1. Rotation and angular momentum
Chapters 10, 11
Rotation and angular momentum

2. Rotation of a rigid body
We consider rotational motion of a rigid body about a fixed axis
Rigid body rotates with all its parts locked together and without any change in its shape
Fixed axis: it does not move during the rotation
This axis is called axis of rotation
Reference line is introduced

3. Angular position
Reference line is fixed in the body, is perpendicular to the rotation axis, intersects the rotation axis, and rotates with the body
Angular position – the angle (in radians or degrees) of the reference line relative to a fixed direction (zero angular position)

4. Angular displacement
Angular displacement – the change in angular position.
Angular displacement is considered positive in the CCW direction and holds for the rigid body as a whole and every part within that body

5. Angular velocity Average angular velocity
Instantaneous angular velocity – the rate of change in angular position

6. Angular acceleration Average angular acceleration
Instantaneous angular acceleration – the rate of change in angular velocity

7. Rotation with constant angular acceleration
Similarly to the case of 1D motion with a constant acceleration we can derive a set of formulas:

8. Relating the linear and angular variables: position
For a point on a reference line at a distance r from the rotation axis:
θ is measured in radians

9. Relating the linear and angular variables: speed
ω is measured in rad/s
Period

10. Relating the linear and angular variables: acceleration
α is measured in rad/s2
Centripetal acceleration

11. Rotational kinetic energy
We consider a system of particles participating in rotational motion
Kinetic energy of this system is
Then

12. Moment of inertia From the previous slide
Defining moment of inertia (rotational inertia) as
We obtain for rotational kinetic energy

13. Moment of inertia: rigid body
For a rigid body with volume V and density ρ(V) we generalize the definition of a rotational inertia:
This integral can be calculated for different shapes and density distributions
For a constant density and the rotation axis going through the center of mass the rotational inertia for 8 common body shapes is given in Table 10-2 (next slide)

14. Moment of inertia: rigid body

15. Moment of inertia: rigid body
The rotational inertia of a rigid body depends on the position and orientation of the axis of rotation relative to the body

16. Chapter 10
Problem 25
Four equal masses m are located at the corners of a square of side L, connected by essentially massless rods. Find the rotational inertia of this system about an axis (a) that coincides with one side and (b) that bisects two opposite sides.

17. Parallel-axis theorem
Rotational inertia of a rigid body with the rotation axis, which is perpendicular to the xy plane and going through point P:
Let us choose a reference frame, in which the center of mass coincides with the origin

18. Parallel-axis theorem

19. Parallel-axis theorem

20. Parallel-axis theorem

21. Chapter 10
Problem 51
A uniform rectangular flat plate has mass M and dimensions a by b. Use the parallel-axis theorem in conjunction with Table 10.2 to show that its rotational inertia about the side of length b is Ma2/3.

22. Torque
We apply a force at point P to a rigid body that is free to rotate about an axis passing through O
Only the tangential component Ft = F sin φ of the force will be able to cause rotation

23. Torque
The ability to rotate will also depend on how far from the rotation axis the force is applied
Torque (turning action of a force):
SI unit: N*m (don’t confuse with J)

24. τ = F r┴ Torque Torque: Moment arm: r┴= r sinφ
Torque can be redefined as:
force times moment arm
τ = F r┴

25. Newton’s Second Law for rotation
Consider a particle rotating under the influence of a force
For tangential components
Similar derivation for rigid body

26. Newton’s Second Law for rotation

27. Chapter 10
Problem 57
A 2.4-kg block rests on a slope and is attached by a string of negligible mass to a solid drum of mass 0.85 kg and radius 5.0 cm, as shown in the figure. When released, the block accelerates down the slope at 1.6 m/s2. Find the coefficient of friction between block and slope.

28. Rotational work
Work
Power
Work – kinetic energy theorem

29. Corresponding relations for translational and rotational motion

30. ds/dt = d(θR)/dt = R dθ/dt vCM = ds/dt = ωR
Smooth rolling
Smooth rolling – object is rolling without slipping or bouncing on the surface
Center of mass is moving at speed vCM
Point of momentary contact between the two surfaces is moving at speed vCM
s = θR
ds/dt = d(θR)/dt = R dθ/dt
vCM = ds/dt = ωR

31. Rolling: translation and rotation combined
Rotation – all points on the wheel move with the same angular speed ω
Translation – all point on the wheel move with the same linear speed vCM

32. Rolling: translation and rotation combined

33. Rolling: pure rotation
Rolling can be viewed as a pure rotation around the axis P moving with the linear speed vcom
The speed of the top of the rolling wheel will be
vtop = (ω)(2R)
= 2(ωR) = 2vCM

34. fs opposes tendency to slide
Friction and rolling
Smooth rolling is an idealized mathematical description of a complicated process
In a uniform smooth rolling, P is at rest, so there’s no tendency to slide and hence no friction force
In case of an accelerated smooth rolling
aCM = α R
fs opposes tendency to slide

35. Rolling down a ramp

36. Chapter 10
Problem 39
What fraction of a solid disk’s kinetic energy is rotational if it’s rolling without slipping?

37. Vector product of two vectors
The result of the vector (cross) multiplication of two vectors is a vector
The magnitude of this vector is
Angle φ is the smaller of the two angles between and

38. Vector product of two vectors
Vector is perpendicular to the plane that contains vectors and and its direction is determined by the right-hand rule
Because of the right-hand rule, the order of multiplication is important (commutative law does not apply)
For unit vectors

39. Vector product in unit vector notation

40. Torque revisited
Using vector product, we can redefine torque (vector) as:

41. Angular momentum
Angular momentum of a particle of mass m and velocity with respect to the origin O is defined as
SI unit: kg*m2/s

42. Newton’s Second Law in angular form

43. Angular momentum of a system of particles

44. Angular momentum of a rigid body
A rigid body (a collection of elementary masses Δmi) rotates about a fixed axis with constant angular speed ω
For sufficiently symmetric objects:

45. Conservation of angular momentum
From the Newton’s Second Law
If the net torque acting on a system is zero, then
If no net external torque acts on a system of particles, the total angular momentum of the system is conserved (constant)
This rule applies independently to all components

46. Conservation of angular momentum

47. Conservation of angular momentum

48. More corresponding relations for translational and rotational motion

49. Chapter 11
Problem 28
A skater has rotational inertia 4.2 kg·m2 with his fists held to his chest and 5.7 kg·m2 with his arms outstretched. The skater is spinning at 3.0 rev/s while holding a 2.5-kg weight in each outstretched hand; the weights are 76 cm from his rotation axis. If he pulls his hands in to his chest, so they’re essentially on his rotation axis, how fast will he be spinning?

50. Questions?

51. Answers to the even-numbered problems
Chapter 10
Problem 24
0.072 N⋅m

52. Answers to the even-numbered problems
Chapter 10
Problem 30
2.58 × 1019 N⋅m

53. Answers to the even-numbered problems
Chapter 10
Problem 40
hollow

54. Answers to the even-numbered problems
Chapter 11
Problem 16
69 rad/s; 19° west of north

55. Answers to the even-numbered problems
Chapter 11
Problem 18
(a) 8.1 N⋅m kˆ
(b) 15 N⋅m kˆ

56. Answers to the even-numbered problems
Chapter 11
Problem 24
1.7 × 10-2 J⋅s

57. Answers to the even-numbered problems
Chapter 11
Problem 26
(a) 1.09 rad/s
(b) 386 J

58. Answers to the even-numbered problems
Chapter 11
Problem 30
along the x-axis
or 120° clockwise from the x-axis

59. Answers to the even-numbered problems
Chapter 11
Problem 42
26.6°