1. 6/12/2015Physics 253 1 Remaining Schedule Friday April 27Friday April 27 –Review for Quiz 4 Monday April 30Monday April 30 –Quiz 5 –50 points –Crib Sheet Wednesday May 2Wednesday May 2 –Final Review Wednesday May 9 noon-1:50PMWednesday May 9 noon-1:50PM –Final! –100 points –Open book

2. 6/12/2015Physics 253 2 The Final Four Lessons Chapter 11: General RotationChapter 11: General Rotation Vectors, Torque, Angular Momentum (10- 4,11-1, 11-2, 11-3)Vectors, Torque, Angular Momentum (10- 4,11-1, 11-2, 11-3) Angular Momentum & Torque for Systems and Rigid Bodies, (11-4, 11-5, 11-6)Angular Momentum & Torque for Systems and Rigid Bodies, (11-4, 11-5, 11-6) Conservation of Angular Momentum and Special Topics (11-7, 11-8, 11-9, 11-10)Conservation of Angular Momentum and Special Topics (11-7, 11-8, 11-9, 11-10) Chapter 12: Static Equilibrium (12-1, 12-2, 12-3, 12-4)Chapter 12: Static Equilibrium (12-1, 12-2, 12-3, 12-4)

3. 6/12/2015Physics 253 3 Vector Nature of Angular Quantities As we’ve discussed in the past rotation has a sense of direction.As we’ve discussed in the past rotation has a sense of direction. Both angular velocity and acceleration can be treated as vectors once we define their direction.Both angular velocity and acceleration can be treated as vectors once we define their direction. Consider a spinning wheelConsider a spinning wheel –The direction can’t be given by the linear velocity since it points in all directions. –The only special direction is the axis of rotation which is perpendicular to the motion. –This is a natural choice for the direction of angular velocity and acceleration

4. 6/12/2015Physics 253 4 We’ll use the axis of rotation for direction. But there’s still an ambiguity since  could point either way along the axis.We’ll use the axis of rotation for direction. But there’s still an ambiguity since  could point either way along the axis. Enter the Right-hand rule: If your fingers curl around the axis and along the direction of motion, then the thumb points in the direction of the angular velocity.Enter the Right-hand rule: If your fingers curl around the axis and along the direction of motion, then the thumb points in the direction of the angular velocity.

5. 6/12/2015Physics 253 5 If the rotation axis is fixedIf the rotation axis is fixed –Only the magnitude of  can change –Since  =d  /dt, the angular acceleration also points along the axis of rotation. But not necessarily in the same direction as  : For instance if the rotation in the figure is CCW and  is increasing  is increasing and in the same direction or upwardFor instance if the rotation in the figure is CCW and  is increasing  is increasing and in the same direction or upward However, if  is decreasing  points in the opposite direction or downwardHowever, if  is decreasing  points in the opposite direction or downward If the axis is not fixedIf the axis is not fixed –  still points along the axis –But  cannot point along the axis

6. 6/12/2015Physics 253 6 Vectorial Treatment of  and . As you might expect as with linear motion a more precise vector treatment of angular motion will be helpful.As you might expect as with linear motion a more precise vector treatment of angular motion will be helpful. We already have a hint that something more is needed since we don’t have any machinery to deal with axes of rotation that change direction with time.We already have a hint that something more is needed since we don’t have any machinery to deal with axes of rotation that change direction with time. The motion of objects not constrained about a fixed axis is very complicated but immensely rich and rewarding…spinning tops, the earth, hurricanes…The motion of objects not constrained about a fixed axis is very complicated but immensely rich and rewarding…spinning tops, the earth, hurricanes… We’ll look at a few illustrative examples but first we need to define a new vector quantity to deal with angular momentum and torque.We’ll look at a few illustrative examples but first we need to define a new vector quantity to deal with angular momentum and torque.

7. 6/12/2015Physics 253 7 The “Vector Cross Product” Also called the “vector product” or the “cross product”.Also called the “vector product” or the “cross product”. An operation between two vectors A & B that creates a 3 rd vector C.An operation between two vectors A & B that creates a 3 rd vector C. The magnitude of the new vector is given by C=ABsin  where  is the smallest angle between A and BThe magnitude of the new vector is given by C=ABsin  where  is the smallest angle between A and B The direction is perpendicular to the plane of A and B and given by applying the RHR:The direction is perpendicular to the plane of A and B and given by applying the RHR: –Start w/ fingers along A. –Curve fingers toward B. –Thumb in direction of C.

8. 6/12/2015Physics 253 8 Properties of the Cross Product

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10. 6/12/2015Physics 253 10 In Vector Notation

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12. 6/12/2015Physics 253 12 Need a negative sign here!

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14. 6/12/2015Physics 253 14 More Properties

15. 6/12/2015Physics 253 15 Torque is a Vector Product!

16. 6/12/2015Physics 253 16 Formal Definition of Torque

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18. 6/12/2015Physics 253 18 Angular Momentum is a Vector Product Newton’s second law can be written as  F=dp/dt.Newton’s second law can be written as  F=dp/dt. By the linear motion  angular motion analog we expect something similar for   =dL/dt.By the linear motion  angular motion analog we expect something similar for   =dL/dt. That is a vector form of the scalar analogy we developed in Chapter 10, but we were limited to rotation about an axis.That is a vector form of the scalar analogy we developed in Chapter 10, but we were limited to rotation about an axis. Since torque is vector product then L should be too!Since torque is vector product then L should be too!

19. 6/12/2015Physics 253 19 Let’s consider a particle of mass m with momentum p and position vector r with respect to the origin O in an inertial reference frame.Let’s consider a particle of mass m with momentum p and position vector r with respect to the origin O in an inertial reference frame. Let’s also assume the definition of the angular momentum is the vector product: l=r x p.Let’s also assume the definition of the angular momentum is the vector product: l=r x p. Then the magnitude is given by rpsin  and the direction by the RHR.Then the magnitude is given by rpsin  and the direction by the RHR.

20. 6/12/2015Physics 253 20 The connection between  and l.

21. 6/12/2015Physics 253 21 Student Evaluations Now’s your chance to evaluate teaching effectivenessNow’s your chance to evaluate teaching effectiveness Assessment will got to personnel committee and to instructor.Assessment will got to personnel committee and to instructor. More importantly an opportunity to improve instruction.More importantly an opportunity to improve instruction. Your comments/criticisms are very welcome.Your comments/criticisms are very welcome. As usual, one of your colleagues will give you instructions and collect them.As usual, one of your colleagues will give you instructions and collect them.

22. 6/12/2015Physics 253 22 Filling out the Scan-tron “LAST NAME”: Blazey“LAST NAME”: Blazey “INIT.”: G“INIT.”: G First four boxes of “ID Number”: 3130First four boxes of “ID Number”: 3130 “SEC”: 1“SEC”: 1 “DEPT.” : PHYS“DEPT.” : PHYS “COURSE”: 253“COURSE”: 253 “DATE”: 4/20/07“DATE”: 4/20/07