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Note 1 Units
Units
In order to obtain meaningful information for analysis, we must make measurements.
Measurements are meaningful only if they are measured against some standard.
These
standards are called units. The set of units we use here is called SI (Le Système International
d'Unités) or MKS.
The Meter
The meter is is the standard unit for distance. Originally, the circumference of the Earth is
assumed to be forty million meters so a meter is one forty millionths of that distance. So you need
to measure the circumference of the Earth accurately. Unfortunately, the circumference of the
Earth is not uniform.
Now, it is the distance travelled by light in vacuum during a time interval of 1/299,792,458 of a
second.
link
The Second
The second is the standard unit for time. Originally, the Earth day is separated into 24 hours each
of which is separated into 60 minutes and each of which is separated into 60 seconds.
Unfortunately, the Earth day can change for various reasons including large earthquakes.
Now, the second is the duration of 9,192,631,770 periods of the radiation corresponding to the
transition between the two hyperfine levels of the ground state of the cesium 133 atom.
link
The Kilogram
The kilogram is the standard unit of mass. Mass is how much stuff there is. The kilogram is
defined by a chunk alloy of 90% platinum and 10% iridium named the international prototype
kilogram (IPK). The IPK is very stable though it still fluctuates due to contaminations. There are
attempts to make it independent of an artifact.
link
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Dimensional Analysis
Units are like variables in an algebraic equation.
area
⇒ meter ×meter = meter 2
acceleration ⇒ meter/second/second = (meter/second)/second = meter/second 2
density
⇒ kilogram/(meter /meter /meter) = kilogram/meter 3
The basic units are called fundamental units. The combination units are called derived units.
Unit conversion is the multiplication by fractions (conversion factors) that are equal to one where
the numerator has the unit you want and the denominator has the unit you have.
How many miles per hour is 1 meter per second?
1
m ⎛⎜ 3600
⎜
s ⎜⎝ hr
s ⎞⎟⎛⎜ 100 cm ⎞⎟⎛⎜ 1 in ⎞⎟⎛⎜ 1 ft ⎞⎟⎛⎜ 1 mi ⎞⎟
mi
⎟⎟⎜
⎟
⎟
⎟
⎟ = 2.2369
⎟⎠⎜⎝ 1 m ⎟⎟⎠⎜⎜⎝ 2.54 cm ⎟⎟⎠⎜⎜⎝ 12 in ⎟⎟⎠⎜⎜⎝ 5280 ft ⎟⎟⎠
hr
The conversion factor from meters per second to miles per hour is
⎛ 3600
⎜⎜
⎝⎜ hr
s ⎞⎟⎛⎜ 100 cm ⎞⎟⎛⎜ 1 in ⎞⎟⎛⎜ 1 ft ⎞⎟⎛⎜ 1 mi ⎞⎟
mi/hr
⎟⎟⎜
⎟⎟⎜
⎟⎟⎜
⎟⎟⎜
⎟⎟ = 2.2369
⎟⎠⎜⎝ 1 m ⎟⎠⎜⎝ 2.54 cm ⎟⎠⎜⎝ 12 in ⎟⎠⎜⎝ 5280 ft ⎟⎠
m/s
How many seconds are there in 1 year?
⎛ 365.25 days ⎞⎟⎛ 24 hr ⎞⎟⎛ 3600 s ⎞⎟
1 yr ⎜⎜
⎟⎟⎜⎜
⎟⎟⎜⎜
⎟⎟ = 31,558, 000 s ≅ π ×107 s
⎜⎝
⎜
⎜
⎟
⎟
1 yr
⎠⎝ 1 day ⎠⎝ 1 hr ⎟⎠
The conversion factor from second to year is
⎛ 365.25 days ⎞⎟⎛ 24 hr ⎞⎟⎛ 3600 s ⎞⎟
s
⎜⎜
⎟⎟⎜⎜
⎟⎟⎜⎜
⎟⎟ = 31,558, 000
⎟⎠⎜⎝ 1 day ⎟⎠⎜⎝ 1 hr ⎟⎠
1 yr
yr
⎝⎜
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Scientific Notation
Scientific notations are useful for expressing excessively large and small numbers. It also
expresses the number of significant digits without ambiguity. Finally, it is useful for estimating the
results of calculations.
The number 1 billion can be written in the following manners.
one significant digit
1,000,000,000
1x109
four significant digits
1,000,(000,000)
1.000×109
Prefixes are short hand for representing the exponent for convenience
103
kilo
10–3
milli
106
mega
10–6
micro
109
giga
10–9
nano
1012
tera
10–12 pico
1015
peta
10–15 femto
Measuring the size of molecules, we would use the nanometer.
Measuring the size of cells, we would use the micrometer.
Measuring the size of people, we would use the meter.
Measuring the size of cities, we would use the kilometer.
Measuring the Size of planets, we would use the gigameter.
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