Phys. 121: Tuesday, 21 Oct.● 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! Forinstance, the weight of an object acts like it pointsdownward from the center of mass (for instance,remember the front-wheel-drive problem: 70 percentof the normal force was on the front tires, because thecenter of mass was close to the front tires).

Center of mass for continuouslydistributed stuff:

Clickers: a solid sphere is sliced in half, andrearranged as shown. The center of mass ofthe 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 anextended object. In force diagrams, it's as if theentire 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 vhits a rack of 15 balls at rest. All the balls havethe same mass. How fast is the center of massof 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 billiardbilliard balls remains constant after they collidewith one another. Why isn't this still true after they collide with the bumper at the edge of thetable?• 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 ofmomentum is a vector equation: true for eachindividual component of momentum in eachdirection.

Example: find the angle θ that billiard ball2 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 byangular acceleration, or ω/α, has thesame 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.

Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A.

Clickers:

A. The angular velocity of A is twice that of B.

B. The angular velocity of A equals that of B.

C. The angular velocity of A is half that of B.

D. I can't be bothered with this; I've got a new Friend

request on Facebook right now.

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.

v⃗

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

Most of our motion equations have angularversions 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 increasean object's angularvelocity, exactly as aregular force acts toincrease an object'sregular velocity.

Torque is alwaysdefined with respect tosome axis: the axis ofrotation.

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 ONLYwhen I does not change!

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

A. mr.B. m2r.C. m2r2.D. m2r4.E. mr2.

Reading Question 12.3

Slide 12-13

Clickers:

Rotational kinetic energy is not any differentfrom ordinary kinetic energy! It's just thetotal ordinary kinetic energy appropriate forsomething that's rotating rigidly (that is, withconstant angular velocity ω).

If something is both rotating and moving, thenadd the K.E. of the center of mass to this to findthe total kinetic energy. (Use the center of massas the rotation axis as well.)