1. Physics 218, Lecture XXII1 Physics 218 Lecture 22 Dr. David Toback

2. Physics 218, Lecture XXII2 Checklist for Today Things due Monday –Chapters 12 & 13 in WebCT Things that are due Tuesday –Read Chapters 14-16 Things that were due yesterday –Chapter 14 problems –Read Lab hand out on webpage Things due next Monday –Chapter 14 in WebCT Next Tuesday –Exam 3, Chapters 10-13 –Mini-practice exam and bonus points available

3. Physics 218, Lecture XXII3 The Schedule This week (4/7) Mon: Chapter 12 & 13 material due in WebCT Tues: Reading: Chap 14-16 Wed: Recitation on Chap 14, Lab Today: Chap 15, Part 1 Next Week (4/14) Monday: Chapter 14 due in WebCT Tues: Exam 3 (Chaps 10-13) Wed: Recitation on Chap 15, Lab Thurs: Lecture on Chap 15, Part 2 Week after that (4/21) Monday: Chapter 15 & 16 due in WebCT Tues: Reading for Chapter 18 Tues: Lecture on Chapter 18 Wed: Recitation on Chapter 18, Lab Thurs: Last lecture, Chapter 18 Week after that (4/28) No lectures or recitations Week after that (5/5) Final: Monday May 5 th, 1PM-3PM in this room

4. Physics 218, Lecture XXII4 Overview Chapters 12-16 are about Rotational Motion While we’ll do Exam 3 on Chapters 10- 13, we’ll do the lectures on 12-16 in six combined lectures Give extra time after the lectures to Study for the exam The book does the math, I’ll focus on the understanding and making the issues more intuitive

5. Physics 218, Lecture XXII5 Rotational Motion Chapters 12 through 16 in six combined lectures This is the 5 th of the 6 lectures Concentrate on the relationship between linear and angular variables Today: Finish up topics Next time: Hard problems

6. Physics 218, Lecture XXII6

7. 7 Angular Quantities Position  Angle  Velocity  Angular Velocity  Acceleration  Angular Acceleration  Moving forward: –Force  Torque  –Mass –Momentum –Energy

8. Physics 218, Lecture XXII8 Analogue of Mass The analogue of Mass is called Moment of Inertia Example: A ball of mass m moving in a circle of radius R around a point has a moment of inertia F=ma   = 

9. Physics 218, Lecture XXII9 Calculate Moment of Inertia Calculate the moment of inertia for a ball of mass m relative to the center of the circle R

10. Physics 218, Lecture XXII10 Moment of Inertia To find the mass of an object, just add up all the little pieces of mass  To find the moment of inertia around a point, just add up all the little moments

11. Physics 218, Lecture XXII11 Torque and Moment of Inertia Force vs. Torque F=ma   = I  Mass vs. Moment of Inertia

12. Physics 218, Lecture XXII12 Pulley and Bucket A heavy pulley, with radius R, and known moment of inertia I starts at rest. We attach it to a bucket with mass m. The friction torque is  fric. Find the angular acceleration 

13. Physics 218, Lecture XXII13 Angular Quantities Position  Angle  Velocity  Angular Velocity  Acceleration  Angular Acceleration  Force  Torque  Mass  Moment of Inertia  Today we’ll finish: –Momentum –Energy

14. Physics 218, Lecture XXII14 Momentum Momentum vs. Angular Momentum: Newton’s Laws:

15. Physics 218, Lecture XXII15 Angular Momentum First way to define the Angular Momentum L:

16. Physics 218, Lecture XXII16 Angular Momentum Definition Another definition:

17. Physics 218, Lecture XXII17 Angular Motion of a Particle Determine the angular momentum, L, of a particle, with mass m and speed v, moving in circular motion with radius r

18. Physics 218, Lecture XXII18 Conservation of Angular Momentum By Newton’s laws, the angular momentum of a body can change, but the angular momentum for a system cannot change Conservation of Angular Momentum Same as for linear momentum

19. Physics 218, Lecture XXII19 Ice Skater This one you’ve seen on TV Try this at home in a chair that rotates Get yourself spinning with your arms and legs stretched out, then pull them in

20. Physics 218, Lecture XXII20 Problem Solving For Conservation of Angular Momentum problems: BEFORE and AFTER

21. Physics 218, Lecture XXII21 Conservation of Angular Momentum Before

22. Physics 218, Lecture XXII22 Conservation of Angular Momentum After

23. Physics 218, Lecture XXII23 Clutch Design As a car engineer, you model a car clutch as two plates, each with radius R, and masses M A and M B (I Plate = ½MR 2 ). Plate A spins with speed  1 and plate B is at rest. you close them so they spin together Find the final angular velocity of the system

24. Physics 218, Lecture XXII24 Angular Quantities Position  Angle  Velocity  Angular Velocity  Acceleration  Angular Acceleration  Force  Torque  Mass  Moment of Inertia  Today we’ll finish: –Momentum  Angular Momentum L –Energy

25. Physics 218, Lecture XXII25 Rotational Kinetic Energy KE trans = ½mv 2  KE rotate = ½I  2 Conservation of Energy must take rotational kinetic energy into account

26. Physics 218, Lecture XXII26 Rotation and Translation Objects can both Rotate and Translate Need to add the two KE total = ½ mv 2 + ½I  2 Rolling without slipping is a special case where you can relate the two  V =  r

27. Physics 218, Lecture XXII27 Rolling Down an Incline You take a solid ball of mass m and radius R and hold it at rest on a plane with height Z. You then let go and the ball rolls without slipping. What will be the speed of the ball at the bottom? What would be the speed if the ball didn’t roll and there were no friction? Note: I sphere = 2/5MR 2 Z

28. Physics 218, Lecture XXII28 A bullet strikes a cylinder A bullet of speed V and mass m strikes a solid cylinder of mass M and inertia I=½MR 2, at radius R and sticks. The cylinder is anchored at point 0 and is initially at rest. What is  of the system after the collision? Is energy Conserved?

29. Physics 218, Lecture XXII29 Rotating Rod A rod of mass uniform density, mass m and length l pivots at a hinge. It has moment of inertia I=ml/3 and starts at rest at a right angle. You let it go: What is  when it reaches the bottom? What is the velocity of the tip at the bottom?

30. Physics 218, Lecture XXII30 Less Spherical Heavy Pulley A heavy pulley, with radius R, starts at rest. We pull on an attached rope with constant force F T. It accelerates to final angular speed  in time t. A better estimate takes into account that there is friction in the system. This gives a torque (due to the axel) we’ll call this  fric. What is this better estimate of the moment of Inertia? R

31. Physics 218, Lecture XXII31 Person on a Disk A person with mass m stands on the edge of a disk with radius R and moment ½MR 2. Neither is moving. The person then starts moving on the disk with speed V. Find the angular velocity of the disk

32. Physics 218, Lecture XXII32 Same Problem: Forces Same problem but with Forces

33. Physics 218, Lecture XXII33 Next Time Exam 3!!! Covers Chapters 10-13 Get caught up on your homework!!! Mini-practice exam 3 is now available Thursday: - Finish up angular “Stuff”

34. Physics 218, Lecture XXII34

35. Physics 218, Lecture XXII35 Spherical Heavy Pulley A heavy pulley, with radius R, starts at rest. We pull on an attached rope with a constant force F T. It accelerates to an angular speed of  in time t. What is the moment of inertia of the pulley? R

36. Physics 218, Lecture XXII36 Less Spherical Heavy Pulley A heavy pulley, with radius R, starts at rest. We pull on an attached rope with constant force F T. It accelerates to final angular speed  in time t. A better estimate takes into account that there is friction in the system. This gives a torque (due to the axel) we’ll call this  fric. What is this better estimate of the moment of Inertia? R

37. Physics 218, Lecture XXII37 Exam II Mean = 75 –Please check to make sure they added your points correctly AND entered them into WebCT correctly!!! Average on first two exam = 76% –Straight scale so far… Reading quizzes should be passed back in recitation

38. Physics 218, Lecture XXII38 Next Time Chapter 11 –Reading Questions: Q11.X & Q11.X –XXX FIXME!!! –Math, Torque, Angular Momentum, Energy again, but more sophisticated –The material will not be on the 3 rd exam, but will help with the exam. It will all be on the final HW 10 Due Monday Exam 3 is next Thursday, April 22 nd

39. Physics 218, Lecture XXII39 Angular Quantities Position  Angle  Velocity  Angular Velocity  Acceleration  Angular Acceleration  Force  Torque  Today we’ll finish: –Mass –Momentum –Energy

40. Physics 218, Lecture XXII40 Calculating Moments of Inertia Here r is the distance from the axis of each little piece of mass

41. Physics 218, Lecture XXII41 Calculate the Moment of Inertia A pulley has mass M, uniform density, radius R, and rotates around its fixed axis Calculate its moment of inertia R

42. Physics 218, Lecture XXII42 Calculate the Moment of Inertia Better example here… Calculate its moment of inertia R

43. Physics 218, Lecture XXII43

44. Physics 218, Lecture XXII44 Hollow Cylinder Consider a hollow cylinder with uniform density, inner radius R 1, outer radius R 2 and total Mass M. Find the moment of Inertia

45. Physics 218, Lecture XXII45 Parallel-Axis Theorem Quick Trick for calculating Moments I = I cm + Mh 2 Example

46. Physics 218, Lecture XXII46 Old stuff

47. Physics 218, Lecture XXII47

48. Physics 218, Lecture XXII48 Kepler’s 2 nd Law 2 nd Law: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out area in equal periods of time

49. Physics 218, Lecture XXII49 Atwood’s Machine A pulley with a fixed center (at point O), radius R  and moment of inertia I, has a massless rope wrapped around it (no slipping). The rope has two masses, m 1 and m 2 attached to its ends. Assume m 2 >m 1 What is the acceleration of the system? Do some checks.

50. Physics 218, Lecture XXII50 Why does the Bicycle Wheel Turn to the Right?

51. Physics 218, Lecture XXII51 Angular Momentum Again we use the Cross Product: Derivation of  = dL/dt

52. Physics 218, Lecture XXII52 L for a system of many bodies Have to be careful with Angular Momentum –  = dl/dt for a single particle –  =  dl/dt) for a system of many particles –All internal torques cancel because of Newton’s law (all internal forces are equal and opposite) Reference Frame matters. Only true for: –The origin is an inertial Reference Frame –The center of mass

53. Physics 218, Lecture XXII53 L for a Rigid Body Find the angular momentum, L, for this body given that it is rotating around the Z axis with angular velocity 