Physics 106: Mechanics Lecture 05

Wenda Cao

NJIT Physics Department

February 18, 2011

Angular Momentum

Vectors Cross Product Torque using vectors Angular Momentum

February 18, 2011

Vector Basics We will be using vectors a lot in

this course. Remember that vectors have both

magnitude and direction e.g. You should know how to find the

components of a vector from its magnitude and direction

You should know how to find a vector’s magnitude and direction from its components

amF

amF

m

aF

Ways of writing vector notation

y

x

a

sin

cos

aa

aa

y

x

xy

yx

aa

aaa

/tan 1

22

,a

ax

ay

February 18, 2011

y

x

a

z

Projection of a Vector in Three Dimensions

Any vector in three dimensions can be projected onto the x-y plane.

The vector projection then makes an angle from the x axis.

Now project the vector onto the z axis, along the direction of the earlier projection.

The original vector a makes an angle from the z axis.

y

x

a

z

February 18, 2011

Vector Basics You should know how to generalize

the case of a 2-d vector to three dimensions, e.g. 1 magnitude and 2 directions

Conversion to x, y, z components

Conversion from x, y, z components

Unit vector notation:

y

x

a

z

, ,a

cos

sinsin

cossin

aa

aa

aa

z

y

x

xy

z

zyx

aa

aa

aaaa

/tan

/cos1

1

222

kajaiaa zyxˆˆˆ

February 18, 2011

A Note About Right-Hand Coordinate Systems

A three-dimensional coordinate system MUST obey the right-hand rule.

Curl the fingers of your RIGHT HAND so they go from x to y. Your thumb will point in the z direction.

y

x

z

February 18, 2011

Vector Math Vector Inverse

Just switch direction

Vector Addition Use head-tail method, or

parallelogram method

Vector Subtraction Use inverse, then add

Vector Multiplication Three kinds! Multiplying a vector by a

scalar Scalar, or dot product Vector, or cross product

A

A

A

B

BA

A

B

BA

A

B BA

B

kBAjBAiBABA zzyyxxˆ)(ˆ)(ˆ)(

Vector Math by Components

kBAjBAiBABA zzyyxxˆ)(ˆ)(ˆ)(

kAjAiAAB zyxˆˆˆ

ksAjsAisAAsB zyxˆˆˆ

February 18, 2011

Scalar Product of Two Vectors

The scalar product of two vectors is written as It is also called the

dot product

is the angle between A and B

Applied to work, this means

A B

A B cos A B

cosW F r F r

February 18, 2011

Dot Product The dot product says

something about how parallel two vectors are.

The dot product (scalar product) of two vectors can be thought of as the projection of one onto the direction of the other.

ComponentsA

xAAiA

ABBA

cosˆ

cos

B

zzyyxx BABABABA

BA )cos(

)cos( BA

February 18, 2011

Projection of a Vector: Dot Product

The dot product says something about how parallel two vectors are.

The dot product (scalar product) of two vectors can be thought of as the projection of one onto the direction of the other.

ComponentsA

B

zzyyxx BABABABA

Projection is zero

xAAiA

ABBA

cosˆ

cos

February 18, 2011

Derivation How do we show that ? Start with

Then

But

So

zzyyxx BABABABA

kBjBiBB

kAjAiAA

zyx

zyx

ˆˆˆ

ˆˆˆ

)ˆˆˆ(ˆ)ˆˆˆ(ˆ)ˆˆˆ(ˆ

)ˆˆˆ()ˆˆˆ(

kBjBiBkAkBjBiBjAkBjBiBiA

kBjBiBkAjAiABA

zyxzzyxyzyxx

zyxzyx

1ˆˆ ;1ˆˆ ;1ˆˆ

0ˆˆ ;0ˆˆ ;0ˆˆ

kkjjii

kjkiji

zzyyxx

zzyyxx

BABABA

kBkAjBjAiBiABA

ˆˆˆˆˆˆ

February 18, 2011

Cross Product

The cross product of two vectors says something about how perpendicular they are.

Magnitude:

is smaller angle between the vectors Cross product of any parallel vectors = zero Cross product is maximum for

perpendicular vectors Cross products of Cartesian unit vectors:

sinABBAC

BAC

A

B

sinA

sinB

0ˆˆ ;0ˆˆ ;0ˆˆ

ˆˆˆ ;ˆˆˆ ;ˆˆˆ

kkjjii

ikjjkikji

y

x

z

ij

k

i

kj

February 18, 2011

Cross Product Direction: C perpendicular

to both A and B (right-hand rule)

Place A and B tail to tail Right hand, not left hand Four fingers are pointed

along the first vector A “sweep” from first vector

A into second vector B through the smaller angle between them

Your outstretched thumb points the direction of C

First practice ? ABBA

? ABBA

- ABBA

February 18, 2011

More about Cross Product The quantity ABsin is the area of

the parallelogram formed by A and B

The direction of C is perpendicular to the plane formed by A and B

Cross product is not commutative

The distributive law

The derivative of cross product obeys the chain rule Calculate cross product

dt

BdAB

dt

AdBA

dt

d

CABACBA

)(

- ABBA

kBABAjBABAiBABABA xyyxzxxzyzzyˆ)(ˆ)(ˆ)(

February 18, 2011

Derivation How do we show that

? Start with

Then

But

So

kBjBiBB

kAjAiAA

zyx

zyx

ˆˆˆ

ˆˆˆ

)ˆˆˆ(ˆ)ˆˆˆ(ˆ)ˆˆˆ(ˆ

)ˆˆˆ()ˆˆˆ(

kBjBiBkAkBjBiBjAkBjBiBiA

kBjBiBkAjAiABA

zyxzzyxyzyxx

zyxzyx

0ˆˆ ;0ˆˆ ;0ˆˆ

ˆˆˆ ;ˆˆˆ ;ˆˆˆ

kkjjii

ikjjkikji

jBkAiBkA

kBjAiBjAkBiAjBiABA

yzxz

zyxyzxyx

ˆˆˆˆ

ˆˆˆˆˆˆˆˆ

kBABAjBABAiBABABA xyyxzxxzyzzyˆ)(ˆ)(ˆ)(

February 18, 2011

The torque is the cross product of a force vector with the position vector to its point of application

The torque vector is perpendicular to the plane formed by the position vector and the force vector (e.g., imagine drawing them tail-to-tail)

Right Hand Rule: curl fingers from r to F, thumb points along torque.

Torque as a Cross Product

Fr

FrFrrF sin

sum)(vector ri

ii

inet iallall

F

Superposition:

Can have multiple forces applied at multiple points.

Direction of net is angular acceleration axis

February 18, 2011

Calculating Cross Products

mjirNjiF )ˆ5ˆ4( )ˆ3ˆ2(

BA

Solution: i

kj

jiBjiA ˆ2ˆ ˆ3ˆ2

kkkijji

jjijjiii

jijiBA

ˆ7ˆ3ˆ40ˆˆ3ˆˆ40

ˆ2ˆ3)ˆ(ˆ3ˆ2ˆ2)ˆ(ˆ2

)ˆ2ˆ()ˆ3ˆ2(

Calculate torque given a force and its location

(Nm) ˆ2ˆ10ˆ120ˆ2ˆ5ˆ3ˆ40

ˆ3ˆ5ˆ2ˆ5ˆ3ˆ4ˆ2ˆ4

)ˆ3ˆ2()ˆ5ˆ4(

kkkijji

jjijjiii

jijiFr

Solution:

Find: Where:

February 18, 2011

i

kj

Net torque example: multiple forces at a single point

x

y

z

r

1F

3F

2F

3 forces applied at point r :

k )sin( r j 0 i )cos( r r

o30 3 r j 2 F k 2 F i 2 F 321

Find the net torque about the origin:

kk2r jk2r ik2r ki2r ji2r ii2r

)k2 j2 i(2 )kr i(r ) F F F (rF r

zzzxxx

zx

netnet

321

.)3cos(30 )rcos( r .)3sin(30 )rsin( r

oz

oxset

6251

0 i)(2r j2r j)(2r k2r 0 zzxxnet

k5.2 j2.2 3i net

Here all forces were applied at the same point.For forces applied at different points, first calculatethe individual torques, then add them as vectors,i.e., use: sum) (vector Fr i

i alli

i allinet

oblique rotation axisthrough origin

February 18, 2011

Angular Momentum Same basic techniques that were used in

linear motion can be applied to rotational motion. F becomes m becomes I a becomes v becomes ω x becomes θ

Linear momentum defined as What if mass of center of object is not

moving, but it is rotating? Angular momentum

mvp

IL

February 18, 2011

Angular Momentum I Angular momentum of a rotating rigid object

L has the same direction as L is positive when object rotates in CCW L is negative when object rotates in CW

Angular momentum SI unit: kgm2/s

Calculate L of a 10 kg disc when = 320 rad/s, R = 9 cm = 0.09 m

L = I and I = MR2/2 for disc L = 1/2MR2 = ½(10)(0.09)2(320) = 12.96 kgm2/s

IL

L

February 18, 2011

Angular Momentum II Angular momentum of a particle

Angular momentum of a particle

r is the particle’s instantaneous position vector p is its instantaneous linear momentum Only tangential momentum component

contribute r and p tail to tail form a plane, L is

perpendicular to this plane

sinsin2 rpmvrrmvmrIL

)( vrmprL

February 18, 2011

Angular Momentum of a Particle in Uniform Circular Motion

The angular momentum vector points out of the diagram

The magnitude is L = rp sin = mvr sin (90o) = mvr

A particle in uniform circular motion has a constant angular momentum about an axis through the center of its path

O

Example: A particle moves in the xy plane in a circular path of radius r. Find the magnitude and direction of its angular momentum relative to an axis through O when its velocity is v.

February 18, 2011

Angular momentum III Angular momentum of a system of

particles

angular momenta add as vectors be careful of sign of each angular momentum

pr L L...LLL i all

ii i all

i n 2 1 net

p r - p r |L| 2211net

for this case:

221121 prprLLLnet

February 18, 2011

Example: calculating angular momentum for particles

Two objects are moving as shown in the figure . What is their total angular momentum about point O?

m2

m1

221121 prprLLLnet

skgm

mvrmvr

mvrmvrLnet

/ 8.945.2125.31

2.25.65.16.31.38.2

sinsin

2

2211

222111

February 18, 2011

Angular Momentum and Torque

Net torque acting on an object is equal to the time rate of change of the object’s angular momentum

Angular momentum is defined as

Analog in impulset

pF

Lt

t

II

tI

tII

00 )(

t

L

interval time

momentumangular in change

February 18, 2011

SUMMARYTranslation

Force

Linear Momentu

mKinetic Energy

F

vmp

22

1mvK

Rotation

Torque

Angular Momentu

mKinetic Energy

Fr

prl

22

1K

Linear Momentu

m

cmi vMpP

Second Law dt

PdFnet

Angular Momentu

miii LL

for rigid bodies about common fixed axis

Second Law dt

Ld sysnet

Momentum conservation - for closed, isolated systems

Systems and Rigid Bodies

constant Psys

constant Lsys

Apply separately to x, y, z axes