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)