1. Schedule Mechanics Lecture 14, Slide 1 Midterm 3 Friday April 24 Collisions and Rotations Units 10-18 Final Exam Units 1-19 Section 001 Section 002

2. Additional “Fun” with Smartphysics… Mechanics Lecture 16, Slide 2

3. Classical Mechanics Lecture 16 Rotational Dynamics Today’s Concepts: a) Rolling Kinetic Energy b) Angular Acceleration Physics 211 Lecture 16, Slide 3

4. Main Points Mechanics Lecture 16, Slide 4

5. Main Points Mechanics Lecture 16, Slide 5 Rolling without slipping 

6. Disk and String Mechanics Lecture 16, Slide 6

7. Main Points Mechanics Lecture 15, Slide 7

8. Vector Cross Product Mechanics Lecture 15, Slide 8 http://en.wikipedia.org/wiki/Cross_product

9. Work and Energy for Rotations Mechanics Lecture 16, Slide 9

10. Total Kinetic Energy of Rolling Ball Mechanics Lecture 16, Slide 10

11. Disk and String Mechanics Lecture 16, Slide 11

12. Total Kinetic Energy = Translational Kinetic Energy + Rotational Kinetic Energy Energy Conservation Rotational Kinetic Energy Mechanics Lecture 16, Slide 12 H Rolling without slipping 

13. Newton’s Second Law  Mg f a Acceleration of Rolling Ball Mechanics Lecture 16, Slide 13 Newton’s 2 nd Law for rotations Rolling without slipping 

14. Rolling Motion Objects of different I rolling down an inclined plane: v  0 0K  0v  0 0K  0   K   U  M g h R M h v = Rv = R Mechanics Lecture 16, Slide 14

15. Rolling If there is no slipping: In the lab reference frame In the CM reference frame v v v Where v  R  Mechanics Lecture 16, Slide 15

16. Rolling Use v  R and I  cMR 2. Doesn’t depend on M or R, just on c (the shape) Hoop: c  1 Disk: c  1/2 Sphere: c  2/5 etc... c c v So: c c Mechanics Lecture 16, Slide 16

17. Clicker Question A. B. C. Mechanics Lecture 16, Slide 17 A) B) C) A hula-hoop rolls along the floor without slipping. What is the ratio of its rotational kinetic energy to its translational kinetic energy? Recall that I  MR 2 for a hoop about an axis through its CM :

18. A block and a ball have the same mass and move with the same initial velocity across a floor and then encounter identical ramps. The block slides without friction and the ball rolls without slipping. Which one makes it furthest up the ramp? A) Block B) Ball C) Both reach the same height. CheckPoint v v  Mechanics Lecture 16, Slide 18

19. The block slides without friction and the ball rolls without slipping. Which one makes it furthest up the ramp? A) Block B) Ball C) Same v v  B) The ball has more total kinetic energy since it also has rotational kinetic energy. Therefore, it makes it higher up the ramp. Mechanics Lecture 16, Slide 19 CheckPoint

20. Rolling vs Sliding Mechanics Lecture 16, Slide 20 Rolling Ball Sliding Block Ball goes 40% higher! Rolling without slipping 

21. CheckPoint A cylinder and a hoop have the same mass and radius. They are released at the same time and roll down a ramp without slipping. Which one reaches the bottom first? A) Cylinder B) Hoop C) Both reach the bottom at the same time Mechanics Lecture 16, Slide 21

22. A) Cylinder B) Hoop C) Both reach the bottom at the same time A) same PE but the hoop has a larger rotational inertia so more energy will turn into rotational kinetic energy, thus cylinder reaches it first. Which one reaches the bottom first? Mechanics Lecture 16, Slide 22 Hoop: c  1 Disk: c  1/2

23. Mechanics Lecture 16, Slide 23

24. CheckPoint A small light cylinder and a large heavy cylinder are released at the same time and roll down a ramp without slipping. Which one reaches the bottom first? A) Small cylinder B) Large cylinder C) Both reach the bottom at the same time Mechanics Lecture 16, Slide 24 Hoop: c  1 Disk: c  1/2

25. A small light cylinder and a large heavy cylinder are released at the same time and roll down a ramp without slipping. Which one reaches the bottom first? A) Small cylinder B) Large cylinder C) Both reach the bottom at the same time C) The mass is canceled out in the velocity equation and they are the same shape so they move at the same speed. Therefore, they reach the bottom at the same time. Mechanics Lecture 16, Slide 25 CheckPoint Hoop: c  1 Disk: c  1/2

26. Mechanics Lecture 16, Slide 26

27. Mechanics Lecture 16, Slide 27 v f R M 

28. Mechanics Lecture 16, Slide 28 v f R M a

29. v  a R M t Once v  R it rolls without slipping t v0v0  R   Rt   t Mechanics Lecture 16, Slide 29 t v v0v0

30. v  a R M Plug in a and t found in parts 2) & 3) Mechanics Lecture 16, Slide 30

31. v  a R M We can try this… Interesting aside: how v is related to v 0 : Mechanics Lecture 16, Slide 31 Doesn’t depend on 

32. v  f R M Mechanics Lecture 16, Slide 32

33. Three Masses Mechanics Lecture 16, Slide 33

34. Three Masses Mechanics Lecture 16, Slide 34

35. Three Masses Mechanics Lecture 16, Slide 35

36. Three Masses Mechanics Lecture 16, Slide 36

37. Three Masses 2 Mechanics Lecture 16, Slide 37

38. Three Masses 2 Mechanics Lecture 16, Slide 38

39. Atwood's Machine with Massive Pulley: A pair of masses are hung over a massive disk-shaped pulley as shown.  Find the acceleration of the blocks. For the hanging masses use F  ma  m 1 g  T 1  m 1 a  m 2 g  T 2  m 2 a For the pulley use   I  T 1 R  T 2 R (Since for a disk) Mechanics Lecture 16, Slide 39 y x m2gm2g m1gm1g a T1T1 a T2T2  R M m1m1 m2m2

40. We have three equations and three unknowns ( T 1, T 2, a ). Solve for a.  m 1 g  T 1   m 1 a (1)  m 2 g  T 2  m 2 a (2) T 1  T 2 (3) Mechanics Lecture 16, Slide 40 y x m2gm2g m1gm1g a T1T1 a T2T2  R M m1m1 m2m2 Atwood's Machine with Massive Pulley:

41. Torque and Horsepower Demos Mechanics Lecture 15, Slide 41 https://www.youtube.com/watch?v=zQPv23R9VGM Wheelie Contest!!!!

42. Let’s analyze a wheelie Mechanics Lecture 15, Slide 42 Camaro Center of mass Wheel Torque Gravitational Torque Wheelie Axle rotation axis

43. Torque-gears and axles! Mechanics Lecture 15, Slide 43 http://shell.cas.usf.edu/~mccolm/pedagogy/HWanalysissynthesis.html Roll without slipping Newton’s Third Law Linear acceleration Freewwheel hub…

44. Homework 15 Mechanics Lecture 16, Slide 44 Average-all: 72.03% Average-attempted: 90.04%