PHY 113 C Fall 2013 -- Lecture 3 19/3/2013

PHY 113 C General Physics I11 AM-12:15 PM TR Olin 101

Plan for Lecture 3:

Chapter 3 – Vectors

1. Abstract notion of vectors

2. Displacement vectors

3. Other examples

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iclicker question

A. Have you attended a tutoring session yet?B. Have you attended a lab session yet?C. Have you attended both tutoring and lab sessions?

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Mathematics Review -- Appendix B Serwey & Jewett

iclicker question

A. Have you used this appendix?B. Have you used the appendix, and find it helpful?C. Have you used the appendix, but find it unhelpful?

iclicker question

Have you changed your webassign password yet?A. yesB. no

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Question from Webassign #2

1 2 3 4 5 6 7 8 9 10

8

46

2

-8-6 -4-20

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Mathematics Review -- Appendix B Serwey & Jewett

aacbbx

cbxax

24

0

:equation Quadratic

2

2

)cos()sin(

:calculus alDifferenti

1

ttdtd

eedtd

antatdtd

tt

nn

)cos(1)sin(

11

:calculus Integral1

tdtt

edte

natdtat

tt

nn

a

b

c

q

bacacb

q

q

q

tan

sin

cos

:ryTrigonomet

PHY 113 C Fall 2013 -- Lecture 3 89/3/2013

Definition of a vector

1. A vector is defined by its length and direction.

2. Addition, subtraction, and two forms of multiplication can be defined

3. In practice, we can use trigonometry or component analysis for quantitative work involving vectors.

4. Abstract vectors are useful in physics and mathematics.

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Vector addition:

ab

a – b

Vector subtraction:a

-b

a + b

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Some useful trigonometric relations

(see Appendix B of your text)

g

c

b

a

Law of cosines:

a2 = b2 + c2 - 2bc cos

b2 = c2 + a2 - 2ca cos

c2 = a2 + b2 - 2ab cos g

Law of sines:

sin c

sin b

sin a

g

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Some useful trigonometric relations -- continued

(from Appendix B of your text)

g

c

b

a

Law of cosines:

a2 = b2 + c2 - 2bc cos

b2 = c2 + a2 - 2ca cos

c2 = a2 + b2 - 2ab cos g

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Some useful trigonometric relations -- continued

Example:

20o

c=?

15

10

Law of cosines:

a2 = b2 + c2 - 2bc cos

b2 = c2 + a2 - 2ca cos

c2 = a2 + b2 - 2ab cos g

654.60922.43

0922.4320cos151021510 222

c

c o

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Possible realization of previous example:

20o

c=?

15m

10m

Start

South

East

A pirate map gives directions to buried treasure following the indicated arrows. A wily physics students decides to take the easterly direct route after computing the distance c.

treasure

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Quantitative representation of a vector

Cartesian coordinates

i

j

A

Ax

Ay

jiA ˆˆyx AA

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Quantitative representation of a vector

reference direction

AA

q

Note: q can be specified in degrees or radians; make sure that your calculator knows your intentions!

Polar coordinates

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Quantitative representation of a vector

AA

q

Polar & cartesian coordinates

iqcosAAx

qsinAAy

x

y

AA

AA

qqq tan

cossin:note Also

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Vector components:

ax

ay

22yx aa a

jiyxa ˆˆˆˆ yxyx aaaa

yxba

yxbyxaˆˆ

ˆˆ and ˆˆFor

yyxx

yxyx

baba

bbaa

PHY 113 C Fall 2013 -- Lecture 3 189/3/2013

ax

ay

q

y

a = 1 m

Suppose you are given the length of the vector a as shown. How can you find the components?

A. ax=a cos q, ay=a sin qB. ax=a sin y, ay=a cos yC. Neither of theseD. Both of these

iclicker question

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Vector components; using trigonometry

An orthogonal coordinate system

A

kjizyxA ˆˆˆˆˆˆ zyxzyx AAAAAA

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Vector components:

ax

ay

yxba

yxbyxaˆˆ

ˆˆ and ˆˆFor

yyxx

yxyx

baba

bbaa

yxa ˆˆ yx aa

by

bx

yxb ˆˆ yx bb ba

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Examples

Vectors ScalarsPosition r Time tVelocity v Mass mAcceleration a Volume VForce F Density m/VMomentum p Vector components

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Vector componentszyxR ˆˆˆ 1111 zyx

zyxR ˆˆˆ 2222 zyx

zyxRR ˆ(ˆ)(ˆ)( )21212121 zzyyxx

Vector multiplication “Dot” product 1ˆˆ;cos AB xxBA qAB

“Cross” product zyxBA ˆˆˆ;sin|| AB qAB

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Example of vector addition:

a

ba + b

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a

ba + b

gcos2

:Dallas and Chicagobetween Distance22 bababa

o

oooo

74

18021590

g

g

mi 788bag

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Webassign version:

f

Note: In this case the angle f is actually measured as north of east.

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Another example:

ji

BARjiB

jiA

ˆ9.16ˆ7.37

ˆ6.34ˆ0.20

ˆ7.17ˆ7.17

:units kmin ectorsPosition v

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iclicker question

A. Because physics professors like to confuse studentsB. Because physics professors like to use beautiful

mathematical concepts if at all possibleC. Because all physical phenomena can be described by

vectors.D. Because there are some examples in physics that

can be described by vectors

Why are we spending 75 minutes discussing vectors

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Example: Vector addition of velocities

Vb

Vw

Vtotal

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Example: Displacement in two dimensions

(0,0)

(8,5)

43.9)5()8( 22 d