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Remaining Schedule

This course covers the final four lessons of Physics 253, including topics such as general rotation, vectors, torque, angular momentum, conservation of angular momentum, and static equilibrium. Prepare for the final exam.

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Remaining Schedule

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  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-1:50PM • 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) 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 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 theRight-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. 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 a 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 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. 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 3rd 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. Physics 253

  8. Properties of the Cross Product Physics 253

  9. Physics 253

  10. In Vector Notation Physics 253

  11. Physics 253

  12. Need a negative sign here! Physics 253

  13. Physics 253

  14. More Properties Physics 253

  15. Torque is a Vector Product! Physics 253

  16. Formal Definition of Torque Physics 253

  17. 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=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. • Since torque is vector product then L should be too! 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. Physics 253

  20. The connection between t and l. 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. 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 Physics 253

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