angular momentum vector product
TRANSCRIPT
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI
Lecture Capture: http://echo360.uml.edu/danylov2013/physics1fall.html
Lecture 19
Chapter 11
Angular momentumVector product.
11.20.2013Physics I
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Chapter 11
Angular Momentum Vector Cross Product Conservation of Ang. Mom. Ang. Mom. of point particle Rigid Objects
Outline
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Vector Cross Product
C
A
B
If we have two vectors
C A
B ABsinMagnitude
Direction: perp. to both A and B (right hand rule)
A Axi Ay j AzkB Bxi By j Bzk
Then the vector product is
A
B
B
AOrder matters:
A
B
C
AB
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Cross product
The cross product vector increases from 0 to AB as θ increases from 0 to 90
A
B
C
A
B
C A
B ABsin
0BA
θ=30
θ=0
A
B
ABBA 21
θ=90A
B
ABBA
The vector product is zero when vectors are parallel
The vector product increases The vector product is max when vectors are perpendicular
ABBA
ABBA 21
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ConcepTest 1 Vector product
A)
B)
C)
For the unit vectorsFind the following vector products
kji ˆ,ˆ,ˆ
?ˆˆ)2 ji
?ˆˆ)1 ii
0ˆˆ iikji ˆˆˆ
iii ˆˆˆ)1
0ˆˆ)2 ji
0ˆˆ)1 iikji ˆˆˆ)2
x
y
z
0ˆˆ)1 iijji ˆˆˆ)2
C A
B ABsin
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
A Axi Ay j Azk
Vector Cross Product
B Bxi By j Bzk
i i 0 j j 0 k k 0 i j k j k i k i j
A
B (Axi Ay j Azk) (Bxi By j Bzk) AxBx (i i ) AxBy (i j) AxBz (i k)
AyBx ( j i ) AyBy ( j j) AyBz ( j k)
AzBx (k i ) AzBy (k j) AzBz (k k)
(AyBz AzBy )i (AzBx AxBz ) j (AxBy AyBx )k
00ˆˆˆˆ Siniiii 190ˆˆˆˆ Sinjiji
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
What is the vector cross product of the two vectors:
A 1i 2 j 4k
A
B 14i 9 j 1k
Vector Cross Product. Example
A
B [(21) (43)]i [(42) (11)] j [(13) (22)]k
A
B (AyBz AzBy )i (AzBx AxBz ) j (AxBy AyBx )k
B 2i 3 j 1k
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Write Torque as the Cross Product
rF
r
F
Axis of rotation
F
rsinF
Let’s look at a door top view:
Applied force F produces torque sinrF
Now, with vector product notation we can rewrite torque as
Torque direction – out of page (right hand rule)
Direction – out of the page
Direction – into the pageNotation convention:
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Angular MomentumAngular momentum is the rotational equivalent
of linear momentum
?L
vmp
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Angular Momentum of a single particle
L
r pL
r mv
x
z
y
O
r pm
L
r p Suppose we have a particle with-linear momentum -positioned at r
p
Then, by definition: Angular momentum of a particle about point O is
If we have many particles, the total angular momentum is
...321 LLLLLi
i
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Angular Momentum of a single particle. Example
L
r pBy definition: Angular momentum of a particle about point O is
What is the angular momentum of a particle of mass m moving with speed v in a circle of radius r in a counterclockwise direction?
L
r p
rp
L
θ=90
sinrpLL
Let’s rewrite this result slightly.
rv Recall: )( rmrrmvL IL 2mrIwhere It looks like we can get a different
expression for L Next
rmv
)( 2mr
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Angular Momentum of a rigid body
L I
points towardsL
For the rotation of a symmetrical object about the symmetry axis, the angular momentum and the angular velocity are related by (without a proof)
IL
IL
IL
I – moment of inertia of a body
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Two definitions of Angular Momentum
L
r
p
L
L I
L
r p
Rigid symmetrical bodySingle particle
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Angular Momentum and Torque/particles
L
r p
dtpdrp
dtrd
dtLd
FrvmvdtLd
Let’s find relationship between angular momentum and torque for a point particle:
0
dtpdFlawndN
2.vmp
dtLd
Torque causes the particle’s angular momentum to change
For many particles:
netii
iLdtd
dtLd
netdtLd
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Angular Momentum and Torque/rigid body
netdtLd
We got exactly the same expression
Let’s find the same relationship between angular momentum and torque for a rigid body:
Inet
Inet dt
dI
dtId )(
dtLd
Torque causes angular momentum to change
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Example:What is the angular momentum (about the origin) of an object of mass m dropped from rest.
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Example: What is the torque (about the origin) of an object of mass m dropped from rest.
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ConcepTest 2 traffic light/car
A car of mass 1000 kg drives away from a traffic light h=10 m high, as shown below, at a constant speed of v=10 m/s. What is the angular momentum of the car with respect to the light?
A) B) C)
skgmk 2 )ˆ(000,10
x
y
z
h
skgmi 2 ˆ000,100
v
)ˆ(000,10)ˆ()ˆ)(( kkmvhkrSinmvprL
skgmk 2 )ˆ(000,10
r
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Conservation of Angular Momentum
I11 I22
netdtLd
Angular momentum is an important concept because, under certain conditions, it is conserved.
If the net external torque on an object is zero, then the total angular momentum is conserved.
0,0 dtLdthenIf net
constL
IL For a rigid body 21 LL
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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19
Thank youSee you on Monday