Exam II
Physics 101: Lecture 16
Angular Momentum
Today’s lecture will cover Textbook Chapter 8.7-8.9
Physics 101: Lecture 16, Pg 1
Overview
 Review
Krotation = ½ I w2
Torque = Force that causes rotation
Equilibrium
SF=0
St=0
 Today
Angular Momentum L = Iw
DL = 0 if S t = 0
Physics 101: Lecture 16, Pg 2 06
Linear and Angular
Displacement
Velocity
Acceleration
Inertia
KE
N2L
Momentum
Linear
Angular
x
q
v
w
a
a
m
I
½ m v2 ½ I w2
F=ma
t = Ia
p = mv
L = Iw
Today
Physics 101: Lecture 16, Pg 3
10
Define Angular Momentum
Momentum
Angular Momentum
p = mV
L = Iw
conserved if SFext = 0
conserved if Stext =0
Vector
Vector!
units: kg-m/s
units: kg-m2/s
Physics 101: Lecture 16, Pg 4 13
Right Hand Rule
 Wrap
fingers of right hand around direction of
rotation, thumb gives direction of angular
momentum.
 What
is direction of angular momentum for wheel
A) Up B) Down C) Left D) Right
Physics 101: Lecture 16, Pg 5 16
Wheel rim
Act: Two Disks

drop
A disk of mass M and radius R rotates around the
z axis with angular velocity wi. A second
identical disk, initially not rotating, is dropped on
top of the first. There is friction between the
disks, and eventually they rotate together with
angular velocity wf.
A) wf = wi
z
B) wf = ½ wi
C) wf = ¼ wi
z
wi
wf
Physics 101: Lecture 16, Pg 6 20
Act: Two Disks

First realize that there are no external torques
acting on the two-disk system.
Angular momentum will be conserved!
2
1
2
L f  I1 w1  I 2 w2  MR w f
Li  I1 w1  0  MR wi
2
1
2
2
MR wi  MR w f
2
z
z
1
2
wi  w f
2
1
w0
wf
Physics 101: Lecture 16, Pg 7 20
Lecture 16, Pre-flights
You are sitting on a freely rotating bar-stool with your arms
stretched out and a heavy glass mug in each hand. Your friend
gives you a twist and you start rotating around a vertical axis
though the center of the stool. You can assume that the bearing the
stool turns on is frictionless, and that there is no net external torque
present once you have started spinning.
You now pull your arms and hands (and mugs) close to your body.
Physics 101: Lecture 16, Pg 8 22
Bonus Question!
 There
are No External forces acting on the
“student+stool” system.
A) True
B) False
C) What!?
Key is no external torques about vertical axis!
FBD has gravity and
normal force.
Physics 101: Lecture 16, Pg 9 24
Lecture 16, Preflight 1
What happens to the angular momentum as you pull
in your arms?
1. it increases
2. it decreases
3. it stays the same
CORRECT
L
L
1
2
“Since there are no external torques
41%
present, the angular momentum is
13%
46%
0%
10%
20%
30%
40%
conserved”
50%
Physics 101: Lecture 16, Pg 10 26
Lecture 16, Preflight 2
What happens to your angular velocityw as you pull in
1
your arms?
1. it increases
CORRECT
I1
2. it decreases
L
L
3. it stays the same
w2
I2
“Angular velocity increases because the moment of
inertia decreases and since angular momentum
must stay the same, angular velocity must
86%
8%
20%
”
“From past experiences where many of my friends
decided to start spinning me while I was holding a
heavy glass mug, you go faster. ! ”
6%
0%
decrease.
40%
60%
80%
100%
Physics 101: Lecture 16, Pg 11 28
Lecture 16, Preflight 3
What happens to your kinetic energy as
w1 you pull in
your arms?
1. it increases
CORRECT
I1
2. it decreases
L
L
3. it stays the same
1 2
1
1 2 2
2

L
K  Iw 
I w
2I
2
2I
w2
I2
(using L = Iw )
“KE increases since it is inversely related to I according
to the equation KE=L^2/2I.”
63%
7%
“Conservation of energy.”
30%
0%
20%
40%
60%
80%
Physics 101: Lecture 16, Pg 12 30
What about Energy Conservation?
A) Energy isn’t conserved here
B) Energy comes from weights
C) Gravitational energy is being converted to
rotational kinetic energy
D) Energy comes from cookies.
E ) I have no clue….
Physics 101: Lecture 16, Pg 13 33
Turning the bike wheel
A student sits on a barstool holding a bike wheel. The wheel
is initially spinning CCW in the horizontal plane (as viewed
from above) L= 25 kg m2/s She now turns the bike wheel
over. What happens?
A. She starts to spin CCW.
B. She starts to spin CW.
C. Nothing
CORRECT
Start w/ angular momentum L pointing up from wheel. When
wheel is flipped, no more angular momentum from it pointing
up, so need to spin person/stool to conserve L!
Physics 101: Lecture 16, Pg 14 35
Turning the bike wheel (more)
She is holding the bike wheel and spinning counter
clockwise. What happens if she turns it the other ½ rotation
(so it is basically upside down from how it started).
A) Spins Faster
B) Stays same
C) Stops
Physics 101: Lecture 16, Pg 15 37
Turning the bike wheel...

Since there is no net external torque acting on the student-stool
system, angular momentum is conserved.
Remenber, L has a direction as well as a magnitude!
Initially: LINI = LW,I = + 25 kg m2/s
Finally: LFIN = LW,F + LS
= -25 kg m2/s + Ls
Ls = 50 kg m2/s
LS
LW,I = LW,F + LS
LW,I
LW,F
Physics 101: Lecture 16, Pg 16 39
Act 2 Rotations

Puck on ice
A puck slides in a circular path on a horizontal frictionless table.
It is held at a constant radius by a string threaded through a
frictionless hole at the center of the table. If you pull on the
string such that the radius decreases by a factor of 2, by what
factor does the angular velocity of the puck increase?
(a) 2
(b) 4
(c) 8
w
Physics 101: Lecture 16, Pg 17 43
Act 2 Solution

Since the string is pulled through a hole at the center
of rotation, there is no torque: Angular momentum is
conserved.
2
L1 = I1w1 = mR2w1
=
mR2w1 = m
1
w1 = w2
4
m
R
w1
L2 = I2w2 = m  R  w2
2
1 2
R w2
4
w2 =
4w1
m
R/2
w2
Physics 101: Lecture 16, Pg 18 45
Gyroscopic Motion:
 Suppose
you have a spinning gyroscope
in the configuration shown below:
 If the left support is removed, what will
happen??
support
w
pivot
g
Physics 101: Lecture 16, Pg 19 48
Gyroscopic Motion...
 Suppose
you have a spinning gyroscope
in the configuration shown below:
 If the left support is removed, what will
happen?
The gyroscope does not fall down!
w
pivot
g
Physics 101: Lecture 16, Pg 20 45
Gyroscopic Motion...
Bicycle wheel
 ...
instead it precesses around its pivot
axis !
w
pivot
Physics 101: Lecture 16, Pg 21
Summary
 St
L
=Ia
=Iw
Right Hand Rule gives direction
If St = 0, L is conserved
Physics 101: Lecture 16, Pg 22
Practice Problems for Chapt 8
Probs. 13, 17, 25, 29, 31, 33, 35, 37, 43,
49, 53, 55, 57, 61, 65, 67, 69 (hard!), 73,
75, 77, 79, 81, 85, 89, 105, 113
Physics 101: Lecture 16, Pg 23