Rotational motion

Rotational Motion
Rotational Motion- By Aditya Abeysinghe

1

Definitions of some special terms
1.

Angular position (Φ) - The angular position of a particle is the

angle ɸ made between the line connecting the particle to the
original and the positive direction of the x-axis, measured in a
counterclockwise direction

ɸ=l/r

2. Angular displacement (θ) - The radian value of the angle

displaced by an object on the center of its path in circular motion
from the initial position to the final position is called the angular
displacement.

2

θ = θf - θi

3. Angular Velocity (ω)- Angular velocity of an object in circular

motion is the rate of change of angular displacement

ω=θ/t
Unit - rads-1
 Vector direction by
Right hand rule

3

4. Angular acceleration- Angular acceleration of an object in

circular motion is the rate of change of angular velocity
ω

t=t

α = (ω – ω0) / t

r
θ
ω0

t=0


Unit- rads-2
Direction- By right
hand rule

4

Angular equations of movement
α = (ω – ω0) / t

ω = ω0 + αt

(ω + ω0 )/2 = θ / t

θ = (ω + ω0 )t/2

(ω + ω0 )/2 = θ / t , ωt = 2θ + ω0t ,
(ω0 + αt)t = 2θ + ω0t
θ = ω0t + ½ αt2
θ = (ω + ω0 )t/2, θ = (ω + ω0 )(ω - ω0 )
2
α

ω2 = ω02 + 2αθ

5

Therefore the four equations of angular movement are-

1.
2.
3.
4.

ω = ω0 + αt
θ = (ω + ω0 )t/2
θ = ω0t + ½ αt2
ω2 = ω02 + 2αθ

It should be noted that these four equations are
analogous to the four linear equations of motion:
1. V = U + at
2. S = (V + U)t/2
3. S = Ut + ½ at2
4. V2 = U2 + 2as

6

Right hand rule
Take your right hand and curl your fingers along the direction of the
rotation. Your thumb directs along the specific vector you need.
(angular velocity, angular acceleration, angular momentum etc.)

Thus, right hand rule
is used whenever, in
rotational motion, to
measure the
direction of a
particular vector

Direction of
rotation

Axis of rotation

7

Relationship between physical
quantities measured in angular
motion and that in linear motion
1.

Linear displacement- Angular displacement
S

r
θ

S= (2πr / 2π ) × θ = rθ

S = rθ

2. Linear velocity- Angular velocity
S/t=rθ/t

V = rω

8

3.

Linear acceleration- Angular acceleration

α = (ω – ω0) / t
αr = (ω – ω0)r / t
αr = (ωr – ω0r) / t
αr = (V - V0) / t

a = rα

αr = a

Displacement

Velocity

Acceleratiom

Translational motion

S

V

a

Rotational motion

θ

ω

α

Relationship

S = rθ

V = rω

a = rα


9

The period and the frequency
of an object in rotation motion
Period (T) is the time taken by an object in
rotational motion to complete one complete
circle.
Frequency (f) is the no. of cycles an object
rotates around its axis of rotation
Thus, f = 1/ T . However, ω = θ / t
Therefore, ω = 2π / T
Therefore, ω = 2π / (1/f)
Thus, ω = 2πf

10

Moment of Inertia
Unlike in the case of linear movement’s inertia (reluctance to
move or stop) , inertia of circular/rotational motion depends
both upon the mass of the object and the distribution of mass
(how the mass is spread across the object)
Moment of Inertia of a single object-

I = mr2
Axis of rotation

is

r

Moment of Inertia
a scalar quantity

m

Path of the object

11

Radius of gyration
Suppose a body of mass M has moment of Inertia I
about an axis. The radius of gyration, k, of the body
about the axis is defined as

I= Mk2
That is k is the distance of a point mass M
from the axis of rotation such that this point
mass has the same moment of inertia about
the axis as the given body.


12

Moment of Inertia of some common shapes
Body

Axis

Figure

Ring
(RadiusR)

Perpendicular
to the plane
at the center

I

MR2

k

R

Disc (Radius R) Perpendicular
to the plane
at the center

½ MR2 R / √2

Solid Cylinder
(Radius R)

Axis of
cylinder

½ MR2 R / √2

Solid Sphere
(Radius R)

Diameter

⅖ MR2


R√(⅖)
13

Angular Momentum
Angular momentum is the product of the moment of inertia and the
angular velocity of the object.
z

L = r sin θ × P
L = r P Sinθ
y
r

P

θ

x

L= Iω Sinθ

However, P= mv ,
(Linear moment= mass × linear velocity )

Therefore, L= m r v Sinθ
= m r (ω r ) Sinθ (as V= rω)
= mr2 ω Sinθ
Therefore, L = Iω Sin θ (as I = mr2)

14

But in most cases the radius of rotation is perpendicular to the
momentum.
Thus, θ = 90° ,

L = Iω
P

r

Axis of
rotation


15

Torque
The rate of change of angular momentum of an
object in rotational motion is proportional to the
external unbalanced torque. The direction of
the torque also lies in the direction of the
angular momentum.
Torque is called the moment of force and is a
measure of the turning effect of the force about
a given axis.
A torque is needed to rotate an object at rest or
to change the rotational mode of an object.

16

τ = (Iω – Iω0 )/t
τ = I [(ω - ω0 )/t]
Therefore,

τ = Iα

By Newton’s second law
of motion
F = ma
Fr = mra = mr (rα) ( as
a=rα)
Fr= mr2 α
Fr = Iα
( as I= mr2 )
However, from the above
derivation, Iα = τ .
Therefore, Fr = τ
Thus

, τ = Fr

17

This theory can be expressed as:
z
F

θ
r
O

y
d

τ = F×r
x


τ = Fd
τ = Fr Sinθ
18

Relationship between Torque and Angular
Momentum-

τ = dL / dt

τ – Net torque
L- Angular momentum

This result is the rotational analogue of
Newton’s second law:
F = dP / dt

19

Applying Newton’s laws to
rotational motion
1. Newton’s First law and equilibriumIf the net torque acting on a rigid object is zero, it will
rotate with a constant angular velocity.
If the system
Concept of equilibriumd
M

Therefore, τM

3d
m

O

is

at
equilibrium, total
torque around O
should be zero

+ τm = 0

Mgd + [ -(mg(3d))] = 0
Therefore, m = M/3


20

2. Newton’s second law of motion
The rate of change of angular momentum of an object in rotational
motion is proportional to the external unbalanced torque. The direction
of the torque also lies in the direction of the angular momentum.

τ α α
τ = Iα
This is analogous to the linear equivalent, which states,
The rate of change of momentum is directly proportional to the
external unbalanced force applied on an object and that force lies
in the direction of the net momentum.

Thus, torque could be treated as the rotational
analogue of the force applied on an object.

21

Theorem of Parallel Axes
Let ICM be the moment of inertia of a body of mass M about an
axis passing through the center of mass and let I be the moment
of inertia about a parallel axis at a distance d from the first axis.

Then, I = ICM + Md2
Thus the minimum
moment of inertia
for any object is at
the center of
mass, as x in the
above expression
is zero.


22

Theorem of perpendicular axis
The moment of inertia of a body about an axis perpendicular to its
plane is equal to the sum of moments of inertia about two mutually
perpendicular axis in its own plane and crossing through the point
through which the perpendicular axis passes

Iz = Ix + Iy


23

Rotational Kinetic Energy
ω

mn

rn

r1
r2

Ek = Σ ½ mi vi2
But, vi = riω
So, Ek = Σ ½ mi ri2ω2
= ½ (Σ mi ri2 ) ω2
Therefore, Ek = ½ I ω2

m1
m2

Ek = ½ I

2
ω

Rotational kinetic energy is the rotational analogue of the translational kinetic
energy, which is EK = ½ mv2. In fact, rotational kinetic energy equation can
be deduced by substituting v =rω and viceversa.

24

Law of Conservation of Angular
Momentum
Law- If the resultant external torque on a system is
zero, its total angular momentum remains constant.
That is, if τ = 0, dL / dt = 0 , which means that L is a constant
This is the rotational analogue of the law of conservation of linear
momentum.


25

Work done by a torque
dW = τ dθ
Therefore, the total work done in rotating the body from an angular
displacement of θ1 to an angle displacement θ2 is
θ2

W=

∫ τ dθ

θ1
θ2

W= τ ∫ 1. dθ
θ1

Therefore,

W= τ [θ2 - θ1 ]

26

Power
The rate at which work is done by a torque is called
Power

P= dw/dt = τ dθ/dt = τ ω
Therefore, P

=τω


27

Work - Energy Principle
From ω2 = ω02 + 2αθ and W= τθ , it is clear that the work
done by the net torque is equal to the change in
rotational kinetic energy.
τ = Iα and ω2 = ω02 + 2αθ .
Therefore, ω2 = ω02 + 2(τ/I)θ .
Thus, ω2 = ω02 + 2[(τθ)/I] .
Thus, ω2 = ω02 + 2W/I
OR

W= ½ I (ω2 - ω02 ). This is called
the work-energy principle.

28

Relationship between Angular
momentum and Angular velocity
τ = Iα = I dω/dt = d(Iω)/dt
But, τ = dL/dt
Thus, dL/dt = d(Iω)/dt
By integrating both sides,
It can be shown that,

L = Iω

This is the rotational analogue of P = mv


29

Rolling Body
Rolling is a combination of rotational and transitional(linear)
motions.
Suppose a sphere is rolling on a plane surface, the velocity
distribution can be expressed as,


30

This two systems (rotational and transitional) can
be combined together to understand how actually
the sphere above moves in the plane. The final
distribution shows clearly that in reality the ball
always instantly experiences a zero velocity at the
point of contact with the surface and the maximum
velocity is at the top

2V
V

V √2
V

V=0

31

Thus, Etotal = Etransitional + Erotational

E=½

2
mv

+ ½

2
Iω

E = ½ mR2 ω2 + ½ mk2 ω2
(Where k- radius of gyration)

E=

2
½mω

2
(R

+

2)
k


32

A body rolling down an inclined
plane
From the conservation of energy,

Mgh = ½mv2 + ½Iω2
s
h

mg

θ


33

Equilibrium of a rigid body
1.

Transitional Equilibrium

For a body to be in transitional equilibrium, the vector sum
of all the external forces on the body must be zero.
Fext = M acm
and acm must be zero for transitional equilibrium.
2. Rotational Equilibrium

For a body to be in rotational equilibrium, the vector sum of
all the external torques on the body about any axis must be
zero.

τext = Iα

and α must be zero for rotational equilibrium.


34

Summarizing it up!!
Term

Definition

Angular position

The angular position of a particle is the angle ɸ made between
the line connecting the particle to the original and the positive
direction of the x-axis, measured in a counterclockwise direction

Angular
displacement

The radian value of the angle displaced by an object on the
center of its path in circular motion from the initial position to the
final position is called the angular displacement.

Angular velocity

Angular velocity of an object in circular motion is the rate of
change of angular displacement

Angular acceleration

Angular acceleration of an object in circular motion is the rate of
change of angular velocity

Angular equations of
motion

ω = ω0 + αt , θ = (ω + ω0 )t/2 ,
θ = ω0t + ½ αt2 , ω2 = ω02 + 2αθ

35

Right hand rule

Take your right hand and curl your fingers along the direction of the
rotation.Your thumb directs along the specific vector you need

Relationship
between linear
and angular
qualities

S = rθ , V = rω , a = rα
(s- translational displacement , v- translational velocity,
a-translational acceleration )

Period and
frequency

Period (T) is the time taken by an object in circular motion to complete
one complete circle.
Frequency (f) is the no. of cycles an object rotates around its axis of
rotation
Thus, f = 1/ T

Moment of
inertia

Inertia of circular/rotational motion depends both upon the mass of the
object and the distribution of mass.

Radius of
gyration

The radius of gyration, k, of the body about the axis is defined as
I= Mk2

Angular
momentum

Angular momentum is the product of the moment of inertia and the
angular velocity of the object.

36

Torque

Torque is called the moment of force and is a measure of the
turning effect of the force about a given axis.

Theorem of parallel
axes

Let ICM be the moment of inertia of a body of mass M about an
axis passing through the center of mass and let I be the moment
of inertia about a parallel axis at a distance d from the first axis.
Then, I = ICM + Md2

Theorem of
perpendicular axes

The moment of inertia of a body about an axis perpendicular to its
plane is equal to the sum of moments of inertia about two
mutually perpendicular axis in its own plane and crossing through
the point through which the perpendicular axis passes
Iz = Ix + I y

Rotational kinetic
energy

Ek = ½ I ω2

Law of conservation
of angular
momentum

If the resultant external torque on a system is zero, its total
angular momentum remains constant.

Work done

W = τ [θ2 - θ1 ]

Power

The rate at which work is done by a torque is called
Power
P=τω

Work-Energy principle

W= ½ I (ω2 - ω02 ). This is called the work-energy
principle.

Rolling body

Rolling is a combination of rotational and
transitional(linear) motions.

Equilibrium of a rigid body

1.

Transitional Equilibrium
For a body to be in transitional equilibrium, the vector
sum of all the external forces on the body must be zero.
Fext = Macm
and acm must be zero for transitional equilibrium.
2. Rotational Equilibrium
For a body to be in rotational equilibrium, the vector
sum of all the external torques on the body about any axis
must be zero.
τ = Iα and α must be zero for rotational equilibrium.

Rotational motion

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