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Revision Notes: Rotational Motion
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Page 1
Revision notes
Physics
System of particles and rotational motion
Rigid body:
? A rigid body is one in which the distances between different particles of
the body do not change even though forces are acting on them.
? Axis of rotation: The line along which the body is fixed is known as the
axis of rotation. Examples for rotation about an axis are a ceiling fan and a
merry-go-round etc.
? In pure rotation of a rigid body about a fixed axis, every particle of the rigid
body moves in a circle which lies in a plane perpendicular to the axis and
has its centre on the axis. Every point in the rotating rigid body has the
same angular velocity at any time.
Image: Linkage of a rigid body
Centre of mass:
? The centre of mass of a body is a point where the entire mass of the body
can be supposed to be concentrated. The nature of motion executed by the
body shall remain unaffected if all the forces acting on the body were
applied directly at this point.
? For a system of two particles of masses
1
m and
2
m having their position
vector has
1
r and
2
r respectively, with respect to origin of the coordinate
system, the position vector
CM
R of the centre of mass is given by
1 1 2 2
12
CM
m r m r
R
mm
?
?
?
Page 2
Revision notes
Physics
System of particles and rotational motion
Rigid body:
? A rigid body is one in which the distances between different particles of
the body do not change even though forces are acting on them.
? Axis of rotation: The line along which the body is fixed is known as the
axis of rotation. Examples for rotation about an axis are a ceiling fan and a
merry-go-round etc.
? In pure rotation of a rigid body about a fixed axis, every particle of the rigid
body moves in a circle which lies in a plane perpendicular to the axis and
has its centre on the axis. Every point in the rotating rigid body has the
same angular velocity at any time.
Image: Linkage of a rigid body
Centre of mass:
? The centre of mass of a body is a point where the entire mass of the body
can be supposed to be concentrated. The nature of motion executed by the
body shall remain unaffected if all the forces acting on the body were
applied directly at this point.
? For a system of two particles of masses
1
m and
2
m having their position
vector has
1
r and
2
r respectively, with respect to origin of the coordinate
system, the position vector
CM
R of the centre of mass is given by
1 1 2 2
12
CM
m r m r
R
mm
?
?
?
If
12
m m m ?? , then
12
2
CM
rr
R
?
?
Thus, the centre of a mass of two equal masses lies exactly at the centre of
the line joining the two masses.
? For a system of N-particles of masses
1 2 3
, , ...
N
m m m m having their
position vectors as
1 2 3
, , ,...,
N
r r r r respectively, with respect to the origin of
the coordinate system, the position vector
CM
R of the centre of mass is
given by
1 1 2 2 1 1
12
1
...
...
NN
i i i i
N N i i
CM
N
N
i
i
m r m r
m r m r m r
R
m m m M
m
??
?
? ? ?
? ? ?
??
??
?
? The coordinates of centre of mass for a 3-D object is given as:
11
1
NN
i i i i
ii
CM
N
i
i
m x m x
X
M
m
??
?
??
??
?
11
1
NN
i i i i
ii
CM
N
i
i
m y m y
Y
M
m
??
?
??
??
?
11
1
NN
i i i i
ii
CM
N
i
i
m z m z
Z
M
m
??
?
??
??
?
? For a continuous distribution of mass, the coordinates of centre of mass are
given by
1
;
CM
X xdm
M
?
?
1
;
CM
Y ydm
M
?
?
1
CM
Z zdm
M
?
?
? Velocity of centre of mass
1 1 2 2 1 1
12
1
...
...
NN
i i i i
N N i i
CM
N
N
i
i
m v m v
m v m v m v
v
m m m M
m
??
?
? ? ?
? ? ?
??
??
?
? Acceleration of centre of mass
Page 3
Revision notes
Physics
System of particles and rotational motion
Rigid body:
? A rigid body is one in which the distances between different particles of
the body do not change even though forces are acting on them.
? Axis of rotation: The line along which the body is fixed is known as the
axis of rotation. Examples for rotation about an axis are a ceiling fan and a
merry-go-round etc.
? In pure rotation of a rigid body about a fixed axis, every particle of the rigid
body moves in a circle which lies in a plane perpendicular to the axis and
has its centre on the axis. Every point in the rotating rigid body has the
same angular velocity at any time.
Image: Linkage of a rigid body
Centre of mass:
? The centre of mass of a body is a point where the entire mass of the body
can be supposed to be concentrated. The nature of motion executed by the
body shall remain unaffected if all the forces acting on the body were
applied directly at this point.
? For a system of two particles of masses
1
m and
2
m having their position
vector has
1
r and
2
r respectively, with respect to origin of the coordinate
system, the position vector
CM
R of the centre of mass is given by
1 1 2 2
12
CM
m r m r
R
mm
?
?
?
If
12
m m m ?? , then
12
2
CM
rr
R
?
?
Thus, the centre of a mass of two equal masses lies exactly at the centre of
the line joining the two masses.
? For a system of N-particles of masses
1 2 3
, , ...
N
m m m m having their
position vectors as
1 2 3
, , ,...,
N
r r r r respectively, with respect to the origin of
the coordinate system, the position vector
CM
R of the centre of mass is
given by
1 1 2 2 1 1
12
1
...
...
NN
i i i i
N N i i
CM
N
N
i
i
m r m r
m r m r m r
R
m m m M
m
??
?
? ? ?
? ? ?
??
??
?
? The coordinates of centre of mass for a 3-D object is given as:
11
1
NN
i i i i
ii
CM
N
i
i
m x m x
X
M
m
??
?
??
??
?
11
1
NN
i i i i
ii
CM
N
i
i
m y m y
Y
M
m
??
?
??
??
?
11
1
NN
i i i i
ii
CM
N
i
i
m z m z
Z
M
m
??
?
??
??
?
? For a continuous distribution of mass, the coordinates of centre of mass are
given by
1
;
CM
X xdm
M
?
?
1
;
CM
Y ydm
M
?
?
1
CM
Z zdm
M
?
?
? Velocity of centre of mass
1 1 2 2 1 1
12
1
...
...
NN
i i i i
N N i i
CM
N
N
i
i
m v m v
m v m v m v
v
m m m M
m
??
?
? ? ?
? ? ?
??
??
?
? Acceleration of centre of mass
1 1 2 2 1 1
12
1
...
...
NN
i i i i
N N i i
CM
N
N
i
i
m a m a
m a m a m a
a
m m m M
m
??
?
? ? ?
? ? ?
??
??
?
Angular velocity and acceleration:
? Angular velocity: It is defined as the time rate of change of angular
displacement and is given by,
d
dt
?
? ?
? Angular velocity is directed along the axis of rotation. Angular velocity is
a vector quantity. Its SI unit is rad/s and its dimensional formula is
0 0 1
M L T
?
??
??
.
? Relationship between linear velocity and angular velocity
The linear velocity of a particle of a rigid body rotating about a fixed axis
is given by,
vr ? ??
where r is the position vector of the particle with respect to an origin
along the fixed axis.
? As in pure translational motion, all body particles have the same linear
velocity at any instant. Similarly, in pure rotational motion, all body
particles have the same angular velocity at any instant.
? Angular acceleration: It is defined as the time rate of change of angular
velocity, and it is given by
d
dt
?
? ?
? Angular acceleration is a vector quantity. Its SI unit is
2
rads
?
and its
dimensional formula is
0 0 2
M L T
?
??
??
.
Equation of rotational motion:
After a brief introduction to angular velocity and angular acceleration, let us see
how they are related to the kinematic equations.
For a certain initial angular velocity, final angular velocity ( ,
o
?? ) with time t,
the kinematic equations of rotational motion is given as:
Page 4
Revision notes
Physics
System of particles and rotational motion
Rigid body:
? A rigid body is one in which the distances between different particles of
the body do not change even though forces are acting on them.
? Axis of rotation: The line along which the body is fixed is known as the
axis of rotation. Examples for rotation about an axis are a ceiling fan and a
merry-go-round etc.
? In pure rotation of a rigid body about a fixed axis, every particle of the rigid
body moves in a circle which lies in a plane perpendicular to the axis and
has its centre on the axis. Every point in the rotating rigid body has the
same angular velocity at any time.
Image: Linkage of a rigid body
Centre of mass:
? The centre of mass of a body is a point where the entire mass of the body
can be supposed to be concentrated. The nature of motion executed by the
body shall remain unaffected if all the forces acting on the body were
applied directly at this point.
? For a system of two particles of masses
1
m and
2
m having their position
vector has
1
r and
2
r respectively, with respect to origin of the coordinate
system, the position vector
CM
R of the centre of mass is given by
1 1 2 2
12
CM
m r m r
R
mm
?
?
?
If
12
m m m ?? , then
12
2
CM
rr
R
?
?
Thus, the centre of a mass of two equal masses lies exactly at the centre of
the line joining the two masses.
? For a system of N-particles of masses
1 2 3
, , ...
N
m m m m having their
position vectors as
1 2 3
, , ,...,
N
r r r r respectively, with respect to the origin of
the coordinate system, the position vector
CM
R of the centre of mass is
given by
1 1 2 2 1 1
12
1
...
...
NN
i i i i
N N i i
CM
N
N
i
i
m r m r
m r m r m r
R
m m m M
m
??
?
? ? ?
? ? ?
??
??
?
? The coordinates of centre of mass for a 3-D object is given as:
11
1
NN
i i i i
ii
CM
N
i
i
m x m x
X
M
m
??
?
??
??
?
11
1
NN
i i i i
ii
CM
N
i
i
m y m y
Y
M
m
??
?
??
??
?
11
1
NN
i i i i
ii
CM
N
i
i
m z m z
Z
M
m
??
?
??
??
?
? For a continuous distribution of mass, the coordinates of centre of mass are
given by
1
;
CM
X xdm
M
?
?
1
;
CM
Y ydm
M
?
?
1
CM
Z zdm
M
?
?
? Velocity of centre of mass
1 1 2 2 1 1
12
1
...
...
NN
i i i i
N N i i
CM
N
N
i
i
m v m v
m v m v m v
v
m m m M
m
??
?
? ? ?
? ? ?
??
??
?
? Acceleration of centre of mass
1 1 2 2 1 1
12
1
...
...
NN
i i i i
N N i i
CM
N
N
i
i
m a m a
m a m a m a
a
m m m M
m
??
?
? ? ?
? ? ?
??
??
?
Angular velocity and acceleration:
? Angular velocity: It is defined as the time rate of change of angular
displacement and is given by,
d
dt
?
? ?
? Angular velocity is directed along the axis of rotation. Angular velocity is
a vector quantity. Its SI unit is rad/s and its dimensional formula is
0 0 1
M L T
?
??
??
.
? Relationship between linear velocity and angular velocity
The linear velocity of a particle of a rigid body rotating about a fixed axis
is given by,
vr ? ??
where r is the position vector of the particle with respect to an origin
along the fixed axis.
? As in pure translational motion, all body particles have the same linear
velocity at any instant. Similarly, in pure rotational motion, all body
particles have the same angular velocity at any instant.
? Angular acceleration: It is defined as the time rate of change of angular
velocity, and it is given by
d
dt
?
? ?
? Angular acceleration is a vector quantity. Its SI unit is
2
rads
?
and its
dimensional formula is
0 0 2
M L T
?
??
??
.
Equation of rotational motion:
After a brief introduction to angular velocity and angular acceleration, let us see
how they are related to the kinematic equations.
For a certain initial angular velocity, final angular velocity ( ,
o
?? ) with time t,
the kinematic equations of rotational motion is given as:
00
t ? ? ? ??
2
0
1
2
tt ? ? ? ??
22
0
2 ? ? ? ? ??
These equations are valid for uniform angular acceleration.
Moment of inertia:
? Moment of inertia of a rigid body about a given axis of rotation is defined
as the sum of the product of masses of all the particles of the body and the
square of their respective perpendicular distances from the axis of rotation.
It is denoted by symbol I and is given by,
2
1
N
ii
i
I m r
?
?
?
? Moment of inertia is a scalar quantity. Its SI unit is
2
kgm and its
dimensional formula is
1 2 0
M L T
??
??
. It depends upon
o Position of the axis of rotation
o Orientation of the axis of addition
o Shape of the body
o Size of the body
o Distribution of mass of the body about the axis of rotation.
? Radius of gyration: It is defined as the distance from the axis of rotation
at which, if the whole mass of the body were concentrated, the moment of
inertia of the body would be the same as the actual distribution of the mass
of the body. It is denoted by the symbol K.
? Radius of gyration of a body about an axis of rotation may also be defined
as the root mean square distance of the particles from the axis of rotation.
i.e.,
2 2 2
12
...
N
r r r
K
N
? ? ?
?
? The moment of inertia of a body about a given axis is equal to the product
of the mass of the body and square of its radius of gyration about that axis.
i.e.,
2
I MK ? .
? The SI unit of radius of gyration is metre and its dimensional formula is
0 1 0
M LT
??
??
.
Page 5
Revision notes
Physics
System of particles and rotational motion
Rigid body:
? A rigid body is one in which the distances between different particles of
the body do not change even though forces are acting on them.
? Axis of rotation: The line along which the body is fixed is known as the
axis of rotation. Examples for rotation about an axis are a ceiling fan and a
merry-go-round etc.
? In pure rotation of a rigid body about a fixed axis, every particle of the rigid
body moves in a circle which lies in a plane perpendicular to the axis and
has its centre on the axis. Every point in the rotating rigid body has the
same angular velocity at any time.
Image: Linkage of a rigid body
Centre of mass:
? The centre of mass of a body is a point where the entire mass of the body
can be supposed to be concentrated. The nature of motion executed by the
body shall remain unaffected if all the forces acting on the body were
applied directly at this point.
? For a system of two particles of masses
1
m and
2
m having their position
vector has
1
r and
2
r respectively, with respect to origin of the coordinate
system, the position vector
CM
R of the centre of mass is given by
1 1 2 2
12
CM
m r m r
R
mm
?
?
?
If
12
m m m ?? , then
12
2
CM
rr
R
?
?
Thus, the centre of a mass of two equal masses lies exactly at the centre of
the line joining the two masses.
? For a system of N-particles of masses
1 2 3
, , ...
N
m m m m having their
position vectors as
1 2 3
, , ,...,
N
r r r r respectively, with respect to the origin of
the coordinate system, the position vector
CM
R of the centre of mass is
given by
1 1 2 2 1 1
12
1
...
...
NN
i i i i
N N i i
CM
N
N
i
i
m r m r
m r m r m r
R
m m m M
m
??
?
? ? ?
? ? ?
??
??
?
? The coordinates of centre of mass for a 3-D object is given as:
11
1
NN
i i i i
ii
CM
N
i
i
m x m x
X
M
m
??
?
??
??
?
11
1
NN
i i i i
ii
CM
N
i
i
m y m y
Y
M
m
??
?
??
??
?
11
1
NN
i i i i
ii
CM
N
i
i
m z m z
Z
M
m
??
?
??
??
?
? For a continuous distribution of mass, the coordinates of centre of mass are
given by
1
;
CM
X xdm
M
?
?
1
;
CM
Y ydm
M
?
?
1
CM
Z zdm
M
?
?
? Velocity of centre of mass
1 1 2 2 1 1
12
1
...
...
NN
i i i i
N N i i
CM
N
N
i
i
m v m v
m v m v m v
v
m m m M
m
??
?
? ? ?
? ? ?
??
??
?
? Acceleration of centre of mass
1 1 2 2 1 1
12
1
...
...
NN
i i i i
N N i i
CM
N
N
i
i
m a m a
m a m a m a
a
m m m M
m
??
?
? ? ?
? ? ?
??
??
?
Angular velocity and acceleration:
? Angular velocity: It is defined as the time rate of change of angular
displacement and is given by,
d
dt
?
? ?
? Angular velocity is directed along the axis of rotation. Angular velocity is
a vector quantity. Its SI unit is rad/s and its dimensional formula is
0 0 1
M L T
?
??
??
.
? Relationship between linear velocity and angular velocity
The linear velocity of a particle of a rigid body rotating about a fixed axis
is given by,
vr ? ??
where r is the position vector of the particle with respect to an origin
along the fixed axis.
? As in pure translational motion, all body particles have the same linear
velocity at any instant. Similarly, in pure rotational motion, all body
particles have the same angular velocity at any instant.
? Angular acceleration: It is defined as the time rate of change of angular
velocity, and it is given by
d
dt
?
? ?
? Angular acceleration is a vector quantity. Its SI unit is
2
rads
?
and its
dimensional formula is
0 0 2
M L T
?
??
??
.
Equation of rotational motion:
After a brief introduction to angular velocity and angular acceleration, let us see
how they are related to the kinematic equations.
For a certain initial angular velocity, final angular velocity ( ,
o
?? ) with time t,
the kinematic equations of rotational motion is given as:
00
t ? ? ? ??
2
0
1
2
tt ? ? ? ??
22
0
2 ? ? ? ? ??
These equations are valid for uniform angular acceleration.
Moment of inertia:
? Moment of inertia of a rigid body about a given axis of rotation is defined
as the sum of the product of masses of all the particles of the body and the
square of their respective perpendicular distances from the axis of rotation.
It is denoted by symbol I and is given by,
2
1
N
ii
i
I m r
?
?
?
? Moment of inertia is a scalar quantity. Its SI unit is
2
kgm and its
dimensional formula is
1 2 0
M L T
??
??
. It depends upon
o Position of the axis of rotation
o Orientation of the axis of addition
o Shape of the body
o Size of the body
o Distribution of mass of the body about the axis of rotation.
? Radius of gyration: It is defined as the distance from the axis of rotation
at which, if the whole mass of the body were concentrated, the moment of
inertia of the body would be the same as the actual distribution of the mass
of the body. It is denoted by the symbol K.
? Radius of gyration of a body about an axis of rotation may also be defined
as the root mean square distance of the particles from the axis of rotation.
i.e.,
2 2 2
12
...
N
r r r
K
N
? ? ?
?
? The moment of inertia of a body about a given axis is equal to the product
of the mass of the body and square of its radius of gyration about that axis.
i.e.,
2
I MK ? .
? The SI unit of radius of gyration is metre and its dimensional formula is
0 1 0
M LT
??
??
.
? Theorem of perpendicular axes: The moment of inertia of a planar
lamina about an axis perpendicular to its plane is equal to the sum of its
moments of inertia about two perpendicular axis concurrent with a
perpendicular axis and lying in the plane of the body.
Image: Theorem of perpendicular axes
z x y
I I I ??
Where x and y are two perpendicular axes in the plane, and the z-axis is
perpendicular to its plane.
? Theorem of parallel axes: The moment of inertia of a body about any
axis is equal to the sum of the moment of inertia of the body about a parallel
axis passing through its centre of mass and the product of its mass and the
square of the distance between the two parallel axis.
Image: Parallel axis theorem
2
CM
I I Md ??
Where
CM
I is the moment of inertia of the body about an axis (z) passing
through the centre of mass, and d is the perpendicular distance between
two parallel axes.
Read MoreFAQs on Revision Notes: Rotational Motion
| 1. What's the difference between angular velocity and angular acceleration in rotational motion? |  |
Ans. Angular velocity measures how fast an object rotates around an axis, while angular acceleration describes the rate of change of that angular velocity. Both are vector quantities; angular velocity (ω) is measured in rad/s, whereas angular acceleration (α) is measured in rad/s². Understanding this distinction helps solve NEET problems involving spinning objects and changing rotational speeds.
| 2. How do I calculate moment of inertia for different shapes and why does it matter? | |
Ans. Moment of inertia represents an object's resistance to rotational acceleration, calculated differently for each shape-solid sphere, hollow sphere, disc, rod, and ring all have unique formulas. The parallel axis theorem and perpendicular axis theorem simplify calculations for complex geometries. Knowing the correct moment of inertia is essential for predicting rotational motion behaviour in NEET questions.
| 3. Why is torque important in rotational dynamics, and how does it relate to angular motion? | |
Ans. Torque (τ) is the rotational equivalent of force; it causes angular acceleration according to τ = Iα. Torque depends on both force magnitude and perpendicular distance from the axis of rotation. Mastering this relationship between torque and angular acceleration is crucial for solving equilibrium and rotational kinematics problems in exams.
| 4. What's the connection between linear and angular quantities, and when should I use each? | |
Ans. Linear and angular quantities are related through the radius: v = ωr, a = αr, and F = τ/r. Use angular quantities when analysing rotation; switch to linear for motion at specific points on the rotating object. Rolling motion problems require both-they're the most common exam traps where students confuse these conversions.
| 5. How do I solve problems involving conservation of angular momentum and rotational collisions? | |
Ans. Angular momentum L = Iω remains constant when external torque is zero. In collisions or interactions between rotating objects, apply L(initial) = L(final) to find final angular velocities. Refer to mind maps and MCQ tests on EduRev to practise conservation scenarios-they appear frequently in NEET and help clarify energy loss versus momentum retention.
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