Download PHYSICS 231 Lecture 1: Units, Measurement, Dimension

PHYSICS 231
Lecture 1: Units, Measurement, Dimension
“This could be the discovery of the century, depending
of course, on far it goes down”
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Why?
Make No Mistake: Medical Errors Can Be Deadly Serious
The American Hospital Association lists these as some
common types of medication errors:
•incomplete patient information
•unavailable drug information
•miscommunication of drug orders, which can involve
poor handwriting, confusion between drugs with similar
names, misuse of zeroes and decimal points, confusion
of metric and other dosing units, and inappropriate
abbreviations
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example
• Medication errors in US: 44000-98000 deaths/year
• 4th-6th leading cause of death in US
• 3000-6800 are due to medication math errors
• Unit conversion (e.g. grams into milligrams)
• Translation language into equations (e.g. a solution of
50 1-mg drops per liter water into 1 mg/20 ml)
• Unit conversions are also important in many other fields
(packaging, architecture, construction etc etc)
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What is a meter?
• Early 18th Century:
•Length of pendulum with half length of
1s
•Distance from N-pole to Equator via
Paris /107
• 1791: second was chosen and 1874 Alloy
was made
• 1889:more precise by using
Platinum/Iridium Alloy
• 1960:Using wavelength by using Krypton86 radiation
• 1983: distance traveled by light in vacuum
in 1/299792458 s
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What is a second?
• Originally 1/86400 of a mean solar day
• 1960: based on a tropical year
• 1967: 9192631770 periods of radiation
corresponding to the transition between two
hyperfine states of the ground state of 133Cesium
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What is a kilogram?
• End of 18th century: 1 dm3 of water
• 1889: defined to a Platinum-Iridium weight
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Système Internationale (SI)
7 Standard Units: We will use them!!
Quantity
Name
unit
length
meter
m
mass
kilogram kg
time
second
electrical Current
Ampere A
temperature
Kelvin
s
K
amount of substance mole
mol
Luminous intensity
cd
candela
Examples:
Speed: m/s
Acceleration: m/s2
Force: kg·m/s2 (N)
NIST: http://physics.nist.gov/cuu/index.html
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Dimensional Analysis
Dimension should be treated as algebraic quantities!
x=x0+(at2)/2 Correct dimensionally?
[m]=[m]+([m]/[s]2)[s]2… YES!
Think about unit conversions!
X,X0 in ft. a in m/s2
[ft]=[ft]+([m]/[s]2)[s]2… ????
1 m = 3.281 ft
[ft]=[ft]+([ft]/[s]2)[s]2… YES!
GOOD WAY TO CHECK IN EXAMS & LIFE!!
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scales
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Very large & very small numbers
0.000000000000001=10-15=1E-15 femto (f)
0.000000000001=10-12=1E-12 pico (p)
0.000000001=10-9=1E-9 nano (n)
0.000001=10-6=1E-6 micro (μ)
0.001=10-3=1E-3 milli (m)
0.01=10-2=1E-2 centi (c)
0.1=10-1=1E-1 deci (d)
1=100=1E+0
10=101=1E+1 deca (da)
100=102=1E+2 hecto (h)
1000=103=1E+3 kilo (k)
1000000=106=1E+6 mega (M)
1000000000=109=1E+9 giga (G)
10000000000=1012=1E+12 tera (T)
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Greek alphabet
We will use some of those…
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Uncertainty & Significance
1.2411 trillion digits known!!
π=3.1415926535 8979323846 2643383279 5028841971
6939937510 5820974944 5923078164 0628620899
8628034825 3421170679 8214808651 3282306647
0938446095 5058223172 5359408128 4811174502
8410270193 8521105559 6446229489 5493038196...
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Significance & Uncertainty
Circumference=2πR=2·3.1415926... ·4.2 =
Calculator: 26.38937… m
Right answer: 26 m.
R=4.2 m
4.2 means that the true value lies
between 4.1 and 4.3:
2π4.1=25.761…=26 m
2π4.3=27.0176…=27 m
so: Right Answer with error: 26±1
The number of significant figures for a result of a
division or multiplication is the least accurate of the
quantities being divided or multiplied.
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Significance & Uncertainty
For addition and subtraction the number of decimal
places should be equal to the smallest number of
decimal places of any term in the sum:
3.0001 + 0.0025 = 3.0026
NUMBER OF DECIMAL PLACES IS NOT
THE SAME AS THE NUMBER OF
SIGNIFICANT FIGURES!
0.0025: 2 significant figures 4 decimal places
Scientific notation. For example 7·107 or 7E+07
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2D Geometrical objects
object
square
surface
circumference
Length2
4xlength
triangle
½x base x height Side1+side2+side3
circle
πx radius2
2 x π x radius
Radius=diameter/2
rectangle
oval
Length x height
2xlength +
2xheight
πxr1xr2
-
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3D geometrical objects
object
surface
volume
6xlength2
length3
2hw+2lh+2wh
hlw
sphere
4πr2
4/3 x πr3
cylinder
2πrh
πr2h
cube
rectangle
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Coordinate Systems
r
b
Cartesian Coordinates:
(x,y)=(a,b)
Plane polar coordinates:
(r,θ)
θ
a
r = a +b
2
Frame transformation:
2
tan(θ ) = b / a
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TRIGONOMETRY
SOH-CAH-TOA:
sin=opposite/hypotenuse
cos=adjacent/hypotenuse
tan=opposite/adjacent
Pythagorean theorem:
c = a +b
2
2
2
Note that sin,cos,tan are
dimensionless.
2π radians corresponds to 360o
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Be careful…
l
β
h
α
h=l x sinα =l x cosβ
Check carefully which angle is given
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Solving quadratic equations
ax 2 + bx + c = 0
a≠0
− b ± b 2 − 4ac
x=
2a
In general there are 2 solutions. In physics problems,
One of them is often not realistic and is thrown out.
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How to solve a problem?
There are no general rules but here are some pointers:
1) READ problem carefully!
2) Summarize (throw away unnecessary info)
3) Visualize (drawing can often help)
4) Convert units (consistency)
5) Set up equations:
• Plug in numbers if not comfortable with solving sets
of equations
• If confidents, plug in numbers at last moment
6) Check whether answers make sense
7) In exams: once you have solved a problem, check
calculations one more time at the end of exam. Especially
important if you tend to make small mistakes
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Problem:The diameter of the orbit of the earth
around the sun is 4x1011 m. (1) What is the
distance traveled by the earth in 1 year? (2) in the
polar coord. System with the sun in the center,
over what angle does the earth travel in 0.30 year?
(1)
a) 1x1012 m
(2)
a) 108 degrees
b) 2x1012 m
b) 120 degrees
c) 2.5x1012 m
c) 1.88 radians
d) 1.3x1023 m
d) 1.1x102 degrees
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Calculate the length of the shorter of two sides of a
rectangle, which has an area of 24 m2 and a perimeter
(circumference) of 22 m.
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