1. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

2. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

3. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

4. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

5. Examples of rotation: Earth
To find the direction of omega, apply the right hand rule as follows:
With fingers curling in direction of rotation, the thumb gives direction of omega
i.e. direction of earth’s omega is upward

6. rotation of the sky
star trails
centered on Polaris
rotate once a day

7. rotation can be fun

8. sometimes the goal is rotational equilibrium
1st condition for equilibrium:
Fnet = 0
2nd condition for equilbrium:
torque net = 0
i.e., torque ccw = torque cw

9. rotational equilibrium again

10. sometimes the goal is large rotational velocity

11. M31 rotates once every few hundred million years

12. Katrina rotated

13. a generic kidney shaped object rotates about a fixed axis: a thing of beauty is a joy forever!

14. a turbine (driven by moving fluid) rotates about a fixed axis

15. a jet engine rotates about a fixed axis

17. steam driven power plant turbine: imagine this thing rotating at 60 hz generating your electricity

18. electric motors are backwards connected generators: they are still mechanical rotators about a fixed axis

19. electric motors power our elderly friends’ scooters

20. and nuclear subs also (when they are submerged)

21. gears rotate about a fixed axis

22. gear attached to an electric motor (sounds like a good idea to me)

23. nanotechology electric motor gear (the gear teeth are smaller than a red blood cell) rotating about a fixed axis, imaged with an electron microscope (end of examples of rotation)

24. Advanced Rotational Dynamics for AP Physics
+Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

25. Advanced Rotational Dynamics for AP Physics
+Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

26. Advanced Rotational Dynamics for AP Physics
+Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

27. Rotational Kinematics
Θ = angular position wrt arbitrary origin
Δθ = angular displacement (rad)
ω = Δθ / Δt = dθ / dt (rad/s)
α = Δ ω / Δt = d ω / dt = d2θ / dt2 (rad/s2)
s = r θ
v = r ω
a = r α

28. α = constant implies Δθ = ω0t + ½ αt2 ω = ω0 + α t ω2 = ω02 + 2 α Δθ
ω2 = ω α Δθ
Δθ = ½ (ω0 + ω) t
If α is variable, you need calculus

29. Intro Rotational Dynamics
Г = τ = r x F
I = Σ mi ri2 (collection of point masses)
= ∫ r2 dm (continuous matter distribution)
I total = I 1 + I 2 + I 3 + … (composite object)
Fnet = ma becomes Гnet = τnet = I α

30. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

31. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

32. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

33. Earth revolves (translation of cm) and rotates about cm

34. Look Ma, no hands

35. combined translation and rotation
Ktotal = Ktranslational + Krotational
= Kof the cm Karound the cm
= ½ mv ½ I ω2

36. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

37. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

38. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

39. linear and angular velocity and acceleration are proportional

40. rolling without slipping in our golden years

41. rolling without slipping
v = ω r (use with energy conservation)
atangential = α r (use with 2nd laws)
friction acts, but does no work
energy conserved as Wnc = 0

42. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping (pure rolling)
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

43. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping (pure rolling)
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

44. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping (pure rolling)
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

45. linear and angular velocities and accelerations are independent, i. e
linear and angular velocities and accelerations are independent, i.e., he’s not getting very much bang (v) for his buck (ω)

46. don’t try this at home

47. rolling with slipping v ≠ ω r atangential ≠ α r
apply Fnet = ma to find atangential of the cm
apply Гnet = Iα to find α around the cm
to compute t where pure rolling sets in,
set a(t) = α(t) r, where a(t) and α(t) are
solutions of force and torque equations

48. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

49. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

50. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

51. definition of angular momentum:
L = r x p = r x mv (point m moving at v)
L = I ω (rigid body with moment I)

52. law of motion in terms of L:
torque net = ΔL/ Δt = dL / dt

53. Conservation of angular momentum:
if torque net = 0, then dL / dt = 0
and so, L = constant
so, I initial ω initial = I final ω final
(ice skater)

54. easier said than done

55. The agony of defeat (net torque ≠ 0, L not conserved)

56. The thrill of victory (net torque = 0, L conserved)

57. Conservation of energy, including rotation
Simply allow for rotational
as well as translational K

58. Conservation of linear momentum
…is not affected by rotational motion, but applies to the cm of the body as before
(i.e., if Fnet,ext = 0 on system. That is,
if something is nailed down and rotates, linear momentum is not conserved, but L can be)

59. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

60. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

61. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions involving objects having Moment of Inertia

62. Advanced Rotational Dynamics for AP Physics
+Common Examples of Rotation
+Review of Introductory Rotational Dynamics
+Combined translation and rotation
+Rolling without slipping
+Rolling with slipping
+Rotational form of Conservation Laws
+Collisions (and other systems) involving objects having Moment of Inertia

63. …this is rather important on the AP Exam…

64. Unfortunately, few common practical applications exist for this type of problem.
They are simply artificially contrived

65. Any or all 3 conservation laws of energy, linear momentum, and angular momentum may be satisfied! You have to figure out which ones are operating, if they don’t tell you!
If Wnc = 0, then Einitial = Efinal
If Fnet,ext = 0, then pinitial = pfinal
If Гnet = 0, then Linitial = Lfinal

66. The End
That’s it! Now go make a 5!