Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

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Lecture 19

Chapter 11

Angular momentumVector product.

11.20.2013Physics I

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

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

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

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

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

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

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:

Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19

Angular MomentumAngular momentum is the rotational equivalent

of linear momentum

?L

vmp

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

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

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

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

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

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

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.

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.

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

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

Department of Physics and Applied Physics95.141, Fall 2013, Lecture 19

Thank youSee you on Monday