Welcome to Phys 144!Newtonian mechanics and

Relativity

Dr. Jeff Gu, a humbled geophysicist

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Me… Me… Me…--------------An ordinary talent who happens to be doing what he likes to do (or “doing the only thing he is somewhat capable of doing”---My Significant Other).About Me:

1. NO CRIMINAL RECORD, one $200 Speeding Ticket (paid in full)--- otherwise safe driver, like watching/listening all sports, plays a little bit of the flute, decent at table tennis and basketball, reading everything non-scientific (shamefully, that includes Harry Potter series).Background:1. Born and raised in China, went to High School in the US 2. BSc. in Physics, MSc. in geophysics and computer science, PhD in Physics

What got me inWhat got me out

Logistics ---- see Course Description,

furthermore,

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Physics Background Calibration■ Solid understanding of Newtonian mechanics

Problem solving/insight enhancement■ Beyond formula memorization■ Emphasize insights and Evaluation■ Introduce Calculus

Introduce Einstein's Special Relativity■ First step beyond classical (Newtonian) physics■ Will challenge your concepts of space and time!

Main Course Goals

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Why Study Physics? To understand the properties of the universe we live in, i.e., to apprehend space-time, forces, matter, energy, power, interaction of matters

All science is either physics or stamp collecting. Ernest Rutherford

The thrill of being on the brink of discovery is second only to being madly in love.

Can try to use Heisenberg uncertainty principle to talk your way out of a traffic ticket.

When you are courting a nice girl an hour seems like a second. When you sit on a red-hot cinder a second seems like an hour. That's relativity. Albert Einstein

The ‘feel-good’ Reasons:

“Physics is like sex: sure, it may give some practical results,but that’s not why we do it.” ----- Richard Feyman

‘Practical’ benefits:

ON ‘Practical’ benefits:

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What is Physics?What is Physics?

■ It is many things, depends...

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A more practical reason, to get your precious degree In Physics (curriculum)

Newtonian Mechanics

Classical Mechanics

Quantum MechanicsSpecial Relativity

ElectromagnetismThermodynamics

Statistical MechanicsOptics

Condensed matter physics

Nuclear physics Particle physics

Electricity and MagnetismFluids and waves

General Relativity

“Modern” physics

Advanced Quantum mechanics

144

146 281

244

271211

381/481

362351311/411

372

415

472

484485

Ma Ph 468

Computational Physics (Yours Truly’s favorite)234 (a little bit of every thing+ geophys)

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"Do not worry too much about your difficulties in mathematics, I can assure you that mine are still greater." -Albert Einstein

Regarding Calculus: Let go of the fear…

In physics, your solution should convince a reasonable person. In math, you have to convince a person who's trying to make trouble. Ultimately, in physics, you're hoping to convince Nature. And I've found Nature to be pretty reasonable. Frank Wilczek (Nobel price laureate)

God does not care about our mathematical difficulties. He integrates empirically. Albert Einstein

http://www.1728.com/calcprim.htm

However, Yours Truly does: Check following link (for a simple integral/derivative calculator)

(second by Yours Truly)

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Problem Solving Suggestions

Disclaimer: These are general strategies, may not be appropriate in all cases!

●Draw a diagram if appropriate

■ Can be essential in solving some types of problem

■ Be careful: an inaccurate diagram may make thequestion seem impossible or lead to a wrong result

■ Allows you to assemble the information from thequestion as you read it

●Write down what you know and what the question asks you to calculate

■ Helps you identify any missing pieces of informationyou need which is the first step to finding them!

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●Solve things symbolically (i.e, numbers at the end)!■ Quicker: ‘g' times ‘m' is easier than 9.8 times 73.2

kg ■ Mistakes less likely ■ One solution: if a parameter changes, e.g. 'g' onMoon vs. Earth, it is easy to plug in the new value■ Easy to check Units: Replace symbols by their unitsand ensure the result agrees with what you expect■ Easy to understand Special Cases: e.g. whathappens when 'g' goes to zero?●Check units and/or dimensions

■ If you are calculating a length and get units of kilograms something is wrong!

●Check via common sense ■ My nephew said on his quiz paper that a trout for dinner (bought by mom) has the mass of 2 grams, well, there isn’t much to eat! Do an order of magnitude calculation.

Out-of-worldly answers are found here10

Prefix Symbol FactorYotta YZetta ZExa EPeta PTera TGiga GMega MKilo kHecto hDeka

1024

1021

1018

1015

1012

109

106

103

102

da 101

Prefix Symbol FactorDeci dCenti cMilli m Micro Nano nPico pFemto fAtto aZepto zYocto y

10-1

10-2

10-3

10-6

10-9

10-12

10-15

10-18

10-21

10-24

S.I. system allows for prefixes to the unit name to denote multiples of the unit:

Units

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Originally base units derived from objects:■ Platinum-iridium bar defined the metre■ Platinum-iridium cylinder defined the kilogram■ Second defined in terms of earth's rotation■ Derived types: Force = ma = kg m/s2 = N (Newton)

Increasing understanding of physics allowed these to be defined more accurately...■ Second defined as time needed for 9,192,631,770 oscillations

of the electro-magnetic wave emitted from an atom of caesium-133

■ One metre is the distance travelled by light in 1/299,792,458 seconds▴ Defined this way because we now know the speed of light to be a

universal constant (see relativity later)

Units

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Mass...unfortunately, mass is still defined by the platinum-iridium block!■ Why? - we do not really

understand mass▴ No fundamental

understanding about what causes it

▴ No universally constant mass which can accurately scale up to everyday sizes e.g. electron=9.1x10-31kg!

▴ Future particle physics may provide a more physical measure.

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A little bit of a trivia on dimensionality

length scale mass scale time scale

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Unit ConversionSince multiple units exist for the same quantity it is often useful to convert between them■ After a party, your friend is driving his Honda Civic at approx. 40 ms-1 on

Whitemud, what is the next thing that will happen?

SI is what we usually used in this course, but in real life, conversions may be necessary

1 in = 2.54 cm = 0.0254 m1 ft = 30.48 cm = 0.3048 m1 yard = 91.44 cm = 0.9144 m

1 mile = 1.6093 Km = 1609.3 m

1 OZ = 28.35 g1 lb = 0.4536 kg = 453.6 g

oC = (oF - 32) / 1.8K = oC + 273.15

€

40 m/s

Answer: a date with the Police

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€

= 40 10-3 km1

3600hour

€

=144 km/hour

Why we to be careful with Units

NASA lost its Mars Orbiter spacecraft due to a failure to convert from the US version of imperial to metric units

Confusion can arise 1 us gal= 3.7854 litres (check that gas price!!)1 imp gal = 4.546 litres

Sometimes it is more serious than confusion on the gas prices:

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Dimensional AnalysisDimensional analysis is a good way to check the consistency of mathematical relations

A 'dimension' refers to the physical nature of a property■ mass [M], length [L], time [T] etc. (= base units!)

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For all physical equations the dimensions on both sides must match■ Note that the reverse is NOT true: not all equations

that have matching dimensions have physical meaning!▴ e.g. F=ma and F=½ma both pass dimensional analysis

since the ½ is dimensionless but only F=ma has physical meaning with S.I. units

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Order of Magnitude Calculations and UnitsAs an example, my students and I put in seismometers in the field.The station stays out there for months and need to write to a flashcard (4 GB). How long can the flashcard last out there with a sample rate of 20 samples/sec for 3

channels?

“Powers of 10”

Well, here is what I do for a conservativeEstimate (don’t do this in exams, only as a way to quickly get an approximate answer):

Each channel: 20 samples/sec x 4 bytes/sample = 80 bytes/sec ~ 100 bytes/sec

3 channels ~ 300 bytes/sec

1 day: 300 bytes/sec x 4000 sec/hourx 24 hours/day (say 25, easier)

= 3.0 x 107 bytes/day ~ 30 MB/day

How many days: 4048 MB (say 4000) / (30 MB bytes/day)~ 130 days ~ 4-5 months

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Other ‘Review’ Concepts from Chapter 1

Have fun reviewing some of the topics in Chapter 1 (some will be explained next time).

Significant Figures

My policy on sig. figs. during exams: no calculator dumpsI leave the explicit instruction on exams to assume that all numbers givenare taken to be exact, so two or three sig figs should suffice. However your labs, which explicitly involve measured values and error, will have a different policy, and much of your effort in your labs and lab prep willbe devoted to estimating errors and their propagation, and sig figureissues become critical. Your lab T.A. will explain how this works.

The SI System of Units – the main focus in my exams mass (kg), length (m) and time (s)

Reminder:

Next Trigonometry and Vectors (a review)

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Experimental UncertaintiesYou'll cover this in more detail in the labs!

■ Very important concept: without it you cannot believe any result that you hear!

For example, if someone claims that the chance of Obama getting re-elected is 55.28% according to a recent poll of 2000 people and Sarah Paulin is ~45.72%, are you happy with the statement?■ In short, without errors and more premises these numbers are

meaningless!!

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■ Obvious loop holes: who are those surveyed, what demographic, and what’s up with the decimal places (i.e., are these surveys that good or simply computer dumps)?

€

52.28% ±1.50%

€

47.72% ±1.50%

€

52.28% ± 4.50%

€

47.72% ± 4.50%

■ Even take these for face value, need to know errors since there is a big difference between:

and:

Scalars – physical quantities that can be specified uniquely by a magnitude (and an associated unit where appropriate).Scalars encountered in Phys 144:distance, time, speed, mass, work, energy, power, moment of inertia

* Also we will often consider any pure number like 0, 1, 2, π, e ≈ 2.71828… to be scalars.

* The result of any single experimental measurement is a real-valued scalar.

Vectors – physical quantities that require both a magnitude and a direction to specify them (and an associated unit where appropriate)

Vectors to encounter in Phys 144:

displacement, velocity, acceleration, force, momentum, torque, angular momentum

In N-dimensions, N numbers are required to specify a vector. 22

Sometimes care needs to be taken:■ Speed is a scalar

▴ e.g. "He was travelling at 108km/h on the Whitemud when he had the accident"

■ Velocity is a vector: something travelling at a constant speed can have a non-constant velocity▴ e.g. the moon orbits the earth at (approximately) a constant

speed. However the moon's velocity is constantly changing as it is always accelerating towards the earth.

▴ Speed is the magnitude of the velocityWhen Michael Phelps or Usain Bolt get to the finish line, SUPPOSE I am in the same races/meet (I know… chances=0), do I have higher or lower VELOCITY than they do at that instance? What about the average speed?

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Coordinate Systems and Vector RepresentationsThus in two dimensions we need two numbers to specify a direction. The simplest description is with an ordered pair of numbers each of which describes ‘how much’the vector points along a given perpendicular axis. This leads to the component representation of a vector.

These all represent the same vector.They all have the same magnitudeand direction.

Displacement Vector

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rr = (x,y) = x(1,0) + y(0,1) = xˆ i + yˆ j

■ Notations

€

x

y

⎛

⎝ ⎜

⎞

⎠ ⎟

€

x, y( )

€

xˆ i + yˆ j

€

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

€

x, y, z( )

€

xˆ i + yˆ j + z ˆ k 3D

2D

Common Notations (personally, I go with bold or top arrow)

€

A__

= x, y, z( )

€

rA = x, y, z( )

€

A = x, y, z( ) 24

** Vectors can be split into component vectors (decomposition into orthogonal vectors)

€

rA =

r A X +

r A Y +

r A z

** Vectors can be split into component scalars multiplied by the corresponding unit vectors

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rA = AX

ˆ i + AYˆ j + Az

ˆ k

** Vectors can be split into component scalars multiplied by the corresponding unit vectors

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arg A = tan−1A y

A x

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

** Similarly, for 3D,

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Magnitude of A : A = A x

2+ A y

2+ A z

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** magnitude of a given vector (scalar, looks like absolute value, ‘length’)

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Magnitude of A : A = A x

2+ A y

2

2D

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Magnitudes and Standard Unit Vectors

DEFINE:

)1,0,0(ˆ

)0,1,0(ˆ

)0,0,1(ˆ

k

j

i

)1,0(ˆ

)0,1(ˆ

j

i

Vector Representations using standard unit vectors:

)1,0,0()0,1,0()0,0,1(),,( zyxzyx vvvvvvv

kvjviv zyxˆˆˆ

(by vector addition and scalar multiplication)

in 2D in 3D

An adventurer is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 45° east of south, then 280 m at 30° east of north. After a fourth unmeasured displacement she finds herself back where she started. Use the method of components to determine the magnitude and direction of the fourth displacement.

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Given vectors and),( yx AAA

),( yx BBB

),( yyxx BABABA

i.e. to add vectors you add their components.

What about subtraction (A-B) ??

Treat it as adding a negative B to A.

Vector Addition (in this particular system)

Example of vector addition At a road bend:

■ A car travels at N30˚E for 5 km and changes bearing to N60˚E, compute the total displacement vector and its orientation.

5km

30 ˚

60 ˚6km N 0°

E 90°

S 180°

W 270°

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First vector

East = 5sin30° = 2.5km

North = 5cos30° = 2.5 3km

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Second vector

East = 6sin60° = 3 3 km

North = 6cos60° = 3 km 28

Solution: add these vectors first decompose each into its north and east components (x and y axes)

Trigonometry Need to know well: sine, cosine, tangent

The three functions are defined as:

Opposite

Adjacent

Hypotenuse

€

cosθ =Adjacent

Hypotenuse

€

sinθ =Opposite

Hypotenuse

€

tanθ =Opposite

Adjacent€

θ

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€

θ =tan−1 Opposite

Adjacent= sin−1 Oppose

Hypotenuse= cos−1 Adjacent

HypotenuseWarning: Careful with your calculators, say for inverse tan, it always goes from 0-90 deg. For those of you who program, also need to think of radians.

Also the inverse functions:

Vector AdditionWe can now add each of the components of the same direction together since they are parallel vectors:

€

East = (2.5 + 3 3) km

€

North = (3 +2.5 3) km

€

distance = (2.5 + 3 3)2 + (3 +2.5 3)2 =10.6 km

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Bearing = 90° - tan3 +2.5 3

2.5 + 3 3

⎛

⎝ ⎜

⎞

⎠ ⎟= 46.4° East

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■ Now use the previous example of combining two perpendicular vectors to get the final distance and direction from the starting point

Given vectors and),( yx AAA

),( yx BBB

),( yyxx BABABA

i.e. to add vectors you add their components.

What about subtraction (A-B) ??

Treat it as adding a negative B to A.

Vector Addition (in this particular system)

Example

amF

When we write we really mean: ),(),( yxyx mamaFF That is

yy

xx

maF

maF

Scalar MultiplicationGiven a vector and a scalar , scalar multiplication is DEFINED as

€

s

),( yx sAsAAs ),( yx AAA

Multiplication(a) Dot Product (or ‘scalar product’)

€

rA ⋅

r B =

r A ⋅

r B cosθ

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In component/matrix form:

€

rA ⋅

r B = Ax Ay Az( )

1 0 0

0 1 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Bx

By

Bz

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟= AxBx + AyBy + AzBz

Special case: one of the vectors is a unit vector (magnitude=1), dot product gives the component of the ‘other’ vector in the direction of the unit vector,

€

rA ⋅ ˆ x =

r A ˆ x cosθ =

r A cosθ

(b) vector Product (or ‘cross-product’, a vector)The resultant vector from a cross product is perpendicularto both vectors (following a right-handed rule),

The direction… The magnitude…

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€

rA ×

r B =

r A

r B sinφ

€

rA ×

r B ⊥

r A and

r B

Cross product in component form,

€

r A ×

r B =

ˆ x ˆ y ˆ z

Ax Ay Az

Bx By Bz

=

ˆ i ˆ j ˆ k

Ax Ay Az

Bx By Bz

To compute the determinant

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rA ×

r B = AyBz − AyBz( )ˆ i − AxBz − AzBx( ) ˆ j + AxBy − AyBx( ) ˆ k

Note: the alternating sign for the middle element! 33

If A

B

and are parallel ABBA

(product of magnitudes)

If A

B

and are perpendicular 0 BA

Other properties of the dot product:

ABBA

Where will we see it? Work: sFW

CABACBA

)( (commutative and distributive)

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rA ×

r B = −

r B ×

r A

Some simple properties:

Basic Calculus

Differentiation: if y=f(x) then the first derivative of y is simply the slope (gradient) at each position.

€

f (x) = x n + c (c = constant)(1) polynomials:

€

df

dx= nx n−1

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€

f (x) = g(x)h(x) ( f , g, h are func. of x)(2) Product rule:

€

df

dx=

dg

dxh + g

dh

dx

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f (x) = (x n + c)m (c = constant, m,n = integer)(3) Chain rule:

Deductive thinking: how can I find derivative of g/h?

€

df

dx= (m)(x n + c)m−1(n)x n−1 = mnx n−1(x n + c)m−1

i.e.,

€

df (g(x))

dx= f '(g(x))g'(x)

(4) Trigonometric functions:

€

d(sinθ)

dx= cosθ

€

d(cosθ)

dx= −sinθ

(5) log/exponential functions:

€

d(ln(x))

dx=

1

x

€

d(ex )

dx= ex

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€

x n

a

b

∫ dx =x n +1

n +1

⎡

⎣ ⎢

⎤

⎦ ⎥a

b

=bn +1

n +1-

an +1

n +1

(2) Polynomials with limits:

Integration: the area between a and b along x-axis and a line defined by a function f(x) is given by the integral of f(x) between a and b.

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x n∫ dx =x n +1

n +1+ c (c = constant)(1) polynomials:

Can do similar operations for trig and log functions. Basic rule of thumb: Differentiation and integration are opposite operations. Verify your integration result by differentiating it. Don’t forget the constant for ‘Open’ integrals (no limits).

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