1. Chapter 6 Angular Momentum
Physics Beyond 2000
Chapter 6
Angular Momentum

2. Rotational Motion
The body is rigid. (i.e. It does not suffer deformation by external forces.)
The forces on the body may act at different points.

3. The Kinematics of Rotation
Axis of rotation – the body is rotating about a fixed axis.
side view
axis of rotation

4. The Kinematics of Rotation
Axis of rotation – the body is rotating about a fixed axis. ω
axis of
rotation
top view

5. The Kinematics of Rotation
Angular displacement – The reference line moves an angle Δθ about the axis of rotation. ω Δθ
axis of
rotation
top view

6. The Kinematics of Rotation
Average angular speed - ω Δθ
axis of
rotation
top view

7. The Kinematics of Rotation
Instantaneous angular speed - ω Δθ
axis of
rotation
top view

8. The Kinematics of Rotation
Example 1 – Find the angular speed.

9. The Kinematics of Rotation
Average angular acceleration -

10. The Kinematics of Rotation
Instantaneous angular acceleration -

11. The Kinematics of Rotation
Constant angular acceleration α
ωo = initial angular velocity
ω = final angular velocity
θ = angular displacement
t = time taken

12. The Kinematics of Rotation
Constant angular acceleration α
ωo = initial angular velocity
ω = final angular velocity
θ = angular displacement
t = time taken

13. The Kinematics of Rotation
Constant angular acceleration α
ωo = initial angular velocity
ω = final angular velocity
θ = angular displacement
t = time taken

14. The Kinematics of Rotation
Constant angular acceleration α

15. The Kinematics of Rotation
Note that the following quantities, except time t, are vectors.

16. The Kinematics of Rotation
We may use + and – signs to indicate the direction of the vectors.

17. The Kinematics of Rotation
Example 2 – to find the angular acceleration.
The negative sign of α indicates that it is in opposite direction to the positive angular velocity. ω α O

18. Linear Acceleration
When the object is rotating, it has two components of linear acceleration.
Tangential acceleration at
It is the linear acceleration along the tangent.
Radial acceleration ar
It is the centripetal acceleration pointing radially inwards.

19. Tangential acceleration
at = r. α
It changes the angular velocity. at r O

20. Radial acceleration at r ar O

21. Linear velocity and angular velocityvA vB ω rA O A B rB
Points A and B have the same angular velocity but different linear velocities.

22. Linear acceleration and angular accelerationvA vB ω rA O A B rB
Points A and B have the same angular acceleration but different linear tangential accelerations. aA aB

23. Example 3
Find the tangential acceleration. A r
equator

24. Kinetic energy of a rotating object
A rigid body of mass M is rotating about a fixed axis at angular speed ω.
Treat the body as a composition of N particles. ω
axis of rotation

25. Kinetic energy of a rotating object
The ith particle has mass mi and speed vi
The distance of the ith particle from the axis of rotation is ri
Note that ω vi mi ri

26. Kinetic energy of a rotating object
The kinetic energy of the ith particle is ω vi mi ri

27. Kinetic energy of a rotating object
The kinetic energy of all N particles is ω vi mi ri

28. Kinetic energy of a rotating object
The rotational kinetic energy Kr of the rigid body is ω vi mi ri

29. Kinetic energy of a rotating object
Define ω vi mi ri
I is called the moment of
inertia of the body about
this axis of rotation.

30. Moment of inertia The value of I depends on the mass of the body
the way the mass is distributed
the axis of rotation ω mi ri
axis of
rotation

31. Example 4
Find the moment of inertia and thus the rotational kinetic energy.
Change the axis of rotation and find the moment of inertia.
axis of rotation

32. Radius of gyration For a rotating body, its I  M.
So we cab write I = M.k2.
The k is known as the radius of gyration of the body about the given axis.

33. Example
Find the radius of gyration k.

34. Experiment to determine the moment of inertia of a flywheel
Supplement Ch.6
The gravitational potential
energy of the weight is converted
into the rotational kinetic energy
of the flywheel and the kinetic
energy of the weight.
However there is loss of energy
due to friction.

35. Table for Moment of Inertia
Hoop about cylindrical axis
I = MR2
Hoop about any diameter
I = MR2
M = mass of the hoop
R = radius of the hoop

36. Table for Moment of Inertia
Solid Cylinder about cylindrical axis
I = MR2
Solid Cylinder about central diameter
I = MR2 + ML2
M = mass of the cylinder
R = radius of the cylinder
L = length of the cylinder

37. Table for Moment of Inertia
Thin Rod about axis through centre perpendicular to its length
I = ML2
Thin Rod about axis through one end and perpendicular to its length
I = ML2
M = mass of the rod
L = length of the rod

38. Table for Moment of Inertia
Solid sphere about any diameter
I = MR2
Hollow sphere about any diameter
I = MR2
M = mass of the sphere
R = radius of the sphere

39. Parallel Axes Theorem m is the mass of the object
G is the centre of gravity
of the object G
IG is the moment of inertia
about the axis through the
centre of gravity h
IP is the moment of inertia
about the axis through the
point P. G P
New axis of rotation

40. Parallel Axes Theorem G Note that the two axes are parallel. h G P
New axis of rotation

41. Parallel Axes Theorem
Example 7 IG G IP h G P
New axis of rotation

42. Perpendicular Axes Theorem
For a lamina lying in the x-y plane, the moment
of inertia IX , IY and IZ, about three mutually
perpendicular axes which meets at the same
point are related by
IZ = IX + IY

43. Perpendicular Axes Theorem
IZ = IX + IY
Example 8

44. Moment of force Γ
Moment of force (torque) It is the product of a force and its perpendicular distance from a point about which an object rotates.
Unit: Nm F O
axis of rotation

45. Moment of force Γ F
axis of rotation O
Top view O F
Γ = F  d

46. Moment of force Γ The force F acts at point P of the object.
The distance vector from O, the point of rotation, to P is r.
θis the angle between the force F and the distance vector.
 Γ = F.r.sinθ
Top view O F θ r P
Γ = F  d

47. Moment of force Γ Moment of force Γis a vector.
In the following diagram, the moment of force is an anticlockwise moment.
It produces an angular acceleration α in clockwise direction.
Top view O F r P θ α
Γ = F  d

48. Moment of force on a flywheel
A force F acts tangentially on the
rim of a flywheel. F r
Γ = F × r

49. Work done by a torque Suppose a force F acts at right angle to the
distance vector r. F r O

50. Work done by a torque What is the moment of force about O? Γ= F × r O

51. Work done by a torque The moment of force turns the object through
an angle θ with a displacement s. F F r θ O
Γ= F × r s

52. Work done by a torque What is the work done by the force? W= F × s F Oθ O s
W= F × s

53. Work done by a torque Express the work done by the force in terms
of Γ and θ.
Use
F =
and
s = r. θ F r θ O s
W = F × s
= Γ× θ

54. Example 9
Work done against the moment of friction is equal to the loss of rotational kinetic energy
of the flywheel.

55. Torque and Angular acceleration
 = . 
 is the torque
I is the moment of inertia
 is the angular acceleration
Compare to F = m.a in linear motion.

56. Torque and Angular acceleration
In an angular motion with uniform angular acceleration :

57. Example 10
Torque and angular acceleration

58. Conditions for equilibrium
A body will be in static equilibrium, if
1. net force is zero
2. net moment of force about any point is zero

59. Angular momentum L
The angular momentum L of an object about an axis is the product of the angular velocity  and its moment of inertia.
L = I.
Unit of L: kg m2 s-1 or Nms.
L is a vector. Its direction is determined by the direction of the angular velocity .

60. Angular momentum of a rotating point mass
A point mass m is rotating tangentially at speed v at a distance r from an axis.
From I = mr2 , L = I and v = r   v r m
axis of
rotation

61. Example 11
Find the angular momentum of a solid sphere.

62. Newton’s 2nd law for rotation
The torque  acting on a rotating body is equal to the time rate of change of the angular momentum.

63. Newton’s 2nd law for rotation
If the net torque is zero, the angular momentum is a constant. The angular acceleration is zero.

64. Example 12
Find the change in angular momentum.

65. Torsional pendulum A disk is suspended by a wire.
The wire is twisted through an angle θ
The restoring torque is
Γ= c. θ where c is the torsional constant.
wire
disk

66. Torsional pendulum The restoring torque is
Γ= c. θ where c is the torsional constant.
Prove that the torsional oscillation is a SHM with the equation
wire
disk

67. Torsional pendulum
and
wire
disk

68. Typical examples of second law
Flywheel with moment of inertia I. r
mass m
axis α
Find the angular acceleration α
in terms of I, m and r.

69. Typical examples of second law
Flywheel with moment of inertia I.
mass m T mg a r
axis α T
T.r = I. α
a = r. α
mg – T = ma

70. Typical examples of second law
Smooth pulley with moment of inertia I and radius r. α
Find the linear acceleration
a of the two masses in terms
of m1, m2, I and r. r m1 m2 a a

71. Typical examples of second law
Smooth pulley with moment of inertia I and radius r. m2 a T2
m2g α m1 a T1
m1g r T2 T1
T2.r-T1.r = I.α
T1-m1g = m1a
m2g-T2 = m2a
a = rα

72. The law of conservation of angular momentum
If external net torque = 0, the sum of angular momentum of the system is zero.
If Γ=0,

73. The law of conservation of angular momentum
For a system with initial moment of inertia I1 and initial angular velocity ω1, its initial angular momentum is I1ω1.
If the system changes its moment of inertia to I2 and angular velocity ω2, its final angular momentum is I2ω2.
If there is not any external net torque, then
I1ω1 = I2ω2