together and without any change in its shape.
5.2 Centre of Mass
Centre of mass of a system is a point that moves as though all the mass were  
concentrated there and all external forces were applied there.
(1) Position vector of centre of mass for n particle system : If a system 
consists of n particles of masses m
1
, m
2
, m
3
 ........ m
n
, whose positions 
vectors are  respectively then position vector of centre 
of mass
If two masses are equal i.e., m
1
 = m
2
, then position vector of centre of  
 mass 
(2) Important points about centre of mass
(i) The position of centre of mass is independent of the co-ordinate 
system chosen.
(ii) The position of centre of mass depends upon the shape of the body 
and distribution of mass.
(iii) In symmetrical bodies in which the distribution of mass is 
homogenous, the centre of mass coincides with the geometrical 
centre or centre of symmetry of the body. Centre of mass of cone 
or pyramid lies on the axis of the cone at point distance  from 
the vertex where h is the height of cone.
(iv) The centre of mass changes its position only under the translatory 
motion. There is no effect of rotatory motion on centre of mass of 
the body.
(v) If the origin is at the centre of mass, then the sum of the moments 
of the masses of the system about the centre of mass is zero i.e., 
.
(vi) If a system of particles of masses m
1
, m
2
, m
3
, ...... move with velocity  
v
1
, v
2
, v
3
, ......... then the velocity of centre of mass
(vii)  If a system of particles of masses m
1
, m
2
, m
3
, ...... move with 
accelerations a
1
, a
2
, a
3
, .......... then the acceleration of centre of 
mass
(viii)  I f  is a position vector of centre of mass of a system then velocity 
of centre of mass  = .
(ix) Acceleration of centre of mass  = .
(x) Force on a rigid body .
(xi) For an isolated system external force on the body is zero 
=
 ?  = constant.
  i.e., centre of mass of an isolated system moves with uniform velocity 
along a straight-line path.
5.6 Equations of Linear Motion and Rotational Motion
Rotational Motion
If angular acceleration is 0, ? = constant and ? = ?t
If angular acceleration a = constant then
(i) 
(ii) 
(iii) ?
2
 = ?
1
 + at
(iv) 
(v) 
(vi) 
If acceleration is not constant, the above equation will not be applicable. In this case
(i) 
(ii) 
(iii) ?d ? = ad?
5.7 Moment of Inertia
Moment of inertia plays the same role in rotational motion as mass plays in  
linear motion. It is the property of a body due to which it opposes any change 
in its state of rest or of uniform rotation.
(1) Moment of inertia of a particle I = mr
2
; where r is the perpendicular 
distance of particle from rotational axis.
(2) Moment of inertia of a body made up of number of particles (discrete 
distribution)
I = 
(3) Moment of inertia of a continuous distribution of mass, dI = dmr
2
 i.e.,
I = 
(4) Dimension : [ML
2
T
0
]
(5) S.I. unit : kgm
2
.
(6) Moment of inertia depends on mass, distribution of mass and on the 
position of axis of rotation.
(7) Moment of inertia is a tensor quantity.
5.8 Radius of Gyration
Radius of gyration of a body about a given axis is the perpendicular distance 
of a point from the axis, where if whole mass of the body were concentrated, 
the body shall have the same moment of inertia as it has with the actual 
distribution of mass.
When square of radius of gyration is multiplied with the mass of the body 
gives the moment of inertia of the body about the given axis.
I = Mk
2
 or k = 
Here k is called radius of gyration.
k = 
Note :
• For a given body inertia is constant whereas moment of inertia is variable.
5.9 Theorem of Parallel Axes
Moment of inertia of a body about a given axis 
I is equal to the sum of moment of inertia of the 
body about an axis parallel to given axis and 
passing through centre of mass of the body I
g
 
and Ma
2
 where M is the mass of the body and 
a is the perpendicular distance between the 
two axes.
I = I
g
 + Ma
2
5.10 Theorem of Perpendicular Axes
According to this theorem the sum of moment 
of inertia of a plane lamina about two mutually 
perpendicular axes lying in its plane is equal to 
its moment of inertia about an axis perpendicular 
to the plane of lamina and passing through the 
point of intersection of first two axis.
I
z
 = I
x
 + I
y
Ig
114
Note : 
• In case of symmetrical two-dimensional bodies as moment of inertia for
all axes passing through the centre of mass and in the plane of body will
be same so the two axes in the plane of body need not be perpendicular
to each other.
5.12 Analogy between Translatory Motion and Rotational 
Motion
Translatory motion Rotatory motion
Mass (m) Moment of Inertia (I)
Linear P = mv Angular L = l ?
Momentum P = Momentum L = 
Force F = ma Torque t = I a
Kinetic energy E = E = 
E = E = 
5.13 Moment of Inertia of Some Standard Bodies and Different 
Axes
Body Axis of Rotation Figure Moment K K
2
/R
2
of inertia
 Ring About an axis
 (Cylindrical Passing through
Shell) C.G. and MR
2
R 1
perpendicular to
its plane
Ring About its diameter