ROTATIONAL MOTION OF RIGID BODY

CHAPTER 8

ROTATION OF A RIGID BODY

Subtopic Duration
8.1 : Rotational Kinematics (1 hour)
8.2 : Equilibrium of a uniform rigid body
8.3 : Rotational dynamics
8.4 : Conservation of angular momentum

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LESSON 8

Learning Outcome:

8.1 Rotational Kinematics 2

a) Define and use:

i) angular displacement,.
ii) average angular velocity,av .
iii) instantaneous angular velocity,.
iv) average angular acceleration,av.
v) instantaneous angular acceleration,.

b) Relate parameters in rotational motion with their
corresponding quantities in linear motion.

c) Solve problem related to rotational motion with
constant angular acceleration.

Overview L8

Rotation of a rigid body

Rotational Equilibrium of Rotational Conservation
kinematics a uniform dynamics of angular
rigid body momentum
Rotational Moment of
linear Torque inertia Angular
momentum
relationship  rF   I
Principle of
Rotation Conditions for conservation
motion with Equilibrium
of angular
uniform F 0 momentum
angular   0
acceleration 3

8.1 Rotational Kinematics L8

8.1.1 Angular displacement,

o is defined as an angle through which a point or line
has been rotated in a specified direction about a
specified axis.

o Consider a point P on a rotating compact disc (CD)
moves through an arc length s on a circular path of
radius r about a fixed axis through point O.

https://youtu.be/fmXFWi-WfyU
https://youtu.be/qPUmcE45U7Y

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θ  s OR s  rθ L8

r

where θ : angle(angular displaceme nt)in radian

s : arclength

r : radius of thecircle

o The S.I. unit : radian (rad).
o Others unit : degree () and revolution (rev).

o Conversion factor :

1rev  2 rad  360o

o Sign convention of angular displacement :
Positive – if the rotational motion is anticlockwise.
Negative – if the rotational motion is clockwise.

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8.1.2 Angular velocity L8

Average angular velocity, av

o is defined as the rate of change of angular

displacement.

Equation : ωav  θ2  θ1  θ
where t2  t1 t

θ2 : final angular displaceme nt in radian
θ1 : initial angular displaceme nt in radian
t : time interv al

Instantaneous angular velocity, 

o is defined as the instantaneous rate of change of

angular displacement.

o Equation :   limit θ  dθ 6
t0 t dt

o vector quantity. L8

o Unit : radian per second (rad s-1)

o Others unit : revolution per minute (rev min1 or rpm)

o Conversion factor:

1 rpm  2 rad s1   rad s1
60 30

Note :
o Every part of a rotating rigid body has the same
angular velocity.

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L8

o Direction of the angular velocity can be

determined by using right hand grip rule where

Thumb : direction of angular velocity

Curl fingers : direction of rotation

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8.1.3 Angular acceleration L8

Average angular acceleration, av

o is defined as the rate of change of angular
velocity.

Equation : ω2  ω1 ω
t2  t1 t
 av  

where ω2 : final angular velocity
ω1 : initial angular v elocity

t : time interv al
Instantaneous angular acceleration, 

o is defined as the instantaneous rate of change of

angular velocity. α  limit ω  dω
Equation :

t0 t dt 9

o Vector quantity. L8
o Unit : rad s2.
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Direction of the angular acceleration
o If the rotation is speeding up,

i.  and  in the same direction.
ii.  and  is positive

o If the rotation is slowing down,

i.  and  have the opposite direction.
ii.  is negative,  is positive

8.1.4 Relationship between linear velocity, v and L8

angular velocity, 

o When a rigid body rotates about rotation axis O, every

particle in the body moves in a circle as shown below.

o Consider at point P, s  rθ ds  r d v  r

y dt dt

  o The direction of the linear velocity
v always tangent to the circular

r Ps path.

O o Every particle on the rigid body has

x the same angular speed

(magnitude of angular velocity)

but the tangential speed is not the

same because the radius of the

circle, r is changing depend on the

position of the particle. 11

8.1.5 Relationship between tangential acceleration, L8

at and angular acceleration,  x

o If the angular speed of the rigid body changes

with time (increases or decreases), angular

acceleration is produced. y

v  r 
at
dv  r d P
r
dt dt s
O
at  r

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8.1.6 Rotational motion with uniform angular L8
acceleration

o Table below shows the symbols and relationship
used in linear and rotational kinematics.

Linear Motion Rotational Motion

Symbol Quantity Relationship

Symbol Quantity

s Displacement s  r θ θ Angular displacement
ω0 Initial angular velocity
u Initial velocity u  r o
ω Final angular velocity
v Final velocity v  r 
a Acceleration a  r  α Angular acceleraton

t Time - t Time

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o Table below shows the analogy of linear motion L8
with rotational motion parameters.
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Linear motion Rotational motion
a  constant
α  constant
v  u  at
s  ut  1 at2 ω  ω0  αt

2 θ  ω0t  1 αt 2
v2  u2  2as 2

s  1 v ut ω2  ω02  2αθ

2 θ  1 ω  ω0  t
2
m
I
p  mv
L  I

 F  ma   I

LESSON 8

Learning Outcome:

8.2 Equilibrium of a uniform rigid body
a) Define torque.
b) Solve problems related to equilibrium of a uniform

rigid body

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8.2 Equilibrium of a uniform rigid body L8
Torque (moment of a force),
8.2.1

o Torque is the tendency of a force to cause or change
the rotational motion od a body.

o Torque also known as the turning effects of the force

on a body.

o Defined as the vector product between lever arm , r

(position vector) and, F.   
r F
    rF sin 

τ : magnitude of thetorque   Fr sin 
  Fd
where F :magnitude of theforce
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r : magnitudeof lev er arm

d  r sin 

 d:apnegrpleenbdeitcwuelaerndirstaanndceF

  Fd L8

oThe magnitude of the torque is defined as the product

of a force and its perpendicular distance from the line of

action of the force to the point (rotation axis).
F
Pivot point

(rotation axis)

 60 o
r 60 o

d Line of action
of a force
d  r sin

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o Vector quantity.
o Unit : Newton. meter (Nm)

Direction of torque : Right hand grip rule.
i. Point the fingers of your right hand in the direction of r.
ii. Curl your fingers toward the direction of vector F.
iii. Your thumb then points in the direction of the torque.

Sign convention of torque:
oPositive if body rotates in anticlockwise.
oNegative if body rotates in clockwise.

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L8

  rF sin 
  Fr sin 
  Fd

o The magnitude of torque depends on how much
force is applied, the length of the lever arm that
connects the axis to the point where the force is
applied, and the angle between the force
vector and the lever arm.

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Case 1: L8

A force is applied perpendicular to a meter rule which is

pivoted at one end.

  Fr sin 

Pivot point   Fd 
(rotation axis) F
or

  Fr

  90o   90o


r r  d (if   90o )

The meter rule has tendency to rotate in clockwise

direction. 20

Case 2: L8

A force is applied at an angle 60o (above x-axis) to a

meter rule which is pivoted at one end.

  Fr sin 

Pivot point   Fr sin 60 
(rotation axis) F
or

  Fr sin 120

60 o


r

The meter rule has tendency to rotate in clockwise
direction.

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Case 3: L8

A force is applied at an angle 60o (below x-axis) to a

meter rule which is pivoted at one end.

  Fr sin 

  Fr sin 60

or

Pivot point   Fr sin 120
(rotation axis)

 60 o
r

The meter rule has tendency to rotate in F
anticlockwise direction.
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8.2.2 Equilibrium of a uniform rigid body L8

o Rigid body is defined as a body with definite shape
that doesn’t change, so that the particles that
compose it stay in fixed position relative to one
another even though a force is exerted on it.

o If the rigid body is in equilibrium, means the body is
translational and rotational equilibrium.

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L8

o Two conditions for a rigid body to be in equilibrium:

i. The vector sum of all external forces acting on a
rigid body must be zero.

 Fx  0, Fy  0

translational equilibrium

ii. The vector sum of all external torques acting on
a rigid body must be zero.

  0axis of rotation

rotational equilibrium

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LESSON 8

Learning Outcome:

8.3 Rotational Dynamics
a) Define and use the moment of inertia of a uniform

rigid body (sphere, cylinder, ring, disc and rod)
b) State and use torque.

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8.3 Rotational dynamics L8

8.3.1 Moment of inertia, I Inertia is the tendency of an
object to resist any attempt to
change its velocity (rest or
linear motion).

o Moment of inertia I, is the tendency of an object to resist any
attempt to change its angular velocity (rest or rotational
motion).

o Figure below shows a rigid body rotates about a fixed axis O

with angular velocity .

o Moment of inertia is defined as the sum of the products of the

mass of each particle and the square of its respective distance

from the rotation axis.

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n L8

I  m1r12  m2r22  m3r32  ...mn rn2  miri2
i1

where

I : moment of inertia of a rigid body about rotation axis

m : mass of particle

r : distance from theparticle to therotation axis

o Scalar quantity.

o Moment of inertia, I in the rotational kinematics
is analogous to the mass, m in linear kinematics.

o S.I. unit : kg m2.

o The factors which affect the moment of inertia, I
of a rigid body:

i. the mass of the body,

ii. the shape of the body,

iii. the position of the rotation axis. 27

Moments of inertia of various bodies L8

o Table below shows the moments of inertia for a number

of objects about axes through the centre of mass.

Shape Diagram Equation

Hoop or ring or CM ICM  MR 2
thin cylindrical

shell

Solid cylinder or CM I CM  1 MR 2
disk 2

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Shape Diagram Equation L8
CM
Uniform rod or I CM  1 ML2
long thin rod 12
with rotation
axis through the
centre of mass.

Solid Sphere CM I CM  2 MR 2
5
Hollow Sphere
or thin spherical CM I CM  2 MR 2
3
shell

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8.3.2 Relationship between torque, and angular L8
acceleration, 

o Figure below shows a force F acts on a rigid body
causes it rotates about a fixed axis O with angular

velocity, .

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o A particle of mass, m1 of distance r1 from the rotation L 8

axis O will experience a nett force F1 . The nett force on

this particle is F1  m1a1 and a1  r1α

F1  m1r1α

o The torque on the mass m1 is 1  r1F1 sin 90

o The total (nett) torque on the rigid body is given by

1  m1r12

  m1r12  m2r22  ...  mnrn2

  n  n
  i 1 mi ri 2 and
 miri2  I

i 1

  I 31

L8

o From the equation, the nett torque acting on the
rigid body is proportional to the body’s angular
acceleration.

o Note : Nett torque ,   I

is analogous to the

Nett force, F  ma

https://youtu.be/tkaZaQ-V4Bk

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LESSON 8

Learning Outcome:

8.4 Conservation of Angular Momentum
a) Define and use angular momentum.
b) State and use principle of conservation of angular

momentum.

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8.4 Conservation of angular momentum L8


8.4.1 Angular momentum, L

o Defined as the product of the angular velocity of
a body and its moment of inertia about the
rotation axis.

OR L  I is analogous to the p  mv

where

L : angular momentum

I : moment of inertia of a body
ω : angular v elocity of a body

• Vector quantity.
• S.I. unit : kg m2 s-1

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8.4.2 Principle of conservation of angular momentum L 8

Statement

o In a closed system, total angular momentum is
constant (conserved).

OR I  constant If  0

 Li   Lf

 I   I  f
i

https://youtu.be/oGzQflqf1VA

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