Welcome to Phys 144!
Newtonian mechanics and
Relativity
Dr. Jeff Gu, a humbled geophysicist
1
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 in
What got me out
2
Logistics ---- see Course Description,
furthermore,
3
Main Course Goals
Physics Background Calibration

■

Problem solving/insight enhancement
■
■
■

Solid understanding of Newtonian mechanics
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!
4
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
The ‘feel-good’ Reasons:
The thrill of being on the brink of discovery is second only to
being madly in love.
All science is either physics or stamp collecting. Ernest
Rutherford
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
‘Practical’ benefits:
ON ‘Practical’ benefits:
Can try to use Heisenberg uncertainty principle
to talk your way out of a traffic ticket.
“Physics is like sex: sure, it may give some practical results,
5
but that’s not why we do it.” ----- Richard Feyman
What is Physics?

What is Physics?
■
It is many things, depends...
6
A more practical reason, to get your precious degree
In Physics (curriculum)
Newtonian Mechanics
144
146
281
Fluids and waves
244
Classical Mechanics
211
Thermodynamics
311/411
Statistical Mechanics
Electricity and Magnetism
381/481
Electromagnetism
271
“Modern” physics
351
362
Special Relativity
372
Quantum Mechanics
Optics
Ma Ph 468
General Relativity
415
Condensed matter physics
472
Advanced Quantum mechanics
484
485
Nuclear physics
Particle physics
234 (a little bit of every thing+ geophys)
Computational Physics (Yours Truly’s favorite)
7
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)
Regarding Calculus: Let go of the fear…
"Do not worry too much about your difficulties in mathematics,
I can assure you that mine are still greater." -Albert Einstein
(second by Yours Truly)
God does not care about our mathematical difficulties.
He integrates empirically.
Albert Einstein
However, Yours Truly does:
Check following link (for a simple integral/derivative calculator)
http://www.1728.com/calcprim.htm
8
Problem Solving Suggestions
Disclaimer: These are general strategies, may not be
appropriate in all cases!
●Write down what you know and what the
question asks you to calculate
■ Helps you identify any missing pieces of information
you need which is the first step to finding them!
Draw a diagram if appropriate
●
■ Can be essential in solving some types
of problem
■ Allows you to assemble the information from the
question as you read it
■ Be careful: an inaccurate diagram may make the
question seem impossible or lead to a wrong result
9
●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' on
Moon vs. Earth, it is easy to plug in the new value
■ Easy to check Units: Replace symbols by their units
and ensure the result agrees with what you expect
■ Easy to understand Special Cases: e.g. what
happens 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 here
10
Units
S.I. system allows for prefixes to the unit name
to denote multiples of the unit:
Prefix Symbol Factor
Yotta
Zetta
Exa
Peta
Tera
Giga
Mega
Kilo
Hecto
Deka
Y
Z
E
P
T
G
M
k
h
da
1024
1021
1018
1015
1012
109
106
103
102
101
Prefix Symbol Factor
Deci
Centi
Milli
Micro
Nano
Pico
Femto
Atto
Zepto
Yocto
d
c
m
n
p
f
a
z
y
10-1
10-2
10-3
10-6
10-9
10-12
10-15
10-18
10-21
10-24
11
Units

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)
12
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.
13
A little bit of a trivia on dimensionality
length scale
mass scale
time scale
14
Unit Conversion

Since 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?
10 -3 km
40 m/s = 40 1
hour
3600
144 km/hour
Answer: a date with the Police


SI is what we usually used in this course, but in real life,
conversions may be necessary

1 in =
1 ft =
1 yard
1 mile
2.54 cm = 0.0254 m
30.48 cm
= 0.3048 m
= 91.44 cm
= 0.9144 m
=
1.6093 Km
=
1609.3 m
oC
K
=
=
1 OZ
1 lb =
(oF - 32) / 1.8
oC + 273.15
= 28.35 g
0.4536 kg
=
453.6 g
15
Why we to be careful with Units

Confusion can arise 1 us gal =
1 imp gal
3.7854 litres (check that gas price!!)
= 4.546 litres
Sometimes it is more
serious than confusion
on the gas prices:

NASA lost its Mars
Orbiter spacecraft
due to a failure to
convert from the
US version of
imperial to metric
units
16
Dimensional Analysis
Dimensional 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!)
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
17
18
Order of Magnitude Calculations and Units
“Powers of 10”
As 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?
Well, here is what I do for a conservative
Estimate (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/hour
x 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)
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~ 130 days ~ 4-5 months
Other ‘Review’ Concepts from Chapter 1
Have fun reviewing some of the topics in Chapter 1 (some will be
explained next time).
Reminder:
The SI System of Units – the main focus in my exams mass (kg), length (m) and
time (s)
Significant Figures
My policy on sig. figs. during exams: no calculator dumps
I leave the explicit instruction on exams to assume that all numbers given
are 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 will
be devoted to estimating errors and their propagation, and sig figure
issues become critical. Your lab T.A. will explain how this works.
Next
Trigonometry and Vectors (a review)
20
Experimental Uncertainties

You'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!!
■ 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)?
■
■
Even take these for face value, need to know errors
since there is a big difference between:
52.28%1.50%
47.72%1.50%
and:
52.28% 4.50%
47.72% 4.50%
21
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 velocity
When 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?
23
Coordinate Systems and Vector Representations
Thus 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
These all represent
axis. This leads to the component
the same vector.
representation of a vector.
They all have the
same magnitude
and direction.
■ Notations
r
r  (x, y)  x(1,0)  y(0,1)  xiˆ  yˆj
2D
x 
  x, y 
y 
3D
xiˆ  yˆj

Common Notations
(personally, I go with bold or

top

r arrow)
A  x, y, z
Displacement Vector
x 
 
y 
 
z 
A  x, y, z
x, y, z xiˆ  yˆj  zkˆ

A  x, y, z
__
24
** Vectors can be split into component vectors (decomposition into
orthogonal vectors)
r r
r
r
A  AX  AY  Az
** Vectors can be split into component scalars multiplied by the
corresponding unit vectors
r
A  AX iˆ  AY ˆj  Az kˆ
** magnitude of a given vector (scalar, looks like absolute value, ‘length’)
Magnitude of A : A  A x  A y
2
2
2D

** Vectors can be split into component scalars
multiplied by the corresponding unit vectors

A
y
1

arg A  tan 
A x




** Similarly, for 3D,
2
Magnitude of A : A  A x  A y  A z
2
2
25
Magnitudes and Standard Unit Vectors
iˆ  (1,0)
in 2D
ˆj  (0,1)
iˆ  (1,0,0)
ˆj  (0,1,0)
DEFINE:
in 3D
kˆ  (0,0,1)
(by vector addition and scalar multiplication)
Vector Representations using
standard unit vectors:

v  (v x , v y , v z )  v x (1,0,0)  v y (0,1,0)  v z (0,0,1)
 v iˆ  v ˆj  v kˆ
x
y
z
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.
26
Vector Addition
 system)
 (in this particular
Given vectors
B  ( Bx , By )
A  ( Ax , Ay ) and
 
A  B  ( Ax  Bx , Ay  By )
i.e. to add vectors you add their components.
What about subtraction (A-B) ??
Treat it as adding a negative B to
A.
27
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.
60 ˚
6km
5km
N 0°
W 270°
E 90°
30 ˚
S 180°
Solution: add these vectors first decompose each into its
north and east components (x and y axes)
First vector
Second vector
East = 5sin30   2.5km
East = 6sin60   3 3 km
North = 5cos30  2.5 3km
North = 6cos60  3 km
28
Trigonometry
Need to know well: sine, cosine, tangent

Hypotenuse
Opposite

Adjacent
The three functions are defined as:
Adjacent
Opposite
cos 
sin   
Hypotenuse
Hypotenuse


Opposite
tan  
Adjacent
Also the inverse functions:
Opposite
Oppose
1
1 Adjacent
  tan
 sin
 cos


Adjacent
Hypotenuse
Hypotenuse
1
Warning: Careful with your calculators, say for inverse tan, it always goes from
29
0-90 deg. For those of you who program, also need to think of radians.
Vector Addition
We 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
Now use the previous example of combining two
perpendicular vectors to get the final distance and direction

 from the starting point
■
distance = (2.5 + 3 3) 2  (3 + 2.5 3) 2 =10.6 km

3 +2.5 3 
Bearing = 90 - tan
= 46.4 East
2.5 + 3 3 
30
Vector Addition
 system)
 (in this particular
Given vectors
B  ( Bx , By )
A  ( Ax , Ay ) and
 
A  B  ( Ax  Bx , Ay  By )
i.e. to add vectors you add their components.
What about subtraction (A-B) ??
Treat it as adding a negative B to
A.
Scalar Multiplication

Given a vector A  ( Ax , Ay ) and a scalar
s
, scalar multiplication is DEFINED as

sA  ( sAx , sAy )
Example
That is
When we write

 we really mean: ( F , F )  (ma , ma )
F  m
a
x
y
x
y
Fx  max
Fy  ma y
Multiplication
(a)
Dot Product (or ‘scalar product’)
r r r r
A  B  A  B cos
31
In component/matrix form:
r r
A  B  Ax Ay
1 0 0Bx 

 
Az 0 1 0By  Ax Bx  Ay By  Az Bz

 
0 0 1Bz 
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,

r
r
r
A  xˆ  A xˆ cos  A cos
(b)
vector Product (or ‘cross-product’, a vector)
The resultant vector from a cross product is perpendicular
to both vectors (following a right-handed rule),

The direction…
The magnitude…
32
r r
r r
A  B  A B sin 
r r r
r
A  B  A and B
Cross product in component form,
xˆ
r r
A  B  Ax
Bx


yˆ
Ay
By
zˆ
iˆ
Az  Ax
Bz Bx
ˆj
Ay
By
kˆ
Az
Bz
To compute the determinant

r r
A  B  Ay Bz  Ay Bz iˆ  Ax Bz  Az Bx ˆj  Ax By  Ay Bx kˆ
Note: the alternating sign for the middle element!
33
Some simple properties:
If
If

A

A
and
and

B

B
are parallel
are perpendicular
 
A
  B  AB
A B  0
(product of magnitudes)
Other properties of the dot product:
   
A B  B  A
Where will we see it?
  
   
A  (B  C)  A  B  A  C
Work:
 
W  F s
(commutative and
distributive)
r r
r r
A  B  B  A
Basic Calculus
Differentiation:
 if y=f(x) then the first derivative of y is simply
the slope (gradient) at each position.
(1) polynomials: f (x)  x n  c (c = constant)
df
 nx n1
dx
34
(2) Product rule:
f (x)  g(x)h(x) ( f , g, h are func. of x)
df dg
dh
 hg
dx dx
dx
Deductive thinking: how can I find derivative of g/h?

(3) Chain rule:


i.e.,
f (x)  (x n  c) m (c = constant, m,n = integer)
df
 (m)(x n  c)m1(n)x n1  mnx n1(x n  c)m1
dx
df (g(x))
 f '(g(x))g'(x)
dx
(4) Trigonometric
functions:

d(sin  )
 cos
dx
(5) log/exponential functions:
d(cos )
 sin 
dx
x
d(e
)
x
d(ln( x)) 1

e

dx 35
dx
x
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.
(1) polynomials:

n 1
x
x n dx 
c
n 1
(2) Polynomials with limits:
b

a


(c = constant)
n 1 b
n 1
n +1

x
b
a
x n dx  
 
n  1a n  1 n  1
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).
36