dr. david toback

Physics 218, Lecture II 1

Dr. David Toback

Physics 218Lecture 2


In Class Quiz

Write down the most important “student case study” from the Frequently Asked Questions handout


Announcements: WebCT• Having trouble getting started? 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

• Still working on Math Quiz figures… sorry about that..

• Finish your “Preliminary Course Materials”


Due dates coming up•Week 1 (This week):

– Lecture: Chapter 1 (Reading, but nothing due)

– Recitation & Lab: Lab 1 (A&B) – Homework due: None

•Week 2 (Next week):– Homework (Monday): Math quizzes– Lecture: Chapter 2– Recitation & Lab: Chapter 1 and Lab 2

•Week 3 (The week after that):– Homework due (Monday): Chapter 1– Lecture: Chapter 3 & 4 – Recitation: Chapter 2 and Lab 3

•Etc..


Chapter 1: Calculus

•Won’t cover the chapter in detail

•This is a chapter that is best learned by DOING

•We’ll cover it quickly– Lots more examples in Chapter

2– Lots of practice in Math Quizzes

on WebCT (when they’re fixed)


Where are we going?We want Equations that describe•Where am I as a function of time?

•How fast am I moving as a function of time?

•What direction am I moving as a function of time?

•Is my speed changing? Etc.


Use calculus to solve

problems!


Motion in One Dimension• Where is the car?

– X=0 feet at t0=0 sec– X=22 feet at t1=1 sec– X=44 feet at t2=2 sec

• Since the car’s position is changing (i.e., moving) we say this car has “speed” or “velocity”

• Plot position vs. time– How do we get the

speed from the graph?


Speed

Questions:•How fast is my position changing?

•What would my speedometer read?


How do we Calculate the speed?

• Define speed: “Change in position during a certain amount of time”

• Math: Calculate from the Slope: The “Change in position as a function of time”

– Change in Vertical divided by the Change in Horizontal

– Speed = XtChange:


Constant SpeedEquation of

Motion for this example is a straight line

Write this as:X = bt

• Slope is constant• Velocity is

constant– Easy to calculate

– Same everywhere

Position time


Moving Car

A harder example:

X = ct2

•What’s the speed at t=1 sec?

Want to calculate the “Slope” here

What would the speedometer say?


Derivatives• To find the slope at time t, just

take the “derivative”• For X=ct2 , Slope = V =dx/dt =2ct• “Gerbil” derivative method

–If X= atn V=dx/dt=natn-1

– “Derivative of X with respect to t”

• More examples– X= qt2 V=dx/dt=2qt– X= ht3 V=dx/dt=3ht2


Common MistakesThe trick is to remember what you

are taking the derivative “with respect to”

More Examples (with a=constant):• What if X= 2a3tn?

– Why not dx/dt = 3(2a2tn)?– Why not dx/dt = 3n(2a2tn-1)?

• What if X= 2a3?– What is dx/dt?– There are no t’s!!! dx/dt = 0!!!– If X=22 feet, what is the velocity? =0!!!


Going the other way: Integrals

•What if you know how fast you’ve been going and how long you’ve been driving

•How can you figure out how far you’ve gone?

•What would your car’s odometer read?


Getting the Displacement from Velocity

• If you are given the speed vs. time graph you can find the total distance traveled from the area under the curve:X=V0t + ½at2

• Can also find this from integrating…

t

ovdtx

Slope is constant =Constant acceleration


Definite and Indefinite Integrals

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Some Integrals

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Our Example

221

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vdtx-x


For Next Week•Before Lecture:

– Read Chapter 2– Math Quizzes due Monday

•In Lecture– Cover Chapter 2

•Recitation, Lab and Homework: – Start Chapter 1 problems and exercises before recitation

– Read your lab materials before lab


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


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


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


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!”


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]


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


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


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.


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


Number of Significant Figures

15 ± 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.


Converting UnitsMultiplying 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!


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


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!!!


• 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?


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…


Example

We have two known displacements D1 and D2. What is the magnitude and angle of the net displacement in this example?


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?


Examples without an axis


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


Addition using Components cont…

Next: add separately in the X and Y directions

2y1yFy

2x1xFx

VVV

VVV

Magnitudes of VF


Drawing the components


Vector Cross Product Cont…

Calculating the cross product is the same as taking the determinant of a Matrix

AB vs. BA

AA

:Check

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