Chapter C2 Vectors Fall 2008 Problems C2B.1, B2, B4, B7, B8, B9, C2S.1 Due Thursday.

Chapter C2Chapter C2

VectorsVectors

Fall 2008Fall 2008

Problems C2B.1, B2, B4, B7, Problems C2B.1, B2, B4, B7, B8, B9, C2S.1 Due Thursday.B8, B9, C2S.1 Due Thursday.

ScalarsScalarsMost physical quantities can be Most physical quantities can be

completely described by a single completely described by a single number. Some examples are:number. Some examples are:MassMassVolumeVolumeTimeTimeChargeChargeHeightHeightScore on testScore on test

Physical quantities that can be described Physical quantities that can be described by a single number are called scalars.by a single number are called scalars.

VectorsVectors

Certain physical quantities need a Certain physical quantities need a direction as well as a size (magnitude) to direction as well as a size (magnitude) to completely describe them. Some completely describe them. Some examples are:examples are: VelocityVelocity ForceForce DisplacementDisplacement MomentumMomentum AccelerationAcceleration

These are called vectors.These are called vectors.

VectorsVectors

1u

12u

1u

2u

1u

adding Vectorsadding Vectors You may move a vector anyplace as long as You may move a vector anyplace as long as

the magnitude and the direction remain the magnitude and the direction remain unchangedunchanged

When adding vectors, 1) move the second When adding vectors, 1) move the second vector, putting the tail of the second vector on vector, putting the tail of the second vector on the head of the first vector.the head of the first vector.

2) Begin the tail of the resultant vector on the 2) Begin the tail of the resultant vector on the tail of the first vector and put the head of the tail of the first vector and put the head of the resultant on the head of the last vector. resultant on the head of the last vector.

3) Be certain to draw the head on the resultant 3) Be certain to draw the head on the resultant vector!vector!

To subtract vectors, change the direction of To subtract vectors, change the direction of the vector with the minus sign and add.the vector with the minus sign and add.

adding Vectorsadding Vectors

1u

2u

3u

21 uu

321 uuu

Vector ComponentsVector Components

u

xu

yuθ

22)( yx uuumagu

This is the length or the magnitude of the vector.

The choice of a coordinate system is arbitrary. In physical problems there is often a particular choice that makes the problem easier.

Vector ComponentsVector Components

u

xu

yuθ

uy=u sinθ

ux=u cosθtanθ=uy/ux

yx uuu 22

zyx uuuu 222

In class exercisesIn class exercises

1u

3u

31 uu

2u

On your paper draw:

31 uu

1u

3u

In class exercisesIn class exercises

1u

3u

31 uu

2u

On your paper On your paper draw:draw:

1u

3u

31 uu

In class exercisesIn class exercises

1u

3u

2u

13 uu

Do on your Do on your

paper:paper:

In class exercisesIn class exercises

1u

3u

2u

321 3 uuu

1u

23u

321 3 uuu

On your paper, draw the On your paper, draw the following vectors.following vectors.

xA ˆ3

yxB ˆ2ˆ4

0

4

3

C

0

2

3

D

Unit vectorsUnit vectors

zuyuxuu zyx ˆˆˆ

z

y

x

u

u

u

u

zyx ˆ,ˆ,ˆ are unit vectors

Note that ux,uy and uz are not vectors

In class exercisesIn class exercises Find the components of the vector a below if Find the components of the vector a below if

its magnitude is 30 and the angle with its magnitude is 30 and the angle with respect to the x axis is 35respect to the x axis is 35º. Raise your hand º. Raise your hand when you have the equation for the vector when you have the equation for the vector written in the form.written in the form.

Find the components of the vector B below if Find the components of the vector B below if its magnitude is 20 and the angle with its magnitude is 20 and the angle with respect to the y axis is 75respect to the y axis is 75º. Raise your hand º. Raise your hand when done.when done.

Aay ax θ

yaxaA yx ˆˆ

75º75º

B

Same problem, continuedSame problem, continued

Add the two vectorsAdd the two vectorsGive the components of the resultantGive the components of the resultantGive the magnitude of the resultantGive the magnitude of the resultantGive the angle of the result with the x Give the angle of the result with the x

axisaxisDraw a figure showing your results.Draw a figure showing your results.

yxB ˆ0.29ˆ8.7

yxC ˆ5.17ˆ2.24

yxA ˆ5.11ˆ4.16

yaxaA yx ˆˆ

C

θ

C = 29.9

Θ =35.9°Φ=54.1°

-24

-17

Now calculate the magnitude and direction of the vector B-2A.

Test will be Monday. If you wish Test will be Monday. If you wish more time you may start early or more time you may start early or stay after class.stay after class.

Hand in the vector lab (even if the Hand in the vector lab (even if the last part is not complete.last part is not complete.

Any questions on the problems? All Any questions on the problems? All problems should be handed in.problems should be handed in.

After a brief discussion of uncertainty After a brief discussion of uncertainty and the unit multiplier methods, we and the unit multiplier methods, we will discuss any questions you have will discuss any questions you have about the old tests.about the old tests.

Use of the unit operatorUse of the unit operator

Given that there are 2.21 lbs/kg, Given that there are 2.21 lbs/kg, calculate the number of nanograms in a calculate the number of nanograms in a ton (2000 lb).ton (2000 lb).

lbslbsngng

lbs2000

kg

lbs21.2

kg

g1000

g

ng9101

ng14109

On Wednesday we will review for the On Wednesday we will review for the test over chapters 1 and 2.test over chapters 1 and 2.Ask any question – ask to see any type Ask any question – ask to see any type

of example worked.of example worked.Hint for test: You not only need to know Hint for test: You not only need to know

how to do the problem, you must clearly how to do the problem, you must clearly show how you arrived at your answer.show how you arrived at your answer.

Thursday – Finish vectors labThursday – Finish vectors lab