1 Remaining Schedule Friday April 27 Review for

1 Remaining Schedule • Friday April 27 – Review for Quiz 4 • Monday April 30 – Quiz 5 – 50 points – Crib Sheet • Wednesday May 2 – Final Review • Wednesday May 9 noon-1: 50 PM – – – 12/18/2021 Final! 100 points Open book Physics 253

2 The Final Four Lessons • Chapter 11: General Rotation • 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) • 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) 12/18/2021 Physics 253

3 Vector Nature of Angular Quantities • 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. • Consider 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 12/18/2021 Physics 253

4 • We’ll use the axis of rotation for direction. But there’s still an ambiguity since w 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. 12/18/2021 Physics 253

5 • If the rotation axis is fixed – Only the magnitude of w can change – Since a=dw/dt, the angular acceleration also points along the axis of rotation. But not necessarily in the same direction as w: • For instance if the rotation in the figure is CCW and w is increasing and in the same direction or upward • However, if w is decreasing a points in the opposite direction or downward • If the axis is not fixed – w still points along the axis – But a cannot point along the axis 12/18/2021 Physics 253

6 Vectorial Treatment of w and a. • 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. • 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. 12/18/2021 Physics 253

7 The “Vector 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. • The magnitude of the new vector is given by C=ABsinq where q 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: – Start w/ fingers along A. – Curve fingers toward B. – Thumb in direction of C. 12/18/2021 Physics 253

8 Properties of the Cross Product 12/18/2021 Physics 253

9 12/18/2021 Physics 253

10 In Vector Notation 12/18/2021 Physics 253

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12 Need a negative sign here! 12/18/2021 Physics 253

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14 More Properties 12/18/2021 Physics 253

15 Torque is a Vector Product! 12/18/2021 Physics 253

16 Formal Definition of Torque 12/18/2021 Physics 253

17 12/18/2021 Physics 253

18 Angular Momentum is a Vector Product • Newton’s second law can be written as SF=dp/dt. • By the linear motion angular motion analog we expect something similar for St=d. L/dt. • 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! 12/18/2021 Physics 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 also assume the definition of the angular momentum is the vector product: l =r x p. • Then the magnitude is given by rpsinq and the direction by the RHR. 12/18/2021 Physics 253

20 The connection between t and l. 12/18/2021 Physics 253

21 Student Evaluations • Now’s your chance to evaluate teaching effectiveness • Assessment will got to personnel committee and to instructor. • More importantly an opportunity to improve instruction. • Your comments/criticisms are very welcome. • As usual, one of your colleagues will give you instructions and collect them. 12/18/2021 Physics 253

22 Filling out the Scan-tron • • “LAST NAME”: Blazey “INIT. ”: G First four boxes of “ID Number”: 3130 “SEC”: 1 “DEPT. ” : PHYS “COURSE”: 253 “DATE”: 4/20/07 12/18/2021 Physics 253