1. Chapter 9 Rotation of rigid bodies

2. Radian Vs Degree

3. Using Radian

4. Example

5. What is  in radians and in degrees?

6. Angular velocity

7. v and ω The dot goes around the circle of radius r=2m once every 10s. What is its v and ω ?

8. Meaning of different variables

9. Period T and frequency f

10. Must remember this: Using this you can convert freely among ω, T, f and v.

11. Example Given r =2m, T=10s, find v and ω.

12. Example Given r =2m, f = 20rpm, find v and ω.

13. Example A car takes 5 minutes to complete one circle on the race track. What is its angular velocity?

14. Equations of Circular Motion (Must Memorize) r v x y (x,y)

15. Units

16. Angular acceleration α

17. Equations of Motion LinearCircular

18. Example: Wheel Radius r=2m Angular acceleration: α=3 rad/s 2 Initial angular velocity ω 0 =0 rad/s After 5s, what is the final angular velocity? What angle has the wheel rotated. How far has the wheel rolled on the ground?

19. Example A cyclist traveling at 5m/s accelerates up to 10m/s in 2s. Each tire has r =0.35m. A small pebble is stuck on the tire. (a) What is the linear acceleration of the bike? (b) What is the angular acceleration of the pebble?

20. Example (Cont.) (c) What is the initial angular velocity? A cyclist traveling at 5m/s accelerates up to 10m/s in 2s. Each tire has r =0.35m. A small pebble is stuck on the tire.

21. Example (Cont.) (d) Through what angle does the pebble revolve? (e) How far around the wheel has the pebble traveled? A cyclist traveling at 5m/s accelerates up to 10m/s in 2s. Each tire has r =0.35m. A small pebble is stuck on the tire.

22. Angular velocity as a vector

23. Angular acceleration as a vector

24. Moment of Inertia m2m2 m3m3 m1m1 1m 2m 5m r is the distance from the axis of rotation.

25. Dependence on axis of rotation The moment of inertia will depends on the axis of rotation.

26. Example 1 To find I, ask yourself: What is the distance r of each mass from the axis of rotation?

27. Example 2 To find I, ask yourself: What is the distance r of each mass from the axis of rotation?

28. Parallel axis theorem

29. The Parallel axis theorem

30. Find I

31. General formula for I Different objects with different geometry will have different formula for I. See the end of the lecture notes for detail.

32. I for extended objects

33. Rotational Energy

34. Example

35. A sphere rolling down from rest. Find velocity at the bottom.

36. Example

37. Example (modified) What is the sphere slides down instead?

38. Moment of inertia calculations Below we will use integration to calculate I for some basic geometrical objects.

39. Example: The Ring R dm ✕ “ ✕ ” represent rotational axis

40. Sequel: The Disk r R dm ✕

41. Another Sequel: The Rod dm r Note that we defined the pivotal point to be on the left ✕

42. Moving the pivotal point r What if I move the pivotal point to the middle? ✕ dm

43. Where you place the rotational axis matters!!! ✕ ✕ ≠