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Physics 218, Lecture II 1
Dr. David Toback
Physics 218Lecture 2
Physics 218, Lecture II 2
Announcements• Having trouble getting started on WebCT? Try:
– ITS Help sessions
– Open access lab/student computing.
– Instructions on faculty.physics.tamu.edu/toback/WebCT
– email to [email protected]
• Check your neo email account for announcements. Two of you had your email bounce
• Any Volume 1 of the 11th edition of Young & Freedman will be fine. You will need Volume 2 for Phys208 and Volume 3 for beyond that. The maroon “University Edition” is also fine.
Physics 218, Lecture II 3
Procedure for Each Week• Week 1 (This week):
– Lecture: Chapter 1 (Reading, but nothing due)– Recitation: Calculus and Lab Techniques– Homework due: None
• Week 2 (Next week):– Homework due (Monday): Math quizzes– Lecture: Chapter 2 (Reading and Lecture Assignment due)– Recitation: Chapter 1 (and Lab 1)
• Week 3 (The week after that):– Homework due (Monday): Chapter 1– Lecture: Chapter 3 (Reading and Lecture Assignment due)– Recitation: Chapter 2 (no lab, but lab 1 is due)
• Etc..
Physics 218, Lecture II 4
Chapter 1: Math n’ Stuff
Won’t cover the entire chapter: •Problem Solving
– Tricks– Methods
•Vectors– Components (Unit vectors)– Addition– Multiplication (dot and cross products)
Physics 218, Lecture II 5
Physics 218, Lecture II 6
Problem Solving Overview
•There are good general problem solving TRICKS
– Units checking– Special case checking– Etc.
•There are good METHODS of problem solving that prepare you for the exams
We’ll use both to solve problems in lecture
Physics 218, Lecture II 7
First Things First!
What’s the first thing you should do when you’re given a a problem?
• Draw a diagram!!!– Usually good for some partial credit
• List givens and wants as variables– Also a good bet for partial credit
Then use reasonable equations and solve with your variables
Trick #1
Physics 218, Lecture II 8
Trick #2: Units• The speed of your car isn’t measured in
seconds, its measured in meters/second (or miles/hour etc.)
• Paying attention to the units will help you catch LOTS of mistakes on exams, quizzes and homework!!– If we ask what the mass of your car is, make
sure your answer is in kg (or lbs etc.)
Trick #2: Every time you finish a problem ALWAYS check the units of your answer!!
Physics 218, Lecture II 9
Tricks #3 and #4
Check Reasonableness:
• Can you find another way to do the same problem that gives the same answer?
• Simple numbers give expected numerical answers? Example: Zero, or infinity
Trick #3
Trick #4
Physics 218, Lecture II 10
How to use the Tricks and Methods
• Next we’ll do an example problem like one of the homework problems in the text book
• Solve this problem using the right method– Draw a diagram– Convert the numbers to variables– Solve to get a formula– Plug in the numbers at the end– Check
• Reasonable numbers? • Silly numbers?• Another way to do the same problem?
Physics 218, Lecture II 11
16 m
Example ProblemYou want to measure the height of a building. You stand 2m away from a 3m pole and see that it’s “in line” with the top of the building. You measure 16 m from the pole to the building. What is the height of the building?
Physics 218, Lecture II 12
Vectors
Vectors: –Why we care about them–Addition & Subtraction–Unit Vectors–Multiplication
Physics 218, Lecture II 13
Why do we care about Vectors?As you may have noticed, the world is not
one-dimensional• Three dimensions: X, Y and Z. Example:
1. Up from us2. Straight in front of us3. To the side from us
– All at 90 degrees from each other. Three dimensional axis.
• Need a way of saying how much in each direction
For this we use VECTORS
Physics 218, Lecture II 14
Vector and Scalar
• Vectors have a magnitude AND a direction–I’m driving 70 miles/hr
SouthEast to Houston
• Scalars are just a number–My speedometer says 70 m/hr
Physics 218, Lecture II 15
Where am I?
My single vector in some funny direction, can be thought of as
two vectors in nice simple directions (like X and Y). This can make things much easier
Let’s say I’m here
You’re here (origin)
I call you on the cell phone. How do I tell you how to get to me?
2 equivalent ways:
1)Travel 11.2 km at an angle of 26.5 degrees
2)Travel 10 km East then 5 km North
Physics 218, Lecture II 16
Vector AdditionTo specify where I am, often
doing the two vector version is easier
Represent Graphically:• Lay down first vector • Lay down second vector
– Put the tail at the head of the first vector
• The “Sum” is where I amAdding vectors is a skill Use this in far more than
just physicsMore on this later…
Physics 218, Lecture II 17
Re-write my location
• Describe my location in terms of the sum of two vectors
• Careful when using the sin and cos
Θ|V|||V
Θ|V|||V
V V V
Y
X
YX
sin
cos
Physics 218, Lecture II 18
Specifying a Vector• Two equivalent ways:
–Components Vx and Vy
–Magnitude V and angle • Switch back and forth
–Magnitude of V |V| = (vx
2 + vy2)½
Pythagorean Theorem–Tan = vy /vx
Either method is fine, but you should pick which is easiest, and be able to use both
Physics 218, Lecture II 19
Unit Vectors
This is how the pros write things!
kV jV iV V
direction z in the 1 means ˆ
directiony in the 1 means ˆ
direction x in the 1 means ˆ
zyx k
j
i
Physics 218, Lecture II 20
Unit Vectors
The pros also use:
zV yV xV V
k as same theis ˆ
j as same theis ˆ
i as same theis ˆ
zyx z
y
x
Physics 218, Lecture II 21
Vector in Unit Vector Notation
jΘ |V| iΘ |V| V
j V i V V
V V V
Θ|V|||V
Θ|V|||V
YX
YX
Y
X
ˆsinˆcos
ˆˆ
sin
cos
Physics 218, Lecture II 22
General Addition Example
Add two vectors using the i-hats, j-hats and k-hats
k km 0 j km 5 i km 10 D
k km 0 j km 5 i km 0 D
k km 0 j km 0 i km 10 D
DD D
R
2
1
21R
Physics 218, Lecture II 23
How do we Multiply Vectors?
• First way: Scalar Product or Dot Product– Why Scalar Product?
• Because the result is a scalar (just a number)
– Why a Dot Product?
• Because we use the notation A.B
• A.B = |A||B|Cos
Physics 218, Lecture II 24
A.B = |A||B|Cos
First Question:
?j i isWhat
? i i isWhat
Physics 218, Lecture II 25
Harder Example
notation?Vector Unit using BA isWhat
j B i B B
j A i A A
YX
YX
Physics 218, Lecture II 26
Vector Cross Product
This is the last way of multiplying vectors we will see
• Direction from the “right-hand rule”
• Swing from A into B!
SinB AC
B A C
Physics 218, Lecture II 27
Vector Cross Product Cont…
Multiply out, but use the Sin to give the magnitude, and RHR to give the direction
)1(sin ˆˆˆ
)1(sin ˆˆˆ
)0(sin 0ˆˆ
jki
kji
ii
Physics 218, Lecture II 28
Cross Product Example
notation?Vector Unit using BA isWhat
j B i B B
j A i A A
YX
YX
Physics 218, Lecture II 29
Results of Math Quizzes
The average of all Math Quizzes taken so far (not the Math Assessment) is about an 8.1 with a standard deviation of just above 1.1.
How to evaluate where you stand. If the average of the scores of all the quizzes you have taken is:
• 95% or above: Well prepared• 85% - 90%: Good, but needs to be better• 80% – 85%: Ok, but really needs some work• 75% - 80%: Hmmmm…maybe get some help• 75% or below: Careful…Definitely get help!
Maybe drop…
Physics 218, Lecture II 30
For Next Week• Before Lecture:
–Read Chapter 2–Math Quizzes due Monday–Lecture Assignment: Q2.8 and Q2.20 (These are
the “Discussion Questions”)• In Lecture
–Cover Chapter 2 –Turn in Lecture Assignment at the beginning
• Recitation, Lab and Homework: –Start HW1 on WebCT before recitation–All Ch. 1 problems due Monday after recitation–Read your lab materials before lab
Physics 218, Lecture II 31
Physics 218, Lecture II 32
Simple Multiplication
• Multiplication of a vector by a scalar– Let’s say I travel 1 km east. What if I had
gone 4 times as far in the same direction?
→Just stretch it out, multiply the magnitudes
• Negatives: – Multiplying by a negative number turns
the vector around
Physics 218, Lecture II 33
Subtraction
Subtraction is easy: • It’s the same as addition but turning
around one of the vectors. I.e., making a negative vector is the equivalent of making the head the tail and vice versa. Then add:
)V(- V V V 1212
Physics 218, Lecture II 34
Where am I?
Traveling East then North is the same as traveling NorthEast
Can think of this the other way: If I had gone NorthEast, the displacement is equivalent to having gone both North and East
My single vector in some funny direction, can be thought of as
two vectors in nice simple directions (like X and Y). This can make things much easier
Physics 218, Lecture II 35
Problem Solving & Diagrams• This class is mostly problem solving (well… you need
to understand the concepts first in order to solve the problems, but we’ll do both).
• In order to solve almost any problem you need a model
• Physicists/engineers are famous for coming up with silly models for complicated problems
• The first step is always:
Trick #2:“Draw a diagram!”
Physics 218, Lecture II 36
Announcement: Free Tutoring
Four foreign graduate students are available to tutor Physics 218 Students without charge. Students desiring help are to e-mail the tutor and arrange a time to meet in Heldenfels 211 on weekdays. The tutors are:
• Sunnam Min, [email protected]
• Xi Wang, [email protected]
• Rongguang Xu, [email protected]
• Hong Lu, [email protected]
Physics 218, Lecture II 37
Components
Let’s do this with the math:
• Break a vector into x and y components (I.e., a right triangle) THEN add them
• This is the sine and cosine game
• Can use the Pythagorean Theorem A2 + B2 = C2
Physics 218, Lecture II 38
Chapter 1: Introduction
This chapter is fairly well written. I won’t lecture on most of it except for the parts which I think are useful in helping you be a better problem solver in general or at least helping you look like a professional
Physics 218, Lecture II 39
Models, theories and Laws
• Models, theories and Laws• Prescriptive vs. Descriptive• What should happen vs. What does
happen when you do an experiment– US law doesn’t allow killing– Physics law shows clearly that it does
happen.
Physics 218, Lecture II 40
Estimating
• Order of Magnitude
• This is a useful thing to be able to do at home
• Let’s say you are at a grocery store and it’s full. How much will it cost you to buy it all?– Estimate using round numbers– 50 items (assuming not lots of little things)– A dollar an item
$50
Physics 218, Lecture II 41
Number of Significant Figures15 ± 1 feet (1 digit in uncertainty, same
“10’s” as last digit)• 15.052 ± 1 feet (Makes you look like an
amateur)• 15 ± 1.05 feet (Same thing)• 15.1 ± 0.1 feet (Ok)• 15 ± 10 feet (Ok)An aside: Personally, I take significant digits
seriously. It makes you look bad when you mess them up. Also, WebCT will do unpredectible things if you don’t use them correctly.
Physics 218, Lecture II 42
Converting Units
Multiplying anything by 1 (no units!) is a GREAT trick! Use it often!!
• 1 meter x 1 = 1 meter• 1 yard x 1 = 1 yard x (3 feet/yard) = 3 feet
(simple! Units cancel out!) • Example:1 football field in feet
– 1 football field x (1) x (1) = 1 football field– 1 football field x (100 yards/1 football field) x (3
feet/yard) = 300 feet– Both are units of length!
Physics 218, Lecture II 43
Significant Figures
• Good test: Write the primary number as 1.5x101 feet (get rid of zeros on either end) which is the “powers of 10 notation” or what we call “scientific notation” – 17526.423 = 1.7526423 x 104
• Then deal with the uncertainty• Usually only one digit in the uncertainty
– Example: Fix 15.052 ± 1 feet → (1.5052 ± 0.1) x 101 feet→ (1.5 ± 0.1) x 101 feet
Physics 218, Lecture II 44
Reference Frames
Frame of reference:
• Need to refer to some place as the origin
• Draw a coordinate axis– We define everything from here
– Always draw a diagram!!!
Physics 218, Lecture II 45
• Vector notation: – In the book,
variables which are vectors are in bold
– On the overheads, I’ll use an arrow over it
• Vectors are REALLY important
• Kinda like calculus: These are the tools!
First the Math: Vector Notation
v
Some motion represented by vectors. What do these vectors represent
physically?
Physics 218, Lecture II 46
Adding vectors in funny directions• Let’s say I walk in some random direction, then in another different
direction. How do I find my total displacement? • We can draw it
• It would be good to have a better way…
Physics 218, Lecture II 47
Example
We have two known displacements D1 and D2. What is the magnitude and angle of the net displacement in this example?
Physics 218, Lecture II 48
Go home with a friend
You are going home with a friend. You live in Houston and your friend lives in San Antonio. First you drive 100 miles SouthEast (known angle ) from Aggieland to Houston, then 300 miles West to San Antonio? Using unit vector notation, what is your displacement from the center of the universe?
Physics 218, Lecture II 49
Examples without an axis
Physics 218, Lecture II 50
Addition using Components
To add two vectors, break both up into their X and Y components…
2y2x2
1y1x1
VVV
VVV
First break each vector into its X and Y
components
21F V V V
Physics 218, Lecture II 51
Addition using Components cont…
Next: add separately in the X and Y directions
2y1yFy
2x1xFx
VVV
VVV
Magnitudes of VF
Physics 218, Lecture II 52
Drawing the components
Physics 218, Lecture II 53
Vector Cross Product Cont…
Calculating the cross product is the same as taking the determinant of a Matrix
AB vs. BA
AA
:Check