Chapter C2Chapter C2

VectorsVectors

Friday, August 24, 27Friday, August 24, 27

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

Chapter C2 problems are due todayChapter C2 problems are due todayTest will be Friday. Ask any Test will be Friday. Ask any

questions about the practice test questions about the practice test today or Wednesday.today or Wednesday.

Tuesday and Thursday labs will be on Tuesday and Thursday labs will be on vectors.vectors.

Uncertainty revisitedUncertainty revisitedWhat you are expected to What you are expected to

know.know.Be able to estimate the uncertainty Be able to estimate the uncertainty

in a particular measurement.in a particular measurement.Be able to estimate the uncertainty Be able to estimate the uncertainty

of the result when quantities of the result when quantities containing uncertainties are used in containing uncertainties are used in calculations.calculations.

ExampleExample

What is the uncertainty in the area of What is the uncertainty in the area of circle whose radius is 20 paces as circle whose radius is 20 paces as determined by my “pacing it off” if I determined by my “pacing it off” if I know my normal pace is .8 m?know my normal pace is .8 m?

Step 1 – What is the uncertainty in my Step 1 – What is the uncertainty in my pace? pace? There is no single correct answer to this, There is no single correct answer to this,

give an answer you can defend)give an answer you can defend)I will guess the uncertainty in my pace is 5 I will guess the uncertainty in my pace is 5

cm/pace.cm/pace.

The 5 cm represents 5/80 = 6%The 5 cm represents 5/80 = 6% The radius of the circle is .8 x 20 = 16 mThe radius of the circle is .8 x 20 = 16 mThe area of the circle is The area of the circle is ππrr22= = ππ161622=804 =804

mm22

Uncertainty is (0.06Uncertainty is (0.0622+ 0.06+ 0.0622))½½ = 8% = 8%8% of 804 is 64 m8% of 804 is 64 m22

Final answer 804±64 mFinal answer 804±64 m22

It would probably be better to write it as It would probably be better to write it as 800±60 m800±60 m22

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

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

1u

3u

31 uu

31 uu

2u

321 3 uuu

13 uu

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

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.º.

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.º.

Aay ax θ

yaxaA yx ˆˆ

75º75º

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

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

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 labWednesday after Labor Day begin C3 Wednesday after Labor Day begin C3

discussiondiscussion