1. Chapter 4 Rotational Motion
Dr Manjunatha S
College of Computer and Information Sciences

2. Wind mill- produces electricity

3. The rotation of a rigid object about a fixed axis

4. 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

5. Translation and Rotation
The motion of a rigid body includes the motion of its center of mass.
This is translational motion
A rigid body can also move while its center of mass is fixed.
This is rotational motion.
vCM 

6. 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.

8. Definition:

9. Definition:
The angular motion between two position of an object in time interval is called angular displacement.
Δθ = θf – θi
SI unit is rad

10. Angular velocity (ω) Definition:
The rate of change of angular displacement of a rigid body is called angular velocity.
SI Unit: rad/s
Relation between velocity and angular velocity:

11. Angular acceleration ()
Definition:
The rate of change of angular velocity of a rigid body is called angular acceleration.
SI unit: rad/s²
Every point on rigid object has same w and a
w is positive or negative

12. 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:

13. Since, the angular displacement is the same in both cases.
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.

14. Relation between acceleration and angular acceleration:
Acceleration, a = 𝑣 𝑡 = 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).

15. Rotational/Linear Correspondence
Rotational Motion
Linear Motion
Constant a
Constant 

16. 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?

17. Example: 3
A wheel rotates with a constant angular acceleration of 3.5 rad/s2.
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?
Through how many revolutions has the wheel turned during this time interval?
What is the angular speed of the wheel at t=2 s?

18. Example 3. contd..

19. Two types of angular acceleration 1. Radial acceleration 2
Two types of angular acceleration 1. Radial acceleration 2. Tangential acceleration
a is angular acceleration

20. Derivation: acent = w2r = v2/r
Similar triangles:

21. Example: 4
A typical CD disc player, the constant speed of the surface at the point of laser lens is 1.3 m/s.
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).
The maximum playing time of standard music disc is 74 min and 33 s. How many revolution does the disc make during that time?

22. Example: 4, contd….
𝜃= 𝜋 = 𝑟𝑒𝑣.𝑠

23. Torque Ft r m
t = r xFt = rF Sinθ
SI unit: N.m

25. Torque and its effect 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

26. Rotational Law of Acceleration
The force law can be combined with rotational motion.
Torque: t = r Ft = r m at = m r 2 a
If torque replaces force, and angular acceleration replaces acceleration, this looks like the law of acceleration.
Where, I is the moment of inertia of the body.

27. 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 = 𝑖=1 𝑛 𝑚 𝑖 𝑟 𝑖 2 =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.

28. Moment of Inertia or Rotational Inertia
Diagram
Axis of Rotation
Name of the body
Sl. No
I = mr2
Passing thro’ its center and perpendicular to its plane
Circular ring of radius R 1
I = 𝟏 𝟐 mr2
Diameter
Circular ring/disc of radius R 2
I = 𝟐 𝟑 mr2
Tangent parallel to diameter 3
I = 𝟐 𝟓 mr2
Sphere 4

29. 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 = 𝒓 𝒙 𝑷 L = r p sinθ where 𝑝 =𝑚 𝑣 , linear momentum of the body. SI unit: kg m2/s Dimensional Formula=M1L2T-1

30. 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

31. Conceptual Question you Who has the greatest tangential velocity?
1. You do 2. Your friend does 3. Same
CORRECT
This is like the example of the "crack-the-whip." 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 = w R (w is the same but R is larger)

32. Conceptual Question Who has the greatest centripetal acceleration? you
1. You do 2. Your friend does 3. Same
you
CORRECT
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 w2 and you have the largest radius.

33. 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
W = F r θ =(F r) θ, t = F r
W = t θ
Work done in rotational motion O F

34. Angular/Rotational Motion
Power
Let t be the time taken by the torque to produce angular displacement θ, then Power, P = 𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 𝑡𝑖𝑚𝑒 = 𝑊 𝑡 = 𝝉𝜽 𝒕 = 𝝉𝛚, 𝐀𝐧𝐠𝐮𝐥𝐚𝐫 𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲, 𝛚= 𝜽 𝒕 P = 𝝉𝛚
Sl. No
Linear Motion
Angular/Rotational Motion 1
W = F S
W = t θ 2
P = 𝑾 𝒕
P = 𝝉𝝎

35. 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
A rolling wheel is rotating with kinetic energy.
The axis of rotation is at the center of mass.
K.Erot = ½ I w2 v w

36. 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

37. 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 q

38. 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.

39. Example : 5

40. Reference: http://hyperphysics.phy-astr.gsu.edu/hbase/circ.html#rotcon