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Wed.
Lab
Fri.
11.7 - .9, (.11) Motion With & Without Torque
L11 Rotation Lab Evals
11.10 Quantization, Quiz 11, Lect Evals
RE 11.d
RE 11.e
Mon. Review for Final (1-11) HW11: Pr’s 39, 57, 64, 74, 78
Sat. 9 a.m. Final Exam (Ch. 1-11)
Using Angular Momentum The measure of motion about a point
Magnitude
sinrp
r
p
p
r
Direction
Orient Right hand so fingers curl from the axis and with motion,
then thump points in direction of angular momentum.
rpL rp
=
Magnitude and Direction
=
pole
polebllr
p
ropeballF
handballF
Fr AfromAabout
)()(where,
net
AaboutAaboutLdt
d)()(
Angular Momentum Principle
Torque
Quantifies motion
about a point
Interaction that
changes motion
about a point
Magnitude
(yet another cross product)
FrAA
sinFrAA
FrA
FrA sin sinFrA
Making sense of the
factors and cross-product
Direction
(yet another cross product)
F
Ar
A
Cloud of Dust
Star
1
2
3
sr1
sr2
sr3
Multi-Particle Angular Momentum Principle
With Multi-Particle Angular Momentum extF1
extF2
extF3
...321 extextextsc
dt
Ld
for rigid objects
i
cmiscmsc LLL
where , etc. extsext Fr 111
cmrotstranssc LLL
cmcmcmrot IL
Example: A uniform solid disk with radius 9 cm has mass
0.9 kg (moment of inertia I = ½MR2). A constant force 12 N
is applied as shown. At the instant shown, the angular
velocity of the disk is 45 radians/s in the –z direction
(where +x is to the right, +y is up, and +z is out of the
page, toward you). The length of the string d is 18 cm.
At this instant, what are the magnitude and direction of
the angular momentum about the center of the disk?
What are the magnitude and direction of the torque
on the disk, about the center of mass of the disk?
The string is pulled for 0.2 s. What are the magnitude and direction
of the angular impulse applied to the disk during this time?
After the torque has been applied for 0.2 s, what are the magnitude and
direction of the angular momentum about the center of the disk?
At this later time, what are the magnitude and
direction of the angular velocity of the disk?
Child runs and jumps on playground merry-go-
round. For the system of the child + disk
(excluding the axle and the Earth), which
statement is true from just before to just after
impact?
K is total (macroscopic) kinetic energy is total (linear) momentum is total angular momentum (about the axle)
a. K,
, and
do not change
b.
, and
do not change
c.
does not change
d. K and
do not change
e. K and
do not change
What is the initial angular momentum of the
child + disk about the axle?
a) < 0, 0, 0 >
b) < 0, –Rmv, 0 >
c) < 0, Rmv, 0 >
d) < 0, 0, –Rmv >
e) < 0, 0, Rmv >
x
y
z (z-axis points
out of page)
What’s the final angular speed of the merry-go-round after the kid jumps on?
The disk has moment of inertia I, and after the
collision it is rotating with angular speed . The
rotational angular momentum of the disk alone
(not counting the child) is
a. < 0, 0, 0 >
b. < 0, –I , 0 >
c. < 0, I , 0 >
d. < 0, 0, –I >
e. < 0, 0, I >
After the collision, what is the speed (in m/s)
of the child?
a. R b. c. R2
d. /R e. 2R
After the collision, what is translational angular
momentum of the child about the axle?
a. < 0, 0, 0 >
b. < 0, –Rm , 0 >
c. < 0, Rm , 0 >
d. < 0, –Rm( R), 0 >
e. < 0, Rm( R), 0 >
x
y
z (z-axis points
out of page)
What’s the final angular speed of the merry-go-round after the kid jumps on?
System: child + merry-go-round
Active environment: none that
change angular momentum
Approximations: negligible
frictional torque at axel
Axis: Axle
tLL AiAfA
..
mvRRmRI cmdisk ,0,0,0,0,0,0 .
iAchildfAchildfArgm LLL .....
mvRmRMR ,0,0,0,0,0,0 22
21
2
21
. MRI cmdisk
mvRRMm ,0,0,0,0 2
21
mvRRMm 2
21
What’s the final angular speed of the merry-go-round after the kid jumps on?
x
y
z (z-axis points
out of page)
RMm
mv
21 R
v
mM
211
1Reasonable?
Example of a “collision” for angular momentum
Zero-Torque Systems
Initial Final
tLL aveirotfrot
..
Demo: spinning dumb bells
0.. irotfrot LL
frotirot LL ..
ffii II
mass farther
from axis: Ii larger
i smaller
mass closer to axis:
If smaller
i larger
Solar system formation Spinning Skater
http://lifeng.lamost.org/courses/astrotoday/
CHAISSON/AT315/HTML/AT31502.HTM
What about the energy?
0E
0.int otherrot EK
other
if
EI
L
I
L.int
22
22
other
if
EII
L.int
2 11
2
Increased kinetic energy
fueled by change in internal
energy (Wheaties)
otherrot EK .int
Increased kinetic energy fueled
by change in internal energy
(gravitational potential)
System: Ball + Earth
Active environment: none
Approximations: negligible drag
Axis: C
Two-Step Example: A blob of clay (mass m) is dropped a distance h to land on and
stick to a wheel (mass M, radius R) horizontally ½ R off axis. What’s the wheel’s
angular speed just after the collision?
R
M
m
h
R/2
Step 1: Ball falls to wheel; use energy
0&BEE
0,int. BEEBB UEEK
02
0.212
1.21 mghmvmv BB
Moment 0
Moment 1
1.BvMoment 2
ghvB 21.
Step 2: Ball & wheel collide; use angular momentum
System: Ball + Wheel Active environment: no torques
Approximations: time interval small enough axel friction and
Earth’s gravitation have negligible effect
iCBfCBfCW LLL ...
By right-hand-rule, all in the +z
direction
2
.. MRII cmringCW
.11..22.2. rmvrmvI BBCW Before collision, Rr21
.1
After collision, Rr .2
RghmRmvI CW 21
22. 2
RghmRRmMR21
22
2 222. RvB
ghmRmM 221
2 RmM
ghm
2
22
2.Bv
R
gh
mM 12
2Reasonable?
2
Multiple Torques You come home for the holidays to find
that your younger sibling has annexed
your room. When you try to push the
door open, your nearest & dearest tries
pushing it closed. Your younger sibling is
stronger than you, but you’ve got
‘physics-knowledge’ on your side. While
the kid leans hard (800 N) against the
middle of the door and straight ‘out’ of
the room, you push against the edge
(310 N) and perpendicular to the door.
Say the door’s open 40°. Who wins?
Ayr
A
N 103yF
40
5090s
AyAs rr
21
N 800sF
net
AALdt
d
sy
sAsyAy FrFr
Your torque is in the +z direction, your sibling’s is in
the –z direction; if the answer is in +z you win!
ssAsyyAy FrFrdt
dLz sinsin:ˆ
ssAsyAy FrFr sin
)50sin(N800N31021
AyAy rr )N306N310(Ayr )N4(Ayr
you win!
0
Multiple Torques - equilibrium
Clearly, as you push the door further
closed, you’ll lose some of the advantage
you had by pushing perpendicular to the
door. At what angle will the door be when
you and your sibling are tied / when the
door’s in equilibrium? Ayr
A
N 103yF
?
90s
AyAs rr
21
N 800sF
net
AALdt
d
sy
sAsyAy FrFr
In equilibrium
ssAsyyAy FrFrz sinsin0:ˆssAsyAy FrFr sin
ssAyyAy FrFr sin021
s
s
y
F
Fsin2 90sin cos
s
y
F
F2cos 1 2.39
800
3102cos 1
N
N
Three Fundamental Principles Angular Momentum:
net
AaboutAaboutLdt
d)()(
(linear) Momentum:
net
Fpdt
d Energy:
net
WE
Example all three together! Say we have a uniform 0.4 kg puck with an 8 cm radius. A
24 cm string is initially wrapped around its circumference. If it’s on a frictionless surface
and a 10 N force is applied to the end of the string until it’s unwound…
R = 0.08m
l = 0.24 m
R
d = ? 2
21 mRI puck
f =?
vcmf = ?
a. What will be its rate of rotation when the string is fully unwound?
Energy Principle
WEtotal
Ftrans rFEK
int
ldFKrF rotcm
FlFdIIFd if
2
212
21
FlI f
2
21
I
Flf
2
m
Fl
RmR
Flf
222
21
?t
Three Fundamental Principles Angular Momentum:
net
AaboutAaboutLdt
d)()(
(linear) Momentum:
net
Fpdt
d Energy:
net
WE
Example all three together! Say we have a uniform 0.4 kg puck with an 8 cm radius. A
24 cm string is initially wrapped around its circumference. If it’s on a frictionless surface
and a 10 N force is applied to the end of the string until it’s unwound…
R = 0.08m
l = 0.24 m
R
d = ? 2
21 mRI puck
f
vcmf = ?
a. What will be its rate of rotation when the string is fully unwound? m
Fl
Rf
2
b. How long was the force applied?
?t
Angular Momentum Principle
(axis through final location of cm) dtL
tLL if
tLLLL rotitransirotftransf
)()( ....
tFrI aFf
tRFI f
Torque and final angular velocity in –z direction
F
ml
F
m
Fl
RRm
F
Rm
RF
mR
RF
It
fff
2
2
2
2
21
RF
It
f
RF
mR f
2
21
F
Rm f
2 F
m
Fl
RRm
2
2
F
ml
Three Fundamental Principles Angular Momentum:
net
AaboutAaboutLdt
d)()(
(linear) Momentum:
net
Fpdt
d Energy:
net
WE
Example all three together! Say we have a uniform 0.4 kg puck with an 8 cm radius. A
24 cm string is initially wrapped around its circumference. If it’s on a frictionless surface
and a 10 N force is applied to the end of the string until it’s unwound…
R = 0.08m
l = 0.24 m
R
d = ? 2
21 mRI puck
f
vcmf = ?
a. What will be its rate of rotation when the string is fully unwound? m
Fl
Rf
2
b. How long was the force applied?
?t
F
mlt
You try:
c. How quickly is the puck finally sliding, vcm f?
d. How far has the puck moved, d?
Mon.
Tues.
Wed.
Lab
Fri.
11.4-.6, (.13) Angular Momentum Principle & Torque
11.7 - .9, (.11) Motion With & Without Torque
L11 Rotation Course Evals
11.10 Quantization, Quiz 11
RE 11.c
EP11
RE 11.d
RE 11.e
Mon. Review for Final (1-11) HW11: Pr’s 39, 57, 64, 74, 78
Sat. 9 a.m. Final Exam (Ch. 1-11)
