Rotational dynamics (MAHARASHTRA STATE BOARD) CLASS XII PHYSICS

1. Class XIIth
Maharashtra state Board
Chapter - 1
Rotational Dynamics
By – Miss. Pooja K More

2. Can you recall?
1. What is circular motion?
2. What is the concept of centre of mass?
3. What are kinematical equations of motion?
4. Do you know real and pseudo forces, their
origin and applications?

4. In Physics, circular motion is a
movement of an object along
the circumference of
a circle or rotation along a circular path
circular motion

5. Difference between Particle and Body
Particle Body

6. Angular displacement and displacement

7. Angular displacement and displacement
• Denoted by ‘’
• Angle through which on
object moves on circular
path
• SI unit is Radian or Degree
• It is also length
• It is distance or minimum
distance between initial and
final position
• SI unit is ‘m’
• Represent by ‘d’ or ‘s’

8. Rate of change of the
position angle of an object
w.r.t time
Rate of change of
displacement per unit time.

9. Angular acceleration ACCELERATION
• Denoted by ‘’
• It is the time rate of
change of angular velocity
• Unit – Radian/ 𝒔𝒆𝒄𝒐𝒏𝒅 𝟐
• Formula –  =
𝒅𝒘
𝒅𝒕
=
𝒘 𝒇𝒊𝒏𝒂𝒍 −𝒘(𝒊𝒏𝒊𝒕𝒊𝒂𝒍)
𝒕
• Denoted by ‘a’
• It is the rate of change of
velocity per unit time.
• Unit – Meter/ 𝒔𝒆𝒄𝒐𝒏𝒅 𝟐
• Formula – a =
𝑽
𝒕
=
𝑽 𝒇𝒊𝒏𝒂𝒍 −𝑽(𝒊𝒏𝒊𝒕𝒊𝒂𝒍)
𝒕

12. It is the movement of
motion of an object along
the circumference of a
circle or rotation along
circle.
Example. – Artificial
satellite (Frame of
reference)
circular motion

13. Types of circular motion

14. • Angular speed and
acceleration is constant
• Angular velocity changes
• Angular speed is also
changes in non - UCM

15. • Circular motion is an essential part of our daily life.
• Every day we come across several revolving or rotating
(rigid) objects.
• During revolution, the object (every particle in the object)
undergoes circular motion about some point outside the
object or about some other object, while during rotation
the motion is about an axis of rotation passing through
the object.
:

16. We can understand this with the help of an example:
Suppose a stone is tied to a thread and is rotated in a
circular path with uniform speed in clockwise direction.
Now, when a stone is reached a certain point say A, then its
speed is directed towards east. And if the stone is released
when it is at A, it will fly off in the east direction. When the
stone is at point B, its speed is directed towards south. And
if the stone is released when it is at point B, it will fly off in
the south direction. That is when a body moves in a circular
path, the direction of speed is not the same at any two
points. And when there is a change in the direction of
speed of the body, its velocity is not uniform.
Therefore, circular motion is accelerated even though the
speed of the body remains constant. Please note that a
force is needed to produce circular motion. The force is
needed to make an object travel in a circular path which is
centripetal force.

17. Some more examples of Uniform Circular Motion are:
1. Artificial satellites move in uniform motion around the earth.
Therefore, the motion of a satellite around the earth is accelerated.
2. The moon moves in a uniform circular motion around the earth. We
know that moon is a natural satellite of the earth.
3. Similarly, we can say that movement of earth around the sun is also
a uniform circular motion. So, the motion of earth around the sun is
accelerated.
4. The tip of a second’s hand of a watch exhibits uniform circular
motion on the circular dial of the watch.
So, we understood that Force is required to make body move in a
circle. And when a body or object moves in a circular path with
constant speed is uniform circular motion.

26. 1.2 Characteristics of Circular Motion:
• It is an accelerated motion
Velocity changes acceleration also
changes
• It is periodic in motion
(Particle repeats its path again and again
during motion)

27. • Uniform Circular Motion
- Acceleration (Radial acceleration)
- Speed and acceleration is contact
- Direction, 𝒂 𝒓 = - 𝒘 𝟐
r (vector)
- Magnitude, 𝒂 𝒓 = - 𝒘 𝟐
r

28. Characteristics of Circular Motion:
1)It is an accelerated motion: As the direction of
velocity changes at every instant, it is an
accelerated motion.
2)It is a periodic motion: During the motion, the
particle repeats its path along the same
trajectory. Thus, the motion is periodic

54. 1.2.1 Kinematics of Circular Motion:
• To describe a circular motion, we use the quantities angular
displacement θ , angular velocity w and angular acceleration 
which are analogous to displacement s , velocity v = ds/dt
and acceleration a = dv/dt used in translational motion.
• Also, the tangential velocity is given by v = w x r where ω is
the angular velocity.
• Here, the position vector r is the radius vector from the centre
of the circular motion. The magnitude of v is v =ω r.
• Direction of ω is always along the axis of rotation and is given
by the right-hand thumb rule.

55. • To know the direction of ω , curl the fingers of the right
hand along the sense of rotation, with the thumb
outstretched.
• The outstretched thumb then gives the direction of ω .

56. • If T is period of circular motion or periodic time and n is the
frequency, w = 2𝝅n =
2 𝝅
𝑻
• Uniform circular motion: During circular motion if the
speed of the particle remains constant, it is called Uniform
Circular Motion (UCM).
• In this case, only the direction of its velocity changes at every
instant in such a way that the velocity is always tangential to the
path.
• The acceleration responsible for this is the centripetal or radial
acceleration 𝒂 𝒓 = - 𝒘 𝟐r

57. • For UCM, its magnitude
is constant and it is
a = 𝒘 𝟐
r =
𝒗 𝟐
𝒓
= vw
• It is always directed
towards the centre of
the circular motion
(along − r ), hence
called centripetal.

58. Illustration: Circular motion of any particle of a fan
rotating uniformly.
Non-uniform circular motion: When a fan is
switched ON or OFF, the speeds of particles of the
fan go on increasing or decreasing for some time,
however their directions are always tangential to their
circular trajectories.

59. • During this time, it is a non-uniform circular motion.
• As the velocity is still tangential, the centripetal or
radial acceleration a, is still there.
• However, for non-uniform circular motion, the
magnitude of a, is not constant.
• The acceleration responsible for changing the
magnitude of velocity is directed along or opposite to
the velocity, hence always tangential and is called as
tangential acceleration 𝒂 𝑻 .

60. • As magnitude of tangential velocity v is
changing during a non-uniform circular motion,
the corresponding angular velocity ω is also
changing at every instant. This is due to the
angular acceleration  =
𝒅𝒘
𝒅𝒕
• Though the motion is non-uniform, the particles
are still in the same plane. Hence, the direction
of α is still along the axis of rotation.

61. • For increasing speed, it is along the direction of
ω while during decreasing speed, it is opposite to
that of ω .

62. • If the angular acceleration α is constant and along the
axis of rotation, all , w and  will be directed along the
axis.
• This makes it possible to use scalar notation and write
the kinematical equations of motion analogous to those
for translational motion.
Example: A fan is rotating at 90 rpm. It is then switched
OFF. It stops after 21 revolutions. Calculate the time taken
by it to stop assuming that the frictional torque is constant.

64. Do you know?
• If the angular acceleration α is along any direction other
than axial, it will have a component perpendicular to the
axis. Thus, it will change the direction of ω also, which
will change the plane of rotation as ω is always
perpendicular to the plane of rotation.

65. If α is constant in magnitude,
but always perpendicular to ω, it
will always change only the
direction of ω and never its
magnitude thereby continuously
changing the plane of rotation.
(This is similar to an
acceleration a perpendicular to
velocity v changing only its
direction).