PHY121 Summer Session I, 2006
Instructor : Chiaki Yanagisawa
• Most of information is available at:
http://nngroup.physics.sunysb.edu/~chiaki/PHY121-06.
It will be frequently updated.
• Homework assignments for each chapter due a week later (normally)
and are delivered through WebAssign. Once the deadline has passed
you cannot input answers on WebAssign.
To gain access to WebAssign, you need to obtain access code and
go to http://www.webassign.net. Your login username, institution
name and password are: initial of your first name plus last name
(such as cyanagisawa), sunysb, and the same as your username,
respectively.
• In addition to homework assignments, there is a reading requirement
of each chapter, which is very important.
• The lab session will start next Monday (June 5), for the first class
go to A-117 at Physics Building. Your TAs will divide each group
into two classes in alphabetic order.
Chapter 1: Introduction
Standards of Length, Mass and Time
 A physical
quantity is measured in a unit which specifies
the scale of the quantity.
• SI units (Systèm International), also known as MKS
A standard system of units for fundamental quantities of
science an international committee agreed upon in 1960.
• Fundamental unit of length : meter (m)
1 m = 100 cm = 1,000 mm, 1 km = 1,000 m,…
1 inch = 2.54 cm = 0.0254 m, 1 foot = 30 cm = 0.30 m
The meter was defined as the distance traveled by light in
vacuum during a time interval of 1/299,792,458 seconds in
1980.
Standards of Length, Mass and Time
 A physical
quantity is measured in a unit which specifies
the scale of the quantity (cont’d)
• Fundamental unit of mass : kilogram (kg)
1 kg = 1,000 g, 1 g = 1,000 mg, 1 ton = 1,000 kg
1 pound = 0.454 kg = 454 g, 1 ounce = 28.3 g
The kilogram is defined as the mass of a specific platinum
iridium alloy cylinder kept at the International Bureau of
Weights and Measures in France.
• Fundamental unit of time : second (s or sec)
1 sec = 1,000 msec = 1,000,000 msec,…
1 hour = 60 min = 3,600 sec, 24 hours = 1 day
The second is defined as 9,192,631,700 times the period
of oscillation of radiation from cesium atom.
Standards of Length, Mass and Time
 A physical
quantity is measured in a unit which specifies
the scale of the quantity (cont’d)
• Scale of some measured lengths in m
Distance from Earth to most remote normal galaxies
Distance from Earth to nearest large galaxy (M31)
Distance from Earth to closest star (Proxima Centauri)
Distance for light to travel in one year (light year)
Distance from Earth to Sun (mean)
Mean radius of Earth
Length of football field
Size of smallest dust particle
Size of cells in most living organism
Diameter of hydrogen atom
Diameter of atomic nucleus
Diameter of proton
4 x 1025
2 x 1022
4 x 1016
9 x 1015
2 x 1011
6 x 106
9 x 101
2 x 10-4
2 x 10-5
1 x 10-10
1 x 10-14
1 x 10-15
Standards of Length, Mass and Time
 A physical
quantity is measured in a unit which specifies
the scale of the quantity (cont’d)
• Scale of some measured masses in kg
Observable Universe
Milky Way Galaxy
Sun
Earth
Human
Frog
Mosquito
Bacterium
Hydrogen atom
Electron
1 x 1052
7 x 1041
2 x 1030
6 x 1024
7 x 101
1 x 10-1
1 x 10-5
1 x 10-15
2 x 10-27
9 x 10-31
Standards of Length, Mass and Time
 Other
systems of units
• cgs
: length in cm, mass in g, time in s
area in cm2, volume in cm3, velocity in cm/s
• U.S. customary : length in ft , mass in lb, time in s
area in ft2 , volume in ft3, velocity in ft/s
 Prefix
10-12
pico- (p)
1012
tera- (T)
10-9
nano- (n)
109
giga- (G)
10-6
10-3
micro- (m) milli- (m)
106
103
mega- (M) kilo- (k)
10-2
centi- (c)
101
deka- (da)
The Building Blocks of Matter
 History
of model of atoms
nucleus (protons and neutrons)
proton
Old view
electrons eSemi-modern view
quarks
nucleus
Modern view
Dimensional Analysis
 In
physics, the word dimension denotes the physical
nature of a quantity
• The distance can be measured in feet, meters,… (different
unit), which are different ways of expressing the dimension
of length.
• The symbols that specify the dimensions of length, mass and
time are L, M, and T.
dimension of velocity [v] = L/T (m/s)
dimension of area
[A] = L2 (m2)
Dimensional Analysis
 In
physics, it is often necessary either to derive a
mathematical expression or equation or to check
its correctness. A useful procedure for this is called
dimensional analysis.
• Dimensions can be treated as algebraic quantities:
dimension of distance [x]
= L (m)
dimension of velocity [v]
= [x]/[t] = L/T (m/s)
dimension of acceleration [a] = [v]/[t] = (L/T)/T
= L/T2
= [x]/[t]2 (m/s2)
Uncertainty in Measurement
 In
physics, often laws in form of mathematics are
tested by experiments. No physical quantity can be
determined with complete accuracy.
• Accuracy of measurement depends on the sensitivity of the
apparatus, the skill of the person conducting the measurement,
and the number of times the measurement is repeated.
• For example, assume the accuracy of measuring length
of a rectangular plate is +-0.1 cm. If a side is measured to be
16.3 cm, it is said that the length of the side is measured to
be 16.3 cm +-0.1 cm. Therefore, the true value lies between
16.2 cm and 16.4 cm.
Significant figure : a reliably known digit
In the example above the digits 16.3 are reliably known
i.e. three significant digits with known uncertainty
Uncertainty in Measurement (cont’d)
•
Area of a plate: length of sides 16.3+-0.1 cm, 4.5+-0.1 cm
The values of the area range between
(16.3-0.1 cm)(4.5-0.1 cm)= (16.2 cm)(4.4cm)=71.28 cm2
=71 cm2 and (16.3+0.1 cm)(4.5+0.1 cm)=75.44 cm2=
75 cm2.
The mid-point between these two extreme values
is 73 cm2 with uncertainty of +-2 cm2 .
Two significant figures! (Note that 0.1 has only one significant
figure as 0 is simply a decimal point indicator.)
Uncertainty in Measurement (cont’d)
 Two
rules of thumb to determine the significant figures
1) In multiplying (dividing) two or more quantities, the number of
significant figures in the final product (quotient) is the same as
the number of significant figures in the least accurate of the
factors being combined, where least accurate means having the
lowest number of significant figures.
•
Area of a plate: length of sides 16.3+-0.1 cm, 4.5+-0.1 cm
16.3 x 4.5 = 73.35 = 73 (rounded to two significant figures).
three (two) significant figures
To get the final number of significant digit, it is necessary
to do some rounding: If the last digit dropped is less than 5,
simply drop the digit. If it is greater than or equal to 5, raise
the last retained digit by one
Uncertainty in Measurement (cont’d)
2) When numbers are added (subtracted), the number of decimal
places in the result should equal the smallest number of decimal
places of any term in the sum (difference).
•
A sum of two numbers 123 and 5.35:
123.xxx
zero decimal places
two decimal places
+ 5.35x
-----------zero decimal places
128.xxx
123 + 5.35 = 128.35 = 128
Uncertainty in Measurement (cont’d)
• More complex example : 2.35 x 5.86/1.57
- 2.35 x 5.89 = 13.842 = 13.8
13.8 / 1.57 = 8.7898 = 8.79
- 5.89 / 1.57 = 3.7516 = 3.75
2.35 x 3.75 = 8.8125 = 8.81
- 2.35 / 1.57 = 1.4968 = 1.50
1.50 x 5.89 = 8.835 = 8.84
A lesson learned :
Since the last significant digit is only one representative
from a range of possible values, this amount of discrepancies
is expected.
Conversion of Units
 Since
we use more than one unit for the same quantity,
it is often necessary to convert one unit to another
• Some typical unit conversions
1 mile = 1,609 m = 1.609 km, 1 ft = 0.3048 m = 30.48 cm
1 m = 39.37 in. = 3.281 ft, 1 in. = 0.0254 m = 2.54 cm
• Example 1.4
28.0 m/s = ? mi/h
Step 1: Conversion from m/s to mi/s:
m  1.00 mi 

2
28.0 m/s   28.0 
  1.74 10 mi/s
s  1609 m 

Step 2: Conversion from mi/s to mi/h:
mi 
s 
min

1.74 10  2 mi/s  1.74 10-2
60
.
0
60
.
0


s
min
h




  62.6 mi/h

Estimates and Order-of Magnitude
 For many problems, knowing the approximate value
of a quantity within a factor of 10 or so is quite useful.
This approximate value is called an order-of-magnitude
estimate.
• Examples
- 75 kg ~ 102 kg (~ means “is on the order of” or “is
approximately”)
- p = 3.1415…~1 (~3 for less crude estimate)
Estimates and Order-of Magnitude
• Example 1.6 : How much gasoline do we use?
Estimate the number of gallons of gasoline used by all cars in the
U.S. each year
Step 1: Number of cars (3.00 108 people)  (0.5 cars/perso n) ~ 108 cars
Step 2: Number of gallons used by a car per year
 10 4 mi/yr

# gal/yr # of mi driven  car


mi
car
mi/gal
10
gal


  103 gal/yr
car
Step 3: Number of gallons consumed per year
gal/yr
# gal ~ (108 cars)  (10 3
)  1011 gal/yr
car
Estimates and Order-of Magnitude (cont’d)
• Example 1.8 : Number of galaxies in the Universe
Information given: Observable distance = 10 billion light year (1010 ly)
14 galaxies in our local galaxy group
2 million (2x106) ly between local groups
1 ly = 9.5 x 1015 m
Volume of the local group of galaxies: Vlg  4 pr 3 ~ (106 ly )3  1018 ly 3
3
Number of galaxies per cubic ly:
# of galaxies # of galaxies 10 galaxies
17 galaxies

~

10
ly 3
Vlg
1018 ly 3
ly 3
Volume of observable universe:
4
Vu  pr 3 ~ (1010 ly ) 3  1030 ly 3
3
Number of galaxies in the Universe:
 # of galaxies 
 17 galaxies  30 3
13




# of galaxies ~ 
V

10
(
10
ly
)

10
galaxies
3
3
 u 

ly
ly




Coordinate Systems
 Locations
in space need to be specified by a coordinate
system
• Cartesian coordinate system
A point in the two dimensional Cartesian system is labeled
with the coordinate (x,y)
Coordinate Systems (cont’d)
• Polar coordinate system
A point in the two dimensional polar system is labeled
with the coordinate (r, q)
Trigonometry
• sinq, cosq, tanq etc.
Pythagorean theorem:
hypotenuse
r 2  x2  y2
side opposite q
side adjacent q
Inverse functions:
y
y
sin q   sin 1  q
r
r
x
x
cos q   cos 1  q
r
r
y
y
tan q   tan 1  q
x
x
Trigonometry (cont’d)
• Example 1.9 : Cartesian and polar coordinates
Cartesian to polar: (x, y)=(-3.50,-2.50) m
r  x2  y2
 (3.50 m) 2  (2.50 m) 2
 4.30 m
y  2.50 m
tan q  
 0.714
x  3.50 m
 q  tan 1 0.714  216
Polar to Cartesian: (r, q)=(5.00 m, 37.0o)
x  r cos q  (5.00 m) cos 37.0  3.99 m
y  r sin q  (5.00 m) sin 37.0  3.01 m
Trigonometry (cont’d)
• Example 1.10 : How high is the building
What is the height of the building?
height
tan 39.0 
46.0 m
 height  (tan 39.0)( 46.0 m)
 (0.810)( 46.0 m)  37.3 m
What is the distance to the roof top?
r  x2  y2
 (37.3 m) 2  (46.0 m) 2
 59.2 m