chapter c2 vectors friday, august 24, 27. scalars most physical quantities can be completely...
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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.
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.
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.
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
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.
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