Mon. 11.4-.6, (.13) Angular Momentum Principle & Torque RE...
Transcript of Mon. 11.4-.6, (.13) Angular Momentum Principle & Torque RE...
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)
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
=
If the Masses Don’t Lie in a Plane
cm
Bp
Dp
Cp
Ap
Ar
Cr
Dr
Br
cmAL
cmCL
cmBL
cmDL
DDCCBBAAcmtotal prprprprL
AAAAcmA prprL
T
rmr A
AA
2
r AAA rmr
Just looking at magnitude for ball A
If all masses, distances, and speeds are the same,
rmrLLLL cmDcmCcmbcmA
cmAL
cmBL sin. cmAzcmA LL
Given the symmetry,
zLLLLL zcmDzcmCzcmBzcmAcmtotalˆ
....
sinrmr
zrmrL cmtotalˆsin4
but sinrr
zmrL cmtotalˆ4 2
axisI
Generally, it’s the moment of inertia about the rotational axis through cm
Rotational Angular Momentum and
Rotational Energy
Recall 2
21 IKrot
now
ILrot
so I
LKrot
2
2
Analogous to 2
21 mvK
vmp
m
pK
2
2
Example: A barbell spins around a pivot at its center at A. The barbell consists
of two small balls, each with mass 500 grams (0.5 kg), at the ends of a very low
mass rod of length 50 cm (0.5 m). The barbell spins clockwise with angular
speed = 120 radians/s.
Rotational Angular Momentum and Kinetic Energy
Special case: rigid body
a) What is the moment of inertia about A?
b) What is the direction of the angular velocity?
c) What is the translational angular momentum?
d) What is the rotational angular momentum?
e) What is the total angular momentum?
f) What is the rotational kinetic energy?
Using Angular Momentum The measure of motion about a point
Effect of a radial force (like gravity or electric) sun
sunEarthr
p
sunEarthF
dt
pdrp
dt
rdpr
dt
dL
dt
d ESEE
SEESESE
sunEarthSEEESE FrpvLdt
d
Parallel Parallel
= 0
constantSEL
Orbit noncircular.py
A B
C
Same interaction Relative to Different Reference Points
Orbit Non-circular L.py
B
Different Angular Momenta
Generally, changing Angular Momenta
Angular Momentum
affect of a non-radial force
pole
polebllr
p
ropeballF
dt
pdrp
dt
rdpr
dt
dL
dt
d bpbb
pb
bpbpb
forcesall
ballpbbbpb FrpvLdt
d
.
Parallel
handballF
Fr AfromAabout
)()(where,
net
AaboutAaboutLdt
d)()(
handballpbropeballpb FrFr
Parallel
Angular Momentum Principle
Torque
analogous to
Momentum Principle
net
Fpdt
d
pole
polebllr
p
ropeballF
handballF
Fr AfromAabout
)()(where,
net
AaboutAaboutLdt
d)()(
Angular Momentum Principle
Torque
analogous to
Momentum Principle
net
Fpdt
d
Quantifies motion
about a point
Interaction that
changes motion
about a point
Quantifies motion Interaction that
changes motion
Example: At t = 15 s, a particle has angular momentum <6, 8, -5> kg · m2/s relative to location A. A
constant torque <13, -14, 18> N · m relative to location A acts on the particle. At t = 15.2 s,
what is the angular momentum of the particle relative to location A?
or
net
AALdt
d tL
avenet
AA
tLL
avenet
AiAfA
..
so
15s-s2.15Nm18,14,13/smkg5,8,6 2
. fAL
s2.0Nm18,14,13/smkg5,8,6 2
. fAL
Nm/s6.3,8.2,6.2/smkg5,8,6 2
. fAL
smm/skgsNm 2
/smkg4.1,2.5,6.8 2
. fAL
A
Ar
F
Fr AfromAabout
)()(
Torque
Magnitude
(yet another cross product)
FrAA
F sinFsinFrAA
FrA
FrA sin
r
sinFrA
Making sense of the
factors and cross-product
F
Large angle, big effect
Small angle, small effect
Large distance, big effect
Small distance, small effect
Angle Distance F
F
Example: a) If rA = 4 m, F = 8 N, and θ = 73°, what is the magnitude of the torque about location A,
including units?
sinFrA
73sinN8m4 Nm6.30
(b) If the force were perpendicular to
but gave the same torque as in the preceding
question, what would its magnitude be?
sinArF
1m4
Nm6.30F N65.7
A yo-yo is in the x-y plane. You pull up on the string with a force of magnitude 0.6 N.
What is the magnitude of the torque (about its center) you exert on the yo-yo?
r = 0.005 m, R= 0.035 m a) 0.005 N· m
b) 0.003 N· m
c) 0.021 N· m
d) 0.035 N· m
e) 0.6 N· m
f) cannot be determined without
knowing the length of the string
r R
x
y
z (z-axis points
out of page)
A
Ar
F
Fr AfromAabout
)()(
Torque
Direction
(yet another cross product)
F
F
Ar
A
x
y
z (z-axis points
out of page)
Orient Right hand so fingers start in direction from axis A to point
of force’s application, then curl in direction of force. The thump
points in direction of torque.
A
Ar
Direction of
torque: +z
Direction of
torque: - z
A
Direction of
torque: no torque
A yo-yo is in the x-y plane. You pull up on the string with a force of magnitude 0.6 N.
What is the direction of the torque (about its center) you exert on the yo-yo?
r = 0.005 m, R= 0.035 m
1) +x
2) –x
3) +y
4) –y
5) +z
6) –z
7) zero magnitude
r R
x
y
z (z-axis points
out of page)
1p
3p
2p
Cloud of Dust
Star
1
2
3
Multi-Particle Angular Momentum Principle:
Cloud of Dust about Star
...321 ssssc LLLL
...232211 prprprL ssssc
11 prdt
dS
dt
pdrp
dt
rdS
S 111
1
netS Frpv .1111
Parallel
focus on one particle
ditto for the others
...232211 prdt
dpr
dt
dpr
dt
dL
dt
dssssc
....33.22.11 netsnetsnetssc FrFrFrLdt
d
extF1
extF2
extF3
where
23133333
32122222
31211111
FFFrFr
FFFrFr
FFFrFr
extsnets
extsnets
extsnets
323133
322122
312111
FFFr
FFFr
FFFr
exts
exts
exts
by reciprocity
sr1
sr2
sr3
323231312121332211 FrrFrrFrrFrFrFrdt
Ldssssssextsextsexts
sc
Cloud of Dust
Star
1
2
3
sr1
sr2
sr3
Multi-Particle Angular Momentum Principle:
Cloud of Dust about Star
....33.22.11 netsnetsnetssc FrFrFrLdt
d extF1
extF2
extF3
3232
3131
2121332211
Frr
Frr
FrrFrFrFrdt
Ld
ss
ss
ssextsextsextssc
extextextsc
dt
Ld321
323231312121 FrFrFr
For central forces (electric, gravitation) ,etc. so , etc. 2121 || Fr
02121 Fr
extnetextextextsc
dt
Ld.321 ...
Note: net torque depends on each force and its point of application.
particlesall
i
extisi Fr.
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
Multi-Particle Angular Momentum Principle
With Multi-Particle Angular Momentum Example: opening a door. Your hands are full so you give the door a quick push with
your foot near the edge. Say you apply a 100 N force for 0.5 s at 75°. The door’s 15kg
and 0.3m wide; what’s its rate of rotation at the end of your push?
System: door
Active environment: your foot,
hinge (but applied at axis-no torque)
Approximations: negligible frictional
torque at hinge 0i
?f
m3.0dAf wr
Axis: hinge
A
N 100fF
75
tLL AfootiAdoorfAdoor
....
tII AfootiAdoorfAdoor
..
Direction: torque & angular velocity
are both “out of the board” (z)
Adoor
Afoot
fI
t
. Adoor
footAf
I
tFr
.
sin
2
31
2
412
121
.
dd
ddddAdoor
wm
wmwmIrod of negligible
radius, about an end
2
31
sin
dd
footd
wm
tFw
dd
foot
wm
tF
31
sin
.3m0kg15
s5.075sinN100
31
srad32
Reasonable?
units?
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?
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)