1. 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: , 10.32, 12.10, 12.13,
12.18, and 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.

2. Hints for today's HW??

4. (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).

5. Center of mass for continuously
distributed stuff:

6. 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

7. 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.

9. External force (gravity): CM moves in a parabola

10. 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.

11. 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.

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

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

14. 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.”

15. 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

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

17. 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

18. 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.

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

20. 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”).

21. 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.

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

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

24. 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

25. 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.)