1. Physics 218, Lecture XXIII1 Physics 218 Lecture 23 Dr. David Toback

2. Physics 218, Lecture XXIII2 Checklist for Today Things due Monday –Chapter 14 in WebCT Things that were due yesterday –Chapter 15 problems as Recitation Prep Things due next Monday –Chapter 15 & 16 in WebCT Next week –Chapter 18 reading

3. Physics 218, Lecture XXIII3 The Schedule This Week (4/14) Monday: Chapter 14 due in WebCT Tues: Exam 3 (Chaps 10-13) Wed: Recitation on Chap 15, Lab Thurs: Last lecture on Chaps 12-16 Next Week (4/21) Monday: Chapter 15 & 16 due in WebCT Tues: –Reading for Chapter 18 –Lecture on Chapter 18 Wed: Recitation on Chapter 18 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 XXIII4 Overview Chapters 12-16 are about Rotational Motion Concentrate on the relationship between linear and angular variables Do them in six combined lectures –Last lecture today –Angular Momentum and Energy The book does the math, I’ll focus on the understanding and making the issues more intuitive

5. Physics 218, Lecture XXIII5

6. 6 Angular Quantities Position  Angle  Velocity  Angular Velocity  Acceleration  Angular Acceleration  Force  Torque  Mass  Moment of Inertia  Today we’ll finish: –Momentum –Energy

7. Physics 218, Lecture XXIII7 Momentum Momentum vs. Angular Momentum: Newton’s Laws:

8. Physics 218, Lecture XXIII8 Angular Momentum Definition Another definition:

9. Physics 218, Lecture XXIII9 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

10. Physics 218, Lecture XXIII10 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

11. Physics 218, Lecture XXIII11 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

12. Physics 218, Lecture XXIII12 Problem Solving For Conservation of Angular Momentum problems: BEFORE and AFTER

13. Physics 218, Lecture XXIII13 Conservation of Angular Momentum Before

14. Physics 218, Lecture XXIII14 Conservation of Angular Momentum After

15. Physics 218, Lecture XXIII15 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

16. Physics 218, Lecture XXIII16 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

17. Physics 218, Lecture XXIII17 Rotational Kinetic Energy KE trans = ½mv 2  KE rotate = ½I  2 Conservation of Energy must take rotational kinetic energy into account

18. Physics 218, Lecture XXIII18 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

19. Physics 218, Lecture XXIII19 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

20. Physics 218, Lecture XXIII20 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?

21. Physics 218, Lecture XXIII21 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?

22. Physics 218, Lecture XXIII22 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

23. Physics 218, Lecture XXIII23 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

24. Physics 218, Lecture XXIII24 Same Problem: Forces Same problem but with Forces

25. Physics 218, Lecture XXIII25 Challenge Exam Announcement of this semester’s 218 Challenge Exam: Monday April 28 th at 6:00PM –Not required (just for fun) –Does not (will not!!!) count as part of your final grade Test your skills against the best Aggies from all the Physics 218 sections (not just this lecture) on Physics 218 material Students who perform well will be recognized as Mechanics Scholars, and honored at a banquet in their honor. Other prizes, including cash. Handout information on my WebSite

26. Physics 218, Lecture XXIII26 Next Week Reading: Chapter 18 “Harmonic Motion” –Last topic for the course Notes for the final: No more “bonus” points Exam is worth 200 points Problem 8 will consist of two items: –5 points for getting 100% on all your homework & WebCT quizzes –5 more points for getting 100 on the Mini-practice Exam final

27. Physics 218, Lecture XXIII27

28. Physics 218, Lecture XXIII28 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

29. Physics 218, Lecture XXIII29 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

30. Physics 218, Lecture XXIII30 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

31. Physics 218, Lecture XXIII31 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

32. Physics 218, Lecture XXIII32 Angular Quantities Position  Angle  Velocity  Angular Velocity  Acceleration  Angular Acceleration  Force  Torque  Today we’ll finish: –Mass –Momentum –Energy

33. Physics 218, Lecture XXIII33 Calculating Moments of Inertia Here r is the distance from the axis of each little piece of mass

34. Physics 218, Lecture XXIII34 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

35. Physics 218, Lecture XXIII35 Calculate the Moment of Inertia Better example here… Calculate its moment of inertia R

36. Physics 218, Lecture XXIII36

37. Physics 218, Lecture XXIII37 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

38. Physics 218, Lecture XXIII38 Parallel-Axis Theorem Quick Trick for calculating Moments I = I cm + Mh 2 Example

39. Physics 218, Lecture XXIII39 Old stuff

40. Physics 218, Lecture XXIII40 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

41. Physics 218, Lecture XXIII41 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.

42. Physics 218, Lecture XXIII42 Why does the Bicycle Wheel Turn to the Right?

43. Physics 218, Lecture XXIII43 Angular Momentum Again we use the Cross Product: Derivation of  = dL/dt

44. Physics 218, Lecture XXIII44 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

45. Physics 218, Lecture XXIII45 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 