Chapter 4
Rotational Motion
Dr Manjunatha S
College of Computer and Information Sciences
Wind mill- produces electricity
The rotation of a rigid object about a fixed axis
Rigid Body
Definition:
The distance between any two constituent
particles remains fixed under external force is
called rigid body.
• Real objects have mass at points other than
the center of mass.
• Each point in an object can be measured from
an origin at the center of mass.
• If the positions are fixed compared to the
center of mass it is a rigid body.
ri
Translation and Rotation
• The motion of a rigid
body includes the
motion of its center of
mass.
• This is translational
vCM
motion
• A rigid body can also
move while its center
of mass is fixed.
• This is rotational
motion.

1. Translational motion:
Linear motion in such a way that every particle of
the rigid body has same velocity.
2. Rotational motion:
A rigid body rotate such that every particle of the
body moves in a circle with its center on axis of
rotation.
Definition:
Definition:
The angular motion between two position of an object
in time interval is called angular displacement.
Δθ = θf – θi
SI unit is rad
Angular velocity (ω)
Definition:
The rate of change of angular displacement of a rigid
body is called angular velocity.
  f   i
Angulardisplacement ,  

t
t f  ti
SI Unit: rad/s
Relation between velocity and angular
velocity:
s r

v
t

t
v  r
 rw
w
t
Angular acceleration ()
Definition:
The rate of change of angular velocity of a rigid body
is called angular acceleration.

 f  i
t
SI unit: rad/s²
 Every point on rigid object has same  and 
  is positive or negative
Example:1
A race car engine at a maximum rate of 12,000 rpm. (revolutions per
minute).
a) What is the angular velocity in radians per second.
b) If helicopter blades were attached to the crankshaft while it turns
with this angular velocity, what is the maximum radius of a blade
such that the speed of the blade tips stays below the speed of sound.
Given: The speed of sound is 343 m/s.
Solution:
Conceptual Question
You and a friend are playing on the merry-go-round. You stand at
the outer edge of the merry-go-round and your friend stands
halfway between the outer edge and the center. Assume the
rotation rate of the merry-go-round is constant.
Who has the greatest angular velocity?
1. You do
2. Your friend does
3. Same
CORRECT
Since, the angular displacement is the same in both cases.
Relation between acceleration and angular
acceleration:
𝑣
𝑡
Acceleration, a = =
𝑟𝑤
𝑡
= r𝛼; 𝒂 = r𝜶
 velocity is +ve, body is rotating in the direction of
increasing θ (ie., anticlockwise)
 ω is -ve, decreasing θ ( ie., clockwise).
Rotational/Linear Correspondence
Linear Motion
Constant a
Constant 
Rotational Motion
Example 2
A pottery wheel is accelerated uniformly from rest to a
rotation speed of 600 rpm in 30 seconds.
a) What was the angular acceleration? (in rad/s2).
b) How many revolutions did the wheel undergo during
that time?
Example: 3
A wheel rotates with a constant angular acceleration of 3.5
rad/s2.
A) If the angular speed of the wheel is 2 rad/s at t=0, through
what angular displacement does the wheel rotate in 2 s?
B) Through how many revolutions has the wheel turned during
this time interval?
C) What is the angular speed of the wheel at t=2 s?
Example 3. contd..
Two types of angular acceleration
1. Radial acceleration
2. Tangential acceleration

ar
 
at v
 is angular acceleration
Δv
Δω
at 
r
 rα
Δt
Δt

Derivation: acent = 2r = v2/r
v s
• Similar triangles: 
v
r
v v s
aavg 

t r t
•
v
av
r
 acent

2
v
  2r 
r


v  r   v /r
Example: 4
A typical CD disc player, the constant speed of the surface at
the point of laser lens is 1.3 m/s.
A) Find the angular speed of the disc in revolution per minute
when information is being read from the innermost track
(r=23 mm) and the outermost track (r=58 mm).
B) The maximum playing time of standard music disc is 74 min
and 33 s. How many revolution does the disc make during
that time?
Example: 4, contd….
𝜃=
1800000
= 280000 𝑟𝑒𝑣. 𝑠
2𝜋
To r q u e
r
Ft
m
t = r xFt = rF Sinθ SI unit: N.m
To r q u e a n d i t s e f f e c t
1. Centripetal Force:
The force radially act towards the center of the circle in
circular motion is called centripetal force.
Fr = mv2/r
2. Centrifugal force:
The force act tangentially at the circle in rotatory
motion is called centrifugal force
Ft = mr𝜶
Rotational Law of Acceleration
• The force law can be combined with rotational
motion.
Torque: t = r Ft = r m at = m r 2 
• If torque replaces force, and angular acceleration
replaces acceleration, this looks like the law of
acceleration.
t  (mr )  I
2
Where, I is the moment of inertia of the body.
Moment of Inertia or Rotational Inertia
The masses of the particles and square of their distances
from the axis of rotation is called moment of inertia.
I = 𝑛𝑖=1 𝑚𝑖 𝑟𝑖2 =mr2
Let, m1, m2, m3..….mn be the masses of the particles and
r1, r2, r3…..rn be their respective distances from their axis
of rotation.
Note:
In linear motion, the mass of the body is a measure of its
inertia.
Moment of Inertia or Rotational Inertia
Sl. No
Name of the
body
Axis of Rotation
Passing thro’ its
center and
perpendicular
to its plane
1
Circular
ring of
radius R
2
Circular
ring/disc of
radius R
Diameter
3
Circular
ring of
radius R
Tangent
parallel to
diameter
4
Sphere
Diameter
Diagram
Moment of
Inertia
I = mr2
I=
𝟏
mr2
𝟐
I=
𝟐
mr2
𝟑
I=
𝟐
mr2
𝟓
Angular momentum or moment of momentum (L)
The angular momentum is defined as the product of linear
momentum of the particle and the perpendicular distance of
the particle from the axis of rotation.
𝑳=𝒓𝒙𝑷
L = r p sinθ
where 𝑝 = 𝑚 𝑣 , linear
momentum of the body.
SI unit: kg m2/s
Dimensional Formula=M1L2T-1
Conservation of angular momentum
Statement:
The product of moment of inertia and angular velocity is
always remains constant.
I ω = constant
If the moment of inertia of the changes from I1 to I2 due
to change in the distribution of mass of the body, then
the angular velocity changes from ω1 and ω2, such that
I1 ω1 = I2 ω2
Conceptual Question
Who has the greatest tangential velocity?
CORRECT
1. You do
2. Your friend does
3. Same
This is like the example of the "crack-thewhip." The person farthest from the pivot
has the hardest job. The skater has to cover
more distance than anyone else. To
accomplish this, the skater must skate faster
to keep the line straight.
vT =  R ( is the same but R is larger)
you
Conceptual Question
Who has the greatest centripetal acceleration?
you
1. You do
CORRECT
2. Your friend does
3. Same
Things toward the outer edge want to "fly off" more
than things toward the middle. Force is greater on
you because you want to fly off more.
Centripetal acceleration is ac = R 2 and you have the
largest radius.
Work and Power in Rotational Motion
Imagine a wheel that can rotate freely at an
axis “O”. If the force F act at tangentially
upward so that the wheel rotate through an
angle θ.
Work Done by the force,
W = F x distance s, s = r θ
O
W = F r θ =(F r) θ, t  F r
W=tθ
Work done in rotational motion
F
Power
Let t be the time taken by the torque to produce angular
displacement θ, then
𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 𝑊 𝝉𝜽
Power, P =
= =
= 𝝉𝛚,
𝑡𝑖𝑚𝑒
𝑡
𝒕
𝐀𝐧𝐠𝐮𝐥𝐚𝐫 𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲, 𝛚 =
𝜽
𝒕
P = 𝝉𝛚
Sl. No
Linear Motion
Angular/Rotational
Motion
1
W=FS
W=tθ
2
𝑾
P=
𝒕
P = 𝝉𝝎
Rotation and Translation
• A rolling wheel is
moving forward with
kinetic energy.
• The velocity is
measured at the
center of mass.
K.ECM = ½ m v2
v
• A rolling wheel is
rotating with kinetic
energy.
• The axis of rotation is
at the center of mass.
K.Erot = ½ I 2

Rolling Energy
• The energy of a rolling wheel
is due to both the translation
and rotation.
• The velocity is linked to the
angular velocity.
• The effective energy is the
same as a wheel rotating
about a point on its edge.
– Parallel axis theorem
KE  KECM  KErot
KE  12 mv 2  12 I 2
KE  (mr  I )
1
2
2
2
Energy Conserved
• A change in kinetic energy is
due to work done on the
wheel.
– Work is from a force
– Force acts as a torque
• Rolling down an incline the
force is from gravity.
– Pivot at the point of contact
• The potential energy is
converted to kinetic energy.
v
R
F = mg

Example : 5
A stationary exercise bicycle wheel starts from rest and
accelerates at a rate of 2 rad/s2 for 5 s, after which the
speed is maintained for 60 s. Find the angular speed
during the 60 s interval and the total number of
revolutions the wheel turns in the first 65 s.
Example : 5
Reference:
http://hyperphysics.phyastr.gsu.edu/hbase/circ.html#rotcon