1. Physics 218, Lecture XVIII1 Physics 218 Lecture 18 Dr. David Toback

2. Physics 218, Lecture XVIII2 Checklist for Today Things that were due Monday: –Chapter 9 HW on WebCT Things that are due yesterday for Recitation –Chapter 10 problems Things due Today: –Read Chapters 12 & 13 Things due next Monday –Chapter 10 in WebCT

3. Physics 218, Lecture XVIII3 The Schedule Rest of this Week: (3/24) Lecture on Chaps 12 & 13 Next Week (3/31) Mon: Chapter 10 & 11 due material in WebCT Tues: Second lecture on Chaps 12 & 13 Wed: Recitation on Chapters 12 & 13, Lab Reading for Thurs: Chapters 14-16 Thurs Lecture: Chap 14 Week after that (4/7) Mon: Chapter 12 & 13 due material in WebCT Tues: Second Lecture on Chap 14 Wed: Recitation on Chap 14, Lab Thurs Lecture: Chap 15, Part 1 Week of 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

4. Physics 218, Lecture XVIII4 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 XVIII5 Rotational Motion Start with Fixed Axis motion The relationship between linear and angular variables Rotating and translating at the same time First kinematics, then dynamics –just like earlier this semester

6. Physics 218, Lecture XVIII6

7. 7 Overview: Rotational Motion Take our results from “linear” physics and do the same for “angular” physics We’ll discuss the analogue of –Position –Velocity –Acceleration –Force –Mass –Momentum –Energy

8. Physics 218, Lecture XVIII8 Rotational Motion Here we’re talking about stuff that goes around and around Start by envisioning: A spinning object like a car tire

9. Physics 218, Lecture XVIII9 Some Buzz Phrases Fixed axis: I.e, an object spins in the same place… an ant on a spinning top goes around the same place over and over again Another example: Earth has a fixed axis, the sun Rigid body: I.e, the objects don’t change as they rotate. Example: a bicycle wheel Examples of Non-rigid bodies?

10. Physics 218, Lecture XVIII10 Overview: Rotational Motion Take our results from “linear” physics and do the same for “angular” physics Analogue of –Position ← –Velocity ← –Acceleration ← –Force –Mass –Momentum –Energy Start here! Chapters 1-3

11. Physics 218, Lecture XVIII11 Axis of Rotation: Definitions Pick a simple place to rotate around Call point O the “Axis of Rotation” Same as picking an origin

12. Physics 218, Lecture XVIII12 An Important Relation: Distance If we are sitting at a radius R relative to our axis, and we rotate through an angle , then we travel through a distance l

13. Physics 218, Lecture XVIII13 Velocity and Acceleration

14. Physics 218, Lecture XVIII14 Motion on a Wheel What is the linear speed of a point rotating around in a circle with angular speed , and constant radius R?

15. Physics 218, Lecture XVIII15 Examples Consider two points on a rotating wheel. One on the inside (P) and the other at the end (b): Which has greater angular speed? Which has greater linear speed? b R1R1 R2R2

16. Physics 218, Lecture XVIII16 Angular Velocity and Acceleration Are  and  vectors?  and  clearly have magnitude Do they have direction?

17. Physics 218, Lecture XVIII17 Right-Hand Rule Yes! Define the direction to point along the axis of rotation Right-hand Rule This is true for  and 

18. Physics 218, Lecture XVIII18 Uniform Angular Acceleration Derive the angular equations of motion for constant angular acceleration

19. Physics 218, Lecture XVIII19 Rotation and Translation Objects can both translate and rotate at the same time. They do both around their center of mass.

20. Physics 218, Lecture XVIII20 Rolling without Slipping In reality, car tires both rotate and translate They are a good example of something which rolls (translates, moves forward, rotates) without slipping Is there friction? What kind?

21. Physics 218, Lecture XVIII21 Derivation The trick is to pick your reference frame correctly! Think of the wheel as sitting still and the ground moving past it with speed V. Velocity of ground (in bike frame) = -  R => Velocity of bike (in ground frame) =  R

22. Physics 218, Lecture XVIII22 Bicycle comes to Rest A bicycle with initial linear velocity V 0 (at t 0 =0) decelerates uniformly (without slipping) to rest over a distance d. For a wheel of radius R: a)What is the angular velocity at t 0 =0? b)Total revolutions before it stops? c)Total angular distance traversed by the wheel? d)The angular acceleration? e)The total time until it stops?

23. Physics 218, Lecture XVIII23 Show Show for constant acceleration that :

24. Physics 218, Lecture XVIII24 Uniform Circular Motion Fancy words for moving in a circle with constant speed We see this around us all the time –Moon around the earth –Earth around the sun –Merry-go-rounds Constant  and Constant R

25. Physics 218, Lecture XVIII25 Uniform Circular Motion - Velocity Velocity vector = |V| tangent to the circle Is this ball accelerating? –Yes! why?

26. Physics 218, Lecture XVIII26 Centripetal Acceleration “Center Seeking” Acceleration vector= V 2 /R towards the center Acceleration is perpendicular to the velocity R

27. Physics 218, Lecture XVIII27 Circular Motion: Get the speed! Speed = distance/time  Distance in 1 revolution divided by the time it takes to go around once  Speed = 2  r/T Note: The time to go around once is known as the Period, or T

28. Physics 218, Lecture XVIII28 Ball on a String A ball at the end of a string is revolving uniformly in a horizontal circle (ignore gravity) of radius R. The ball makes N revolutions in a time t. What is the centripetal acceleration?

29. Physics 218, Lecture XVIII29 The Trick To Solving Problems

30. Physics 218, Lecture XVIII30 Banking Angle You are a driver on the NASCAR circuit. Your car has m and is traveling with a speed V around a curve with Radius R What angle, , should the road be banked so that no friction is required?

31. Physics 218, Lecture XVIII31 Skidding on a Curve A car of mass m rounds a curve on a flat road of radius R at a speed V. What coefficient of friction is required so there is no skidding? Kinetic or static friction?

32. Physics 218, Lecture XVIII32 Conical Pendulum A small ball of mass m is suspended by a cord of length L and revolves in a circle with a radius given by r = Lsin . 1.What is the velocity of the ball? 2.Calculate the period of the ball

33. Physics 218, Lecture XVIII33 Exam 2 Class average for the 2 nd exam (including the 5 points) was 80.1% –Average for first two exams is a 78.7% Straight scale for curve for now Many have asked “should I q-drop?” –Talk to your advisor and read my FAQ! –Generic advice: Drop if you can’t keep up with the homework by yourself

34. Physics 218, Lecture XVIII34 Next Week Monday: Chapter 10 & 11 due material in WebCT Tuesday: Second lecture on Chaps 12 & 13 Wednesday Recitation: Recitation on Chapters 12 & 13 Wednesday Lab: Elastic Collisions

35. Physics 218, Lecture XVIII35

36. Physics 218, Lecture XVIII36 Circular Motion Example A ball of mass m is at the end of a string and is revolving uniformly in a horizontal circle (ignore gravity) of radius R. The ball makes N revolutions in a time t. a)What is the centripetal acceleration? b)What is the centripetal force?

37. Physics 218, Lecture XVIII37 Next Time Exam 2 is Thursday! Bonus Points for getting a 100 on the mini- practice exam BEFORE the in-class exam Next week: Chapter 8HW due Monday morning Lecture next Thursday will cover Chapters 9 and 10: –Reading questions due: Q10.7 & Q10.26

38. Physics 218, Lecture XVIII38 Computer Hard Drive A computer hard drive typically rotates at 5400 rev/minute Find the: Angular Velocity in rad/sec Linear Velocity on the rim (R=3.0cm) Linear Acceleration It takes 3.6 sec to go from rest to 5400 rev/min, with constant angular acceleration. What is the angular acceleration?

39. Physics 218, Lecture XVIII39 Next Time Read Chapter 10 –More on angular “Stuff” –Angular kinematics –Torque –Reading questions: Q10.7 & Q10.26 HW7 Due Monday (released this afternoon) Exam 2 next Thursday on Chapters 4- 7

40. Physics 218, Lecture XVIII40 More definitions Frequency = Revolutions/sec  radians/sec  f =  /2  Period = 1/freq = 1/f

41. Physics 218, Lecture XVIII41 Motion on a Wheel cont… A point on a circle, with constant radius R, is rotating with some speed  and an angular acceleration . What is the linear acceleration?