Phys. 121: Tuesday, 21 Oct. Written HW 8: due by 2:00 pm. HW 6 returned: please pick up in front. Written HW 8: due by 2:00 pm. Written HW 9: 10.29, 10.32, 12.10, 12.13, 12.18, and 12.34. Due in one week. Mastering Phys.: Assign. 7 due Thursday. Lab Switch: this week will be Inelastic Colls. Midterm Grades: I will post grade breakdowns to Moodle soon. Grades will be updated after exam 2 (deadline to change to S/U or withdraw is Nov. 5). Exam 2: will cover chapters 7, 8, 10, and 11, and will be Thursday, 30 October (a week from this Thursday). Study guide, sample formula sheet, and sample exam problems are all available.

Hints for today's HW??

(Chapter 12:) Definition of the center of mass: The center of mass is used a lot in physics! For instance, the weight of an object acts like it points downward from the center of mass (for instance, remember the front-wheel-drive problem: 70 percent of the normal force was on the front tires, because the center of mass was close to the front tires).

Center of mass for continuously distributed stuff:

Clickers: a solid sphere is sliced in half, and rearranged as shown. The center of mass of the two halves is... a) Exactly where they touch b) A bit above where they touch c) A bit below where they touch d) At the top of the figure e) At the bottom of the figure

The center of mass is where you can balance an extended object. In force diagrams, it's as if the entire object's weight acts “at” the center of mass.

External force (gravity): CM moves in a parabola

Clickers: In billiards, a cue ball with speed v hits a rack of 15 balls at rest. All the balls have the same mass. How fast is the center of mass of the balls moving just after they collide? a) It will be at rest. b) It will be moving at v also. c) It will be moving at v/2 d) It will be moving at v/16 e) It is impossible to know without more information.

Clickers: Total momentum of a set of billiard billiard balls remains constant after they collide with one another. Why isn't this still true after they collide with the bumper at the edge of the table? a) No, it is still true. b) Because momentum is not really conserved. c) Because the bumper collision is not elastic. d) Because we need to include the momentum of the table as well. e) Because we need to include the momentum of the table and the earth as well.

In two or more dimensions, conservation of momentum is a vector equation: true for each individual component of momentum in each direction.

Example: find the angle θ that billiard ball 2 goes out along. (Essentially problem 9.70.)

Rotational Motion Objects don't just move from place to place; they can also rotate! For rotational motion, we keep track of how fast the angle that some part of the object makes changes with time. Notation: ө represents angle in radians ѡ represents d(ө )/dt, and is called the “angular velocity.” α represents dω/dt = d²(θ)/dt², and is called the “angular acceleration.”

Clickers: angular velocity divided by angular acceleration, or ω/α, has the same units as... a) Time b) 1/ω c) 1/√α d) All of the above e) None of the above

Rotational Motion Review Rotational kinematics for constant angular acceleration The signs of angular velocity and angular acceleration. 16 16

Clickers: Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. The angular velocity of A is twice that of B. The angular velocity of A equals that of B. The angular velocity of A is half that of B. I can't be bothered with this; I've got a new Friend request on Facebook right now. Answer: B 17 17

Important special case: rigid rotation If something rotates as a “whole”, it has the same angular speed ω everywhere! This means that the speed of any point at a distance r from the rotation axis obeys the equation v = ω r (this follows from the definition of what a radian is!)‏ Here, v is the magnitude of vector . Special case: rolling without slipping: use the radius of the object for r, and the overall speed for v. Add the rotational velocity to the overall velocity (as vectors!) to get the overall velocity of any point.

Note the similarities between linear and angular! (No actual physics here; just mathematical definitions and their consequences.)

Most of our motion equations have angular versions of them which look similar! But we need something to represent the angular versions of force and mass, to do physics! Angular version of “force” is called torque, and torque produces angular acceleration just like regular force produces regular acceleration. Angular version of “mass” (inertia) is angular inertia, called moment of inertia (and sometimes called “rotational inertia”).

Torque acts to increase an object's angular velocity, exactly as a regular force acts to increase an object's regular velocity. Torque is always defined with respect to some axis: the axis of rotation.

Magnitude of Torque = r F sin(angle between), and goes either clockwise or counter.

Rotational Inertia (Moment of Inertia) I = rotational analog of mass: Note: This equation is valid ONLY when I does not change!

Clickers: Reading Question 12.3 A single particle has a mass, m, and it is at a distance, r, away from the origin. The moment of inertia of this particle about the origin is mr. m2r. m2r2. m2r4. mr2. Answer: E Slide 12-13 24 24

Rotational kinetic energy is not any different from ordinary kinetic energy! It's just the total ordinary kinetic energy appropriate for something that's rotating rigidly (that is, with constant angular velocity ω). If something is both rotating and moving, then add the K.E. of the center of mass to this to find the total kinetic energy. (Use the center of mass as the rotation axis as well.)