Prof. Rupak Mahapatra. Dynamics of Rotational Motion

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1 Physics 218 Chapter Prof. Rupak Mahapatra Dynamics of Rotational Motion 1

2 Overview Chapters are about Rotational Motion While we ll do Exam 3 on Chapters 10-13, we ll do the lectures on 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 ng and making the issues more intuitive Dynamics of Rotational Motion 2

3 Rotational Motion Start with Fixed Axis motion on The relationship between linear and angular variables Rotating and translating at the same time First kinematics, then dynamics just like earlier this semester Dynamics of Rotational Motion 3

4 Dynamics of Rotational Motion 4

5 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 Dynamics of Rotational Motion 5

6 Rotational Motion Here we re talking about stuff that goes around and around Start t by envisioning: i i A spinning i object like a car tire Dynamics of Rotational Motion 6

7 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: Ie I.e, the objects don t change as they rotate. Example: a bicycle wheel Examples of Non-rigid bodies? Dynamics of Rotational Motion 7

8 Overview: Rotational Motion Take our results from linear physics and do the same for angular physics Analogue of Position Velocity Start Acceleration Force Mass Momentum Energy here! 1-3 Chapt ters Dynamics of Rotational Motion 8

9 Axis of Rotation: Definitions Pick a simple place to rotate around Call point O the Axis of Rotation Same as picking an origin Dynamics of Rotational Motion 9

10 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 l Circ θr 2 R Dynamics of Rotational Motion 10

11 Velocity and Acceleration Define as the angular velocity t or d dt radians/ sec Define as the angular acceleration d dt or 2 d radians/ sec dt 2 2 Dynamics of Rotational Motion 11

12 Motion on a Wheel What is the linear speed of a point rotating around in a circle with angular speed, and constant radius R? Dynamics of Rotational Motion 12

13 Examples Consider two points on a rotating wheel. One on the inside (P) and the other at the end (b): Which h has greater angular speed? R 1 R 2 b Which has greater linear speed? Dynamics of Rotational Motion 13

14 Angular Velocity and Acceleration Are and vectors? and clearly have magnitude Do they have direction? Dynamics of Rotational Motion 14

15 Yes! Define the direction to point along the axis of rotation Right-hand Rule This is true for and Right-Hand Rule Dynamics of Rotational Motion 15

16 Uniform Angular Acceleration Derive the angular equations of motion for constant angular acceleration 1 t t t 0 2 Dynamics of Rotational Motion 16

17 Rotation and Translation Objects can both translate and rotate at the same time. They do both around their center of mass. Dynamics of Rotational Motion 17

18 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? Dynamics of Rotational Motion 18

19 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 Dynamics of Rotational Motion 19

20 Try Differently: Paper Roll A paper towel unrolls with velocity V Conceptually same thing as the wheel What s the velocity of points: A? B? C? D? C B A Point C is where rolling D part separates from the unrolled portion C B A Both have same velocity there 20 Dynamics of Rotational Motion

21 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? Dynamics of Rotational Motion 21

22 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 rounds Constant and Constant R Dynamics of Rotational Motion 23

23 Uniform Circular Motion - Velocity Velocity vector = V tangent to the circle Is this ball accelerating? Yes! why? Dynamics of Rotational Motion 24

24 Centripetal Acceleration Center Seeking 2 Acceleration vector= V 2 /R towards the center Acceleration is perpendicular to the velocity a rˆ v direction R ( R rˆ ) Dynamics of Rotational Motion 25

25 The Trick To Solving Problems F ma 2 v m ( rˆ ) R Dynamics of Rotational Motion 28

curve with Radius R What angle,, should the road be banked

26 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? Banking Angle Dynamics of Rotational Motion 29

27 Skidding on a Curve A car of mass m rounds a curve on a flat road of radius R at a speed V. V What coefficient of friction is required so there is no skidding? Kinetic or static friction? Dynamics of Rotational Motion 30

28 Conical Pendulum A small ball of mass m is suspended dd 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 Dynamics of Rotational Motion 31

29 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? Dynamics of Rotational Motion 32

30 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? Dynamics of Rotational Motion 33

31 More definitions Frequency = Revolutions/sec ons/s radians/sec f = /2 Period = 1/freq = 1/f Dynamics of Rotational Motion 34

32 Motion on a Wheel cont A point on a circle, with constant tradius R, is rotating with some speed and an angular acceleration. What is the linear acceleration? Dynamics of Rotational Motion 35

33 Dynamics of Rotational Motion 36

34 Angular Quantities Last time: Position i Angle Velocity Angular Velocity Acceleration Angular Acceleration This time we ll start by discussing the vector nature of the variables and then move forward on the others: Force Mass Momentum Energy Dynamics of Rotational Motion 37

35 Angular Quantities Position Angle Velocity Angular Velocity Acceleration Angular Acceleration Moving forward (chapter 14 today): Force Mass Momentum Energy Dynamics of Rotational Motion 38

36 Torque Torque is the analogue of Force Take into account the perpendicular distance from axis Same force further from the axis leads to more Torque Dynamics of Rotational Motion 39

37 Slamming a door We know this from experience: If we want to slam a door really hard, we grab it at the end If we try to push in the middle, we aren t able to make it slam nearly as hard Dynamics of Rotational Motion 40

38 Torque Continued What if we change the angle at which the Force is applied? What is the Effective Radius? Dynamics of Rotational Motion 41

39 Slamming a door We also know this from experience: If we want to slam a door really hard, we grab it at the end and throw perpendicular to the hinges If we try to pushing towards the hinges, the door won t even close Dynamics of Rotational Motion 42

40 Torque Torque is our slamming ability Need some new math to do Torque Write Torque as r F sin r F To find the direction of the torque, wrap your fingers in the direction the torque makes the object twist Dynamics of Rotational Motion 43

41 Vector Cross Product C A B C A B Sin This is the last way of multiplying py vectors we will see Direction from the iht right-hand h rule Swing from A into B! Dynamics of Rotational Motion 44

42 Vector Cross Product Cont ˆi Multiply py out, but use the Sin to give the magnitude, and RHR to give the direction ˆi 0 (sin 0 ) îi ĵj kˆ (i (sin 1 ) îi kˆ ĵj (i (sin 1 ) Dynamics of Rotational Motion 45

43 Cross Product Example A B B A B X X ˆî îi A Y B Y ˆĵ ĵj What is A B using Unit Vector notation? Dynamics of Rotational Motion 46

44 Torque and Force Torque problems are like Force problems 1. Draw a force diagram 2. Then, sum up all the torques to find the total torque Is torque a vector? Dynamics of Rotational Motion 47

45 Example: Composite Wheel Two forces, F 1 and F 2, act on different radii of a wheel, R 1 and R 2, at different angles 1 and 2. 1 is a right angle. If the axis is fixed, what is the net torque on the wheel? F 1 Dynamics of Rotational Motion 48 F 2

46 Angular Quantities Position Angle Velocity Angular Velocity Acceleration Angular Acceleration Moving forward: Force Torque Mass Momentum Energy Dynamics of Rotational Motion 49

47 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 = Dynamics of Rotational Motion 50

48 Calculate Moment of Inertia Calculate the moment of inertia for a ball of mass m relative to the center of the circle R Dynamics of Rotational Motion 51

49 Moment of Inertia To find the mass of an object, just add up all the little l pieces of mass To find the moment m of inertia around a point, just add up all the little moments I mr 2 or I r 2 dm Dynamics of Rotational Motion 52

50 Torque and Moment of Inertia Force vs. Torque F=ma = I Mass vs. Moment of Inertia m I mr 2 or I r 2 dm Dynamics of Rotational Motion 53

51 Pulley and Bucket A heavy ypulley, with radius R, and known moment of inertia I starts at rest. We attach it to a bucket with mass m. The fiti friction torque is fric. Find the angular acceleration Dynamics of Rotational Motion 54

52 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 m of inertia of the pulley? R Dynamics of Rotational Motion 55

53 Less Spherical Heavy Pulley A heavy pulley, with radius R R, 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? Dynamics of Rotational Motion 56

54 Next Time More on angular Stuff Angular g Momentum Energy Get caught up on your homework!!! Mini-practice exam 3 is now available Dynamics of Rotational Motion 57

55 Angular Quantities Position Angle Velocity Angular Velocity Acceleration Angular Acceleration Moving forward: Force Torque Mass Momentum Energy Dynamics of Rotational Motion 58

56 Momentum Momentum vs. Angular Momentum: p mv L I Newton s Laws: F dp dt dl dt Physics 218, Lecture XXII 59

57 Angular Momentum First way to define the Angular Momentum L: I d d ( I ) I dt dt I d ( L ) dt dl dt dl dt Physics 218, Lecture XXII 60

58 Angular Momentum Definition Another definition: L r p Physics 218, Lecture XXII 61

59 Angular Motion of a Particle Determine the angular momentum, L, L of a particle, with mass m and speed v, v moving in circular motion with radius r Physics 218, Lecture XXII 62

60 Conservation of Angular Momentum dl d dt if 0 L Const By Newton s laws, the angular momentum of a body can change, but the angular momentum for a system cannot change Conservation n of Angular Momentum Same as for linear momentum Physics 218, Lecture XXII 63

61 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 Ice Skater L I Physics 218, Lecture XXII 64

62 Problem Solving For Conservation of Angular Momentum problems: BEFORE and AFTER Physics 218, Lecture XXII 65

63 Conservation of Angular Momentum Before Physics 218, Lecture XXII 66

64 Conservation of Angular Momentum After Physics 218, Lecture XXII 67

Clutch Design As a car engineer, you model a car

you close them so they spin together Find the

65 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 Physics 218, Lecture XXII 68

66 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 Physics 218, Lecture XXII 69

67 Rotational Kinetic Energy KE = ½mv 2 trans KE = ½I 2 rotate t Conservation of Energy must take rotational kinetic energy into account Physics 218, Lecture XXII 70

2 Rolling without slipping is a special case where

68 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 Physics 218, Lecture XXII 71

Rolling Down an Incline You take a solid ball of mass m and

hih Z. Z You then let go and the ball rolls without slipping.

What would be the speed if the ball didn t roll and there

69 Rolling Down an Incline You take a solid ball of mass m and radius R and hld hold it at rest on a plane with ih height hih Z. 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? Z Note: I sphere = 2/5MR 2 Physics 218, Lecture XXII 72

70 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? i Is energy Conserved? Physics 218, Lecture XXII 73

71 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? Rotating Rod Physics 218, Lecture XXII 74

Less Spherical Heavy Pulley A heavy pulley, with

We pull R on an attached rope with constant force

72 Less Spherical Heavy Pulley A heavy pulley, with radius R, starts at rest. We pull R 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? Physics 218, Lecture XXII 75

The person then starts moving on the disk with speed V.

73 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 Physics 218, Lecture XXII 76

74 Same Problem: Forces Same problem but with Forces Physics 218, Lecture XXII 77

75 Exam 3!!! Next Time Covers Chapters Get caught up on your homework!!! Mini-practice exam 3 is now available Thursday: - Finish up angular Stuff Physics 218, Lecture XXII 78

76 Physics 218, Lecture XXII 79

77 Dynamics of Rotational Motion 80

78 Example of Cross Product The location of a body is length r from the origin and at an angle from the x- axis. A force F acts on the body purely in the y direction. What is the Torque on the body? x z y Dynamics of Rotational Motion 81

79 Calculate Moment of Inertia 1.Calculate the moment of inertia for a ball of mass m relative to the center of the circle R 2.What about lots of fpoints? For example a wheel Dynamics of Rotational Motion 82

80 Rotating Rod A uniform rod of mass m, length l, and moment of inertia I = ml 2 /3 rotates around a pivot. It is held horizontally and released. Find the angular acceleration and the linear acceleration a at the end. Where, along the rod, is a = g? g Dynamics of Rotational Motion 83

81 Two weights on a bar Find dthe middle moment of inertia for the two different Axes Dynamics of Rotational Motion 84

82 Schedule Changes Please see the handout for schedule changes New Exam 3 Date: Exam 3 Tuesday Nov. 26th Dynamics of Rotational Motion 85

83 Moments of Inertia Dynamics of Rotational Motion 86

Prof. Rupak Mahapatra. Physics 218, Chapter 15 & 16

Prof. Rupak Mahapatra. Physics 218, Chapter 15 & 16 Physics 218 Chap 14 & 15 Prof. Rupak Mahapatra Physics 218, Chapter 15 & 16 1 Angular Quantities Position Angle θ Velocity Angular Velocity ω Acceleration Angular Acceleration α Moving forward: Force Mass