More Math and Units

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More Math and Units

• NATS102-13Lecture 58 SEP 2009

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Announcements
Homework 2 is due today Homework 3 has been
posted on the web site Essay topics are due this
Thursday (Sep. 10)
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Some Simple Geometry
Circles The ratio of the circumference of a
circle (C) to the diameter (D) is called ?
(pi), C/D ?. The quantity is the same for all
circles ?3.1415926535897932384626433832795028841
971693993751.... The area (A) of a circle is
related to the diameter by A 1/4 ?D2 A
?R2 Sometime radius (R) is used in place of
diameter. The radius of a circle or sphere is
equal to half its diameter RD/2
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Digression on ?
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Archimedes Antiquitys Greatest Scientist
The discovery Archimedes was most proud of
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Spheres
The volume (V) of a sphere is equal to V 4/3
?R3 or V 1/6 ?D3 We measure volume in units
of length cubed, for example meters cubed, which
is usually denoted as m3, though you might
sometimes see it spelled out as meters cubed. We
can also measure the area on the surface of a
sphere, called the surface area (A), A 4?R2 or
A ?D2
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Visualize taking each little segment in this
drawing, laying it flat, measuring its area, and
adding them all together. This would give you
the surface area.
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Volume
D
L
Find the length of the sides of a cube that has a
volume equivalent to the volume of the Earth.
Use 12,756 km for the diameter of the Earth.
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Take a Guess

• The length of one side of the cube is
• Roughly 100 km
• Roughly 1000 km
• Roughly 10,000 km
• Roughly 100,000 km

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Calculation
For a Sphere D 12,756 km 1.3?104 km
(Diameter of Earth) V (?/6)D3 V
(?/6)(1.3?104)3 1.2?1012 km3 Or, R
12,756/2 km 6.4?104 km
V(4/3)?R3(4/3) ?(6.4?104)3 km31.2?1012 km3 For
a Cube VL3, L V1/3 L (1.2?1012
km3)1/31.1?104 km
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Order of Magnitude Estimatesand Fermi Problems
General Methodology Break the question into
smaller pieces and estimate the answer to each
smaller piece.
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Enrico Fermi Born Rome, Sept 29, 1901 Died
Chicago, Nov 28, 1954 Fermi was one of the
leading physicists of the 20th century. He was
unusual in that he excelled as a theorist and as
an experimentalist.
He led the team that achieved the first sustained
nuclear reaction, thereby leading the way to
nuclear power plants. An ardent foe of fascism
he fled his native Italy in 1939 for the US where
he had a leading role in the Manhattan project.
He later made fundamental contributions to
sub-atomic physics. Element 100 on the periodic
table (Fermium) is named in his honor. Fermi was
known for being able estimate the solutions to
complex problems quickly and accurately. The
order of magnitude problems were discussing are
sometimes called Fermi problems.
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Order of Magnitude Estimates

• Example
• How many saguaro cactus are there in Saguaro East
  Park?
• Guess at dimensions of park
• Convert dimensions to area
• Estimate area per saguaro
• Divide to get of saguaros.

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Order of Magnitude Estimates

• Example
• How many saguaro cactus are there in Saguaro East
  Park?
• Guess at dimensions of park 5 miles x 5 miles
• Convert dimensions to area 7?108 square feet
• Estimate area per saguaro 20 ft ? 20 ft 400
  ft2
• Divide to get of saguaros 7?108/4?102 2?106

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Example Piano Tuners in Chicago

• How many piano tuners are there in Chicago?
• How many people in Chicago?
• How many people per household?
• How many households have pianos?
• How often are pianos tuned?
• How long does it take to tune a piano (include
  travel)?
• How many hours does a piano tuner work per year?

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Example Piano Tuners in Chicago

• How many people in Chicago? 5,000,000 5?106
• How many people per household? 2
• How many households have pianos? 1/20
• How often are pianos tuned? 1 per year
• How long does it take to tune a piano (include
  travel)? 2 hours
• How many hours does a piano tuner work per year?
  2000
• of Tunings/year (5?106/2) ?(1/20)125,000
• Average tuner does 2000 hours/2 hours 1000
  tunings per year
• There must be 125 tuners to do 125,000
  tunings/year.

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Units
All physical quantities are associated with
units. The simplest example is distance, which
can be measured in a variety of units including
inches, feet, miles, meters, kilometers, etc.
You cant say my house is 10 from your house.
The value needs a unit to make sense, so we say
my house is 10 feet from your house.
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Units of Distance
inches feet, yards, miles, fathoms, nautical
miles, furlongs,
meters, kilometers, micrometers
(microns), angstroms, nanometers, lightyears,
parsecs,
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Units for Time
Another simple example is time. We dont say
the class lasts 75, but the class last 75
minutes. Units of time are seconds, minutes,
hours, days, years, etc. Ever wonder why
different cultures have different units for
distance, different languages, different
currencies, but everyone uses hours, minutes, and
seconds? Days and years are easy to understand,
but why hours, minutes, and seconds?
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Time and Babylon
The history of the second, minute, and hours can
be based to the ancient Summerians that ruled the
area around modern day Iraq though city states
such as Babylon, Ur, and Uruk. The Summerians
were accomplished mathematicians and used a base
60, rather than base 10, number system. Why base
60? We dont know. The concept of a week
consisting of 7 days also originated with the
Summerians, as well as the division of a year
into 12 months.
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The Babylonian Empire in 1700 BC
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Derived Units
Some physical quantities are related in a
specific way to other physical quantities. If
so, this defines the relationship between their
units. The simplest example is velocity, which
is distance divided by time. So, if a car is
traveling 60 mph (miles per hour) this means that
if it were going that speed constantly for one
hour it would travel 60 miles. Velocity doesnt
have a separate unit, rather the unit for
velocity is related in a specific way to the
units for distance and time.
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Signifying units
Units can be written in a variety of ways. For
example, we can write 60 miles per hour or 60
mph. Often we abbreviate the symbols for common
units. 10 meters per second is often written as
10 m/s or 10 m s-1. 10 meter per second per
second can be written as 10 meters per second
squared 10 m/s2 10 m s-2.
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Acceleration
Acceleration is the change in velocity with time.
So, if a car is traveling 40 mph and accelerates
to 60 mph in one minute, the acceleration is to
the change in velocity divided by the time, or 20
mph/ 1 minute. This brings up another point.
When using physical quantities in calculations
you have to make sure the units are the same. In
the example above we have miles per hour per
minute. It is usual easer to stick to one unit
for each quantity, I.e. either minutes or hours,
but not both. So, 20 mph/minute 20/60
miles/min2
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What are possible units for velocity?

• Feet / sec
• Meters / sec
• Inches / year
• Meters / sec / sec
• a, b, or c

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What are possible units for acceleration?

• Feet / sec
• Meters / sec
• Feet / sec / sec
• Meters / sec / sec
• c or d

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Which of the following is true?

• You can have acceleration not equal zero, but
  velocity equal to zero
• You can have acceleration equal to zero, but
  velocity not equal to zero
• You can accelerate without changing your speed
• a and b
• a, b, and c

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The Metric System
In a spirit of pan-European unity that predated
the euro by 200 years, a group of European
scientists defined the metric system in 1795.
The idea was to get away from a system where the
unit of distance differed from country to country
according to the size of the monarchs feet. (By
1795, serious people had begun to think that such
a system was embarrassing.) So, what should be
the standard length? There have been several.
There were two goals 1) uniformity for all
nations and 2) a base ten system.
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History of the Meter

• The original definition (1791) was 10-7 of the
  distance from the pole to the equator.
• In 1889 the definition was changed and 1 meter
  was defined as the length of a platinum-iridium
  rod kept at a specific temperature in a lab in
  Paris.
• In 1960 the meter was redefined again to be based
  on the wavelength of light emitted by Kyrpton-86
• In 1983 the meter was redefined as the the length
  of the path traveled by light in vacuum during a
  time interval of 1/299 792 458 of a second.

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Good book to read.
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Important Physical Quantities and SI Units

• Time s (seconds)
• Distance m (meters)
• Mass kg (kilograms)
• Velocity m/s
• Acceleration m/s2
• Force kg m/s2 (Newtons)
• Area m2
• Volume m3
• Density mass/volume kg/m3
• Temperature Kelvins

Life is easier if we adopt a common set of units.
Scientists have adopted the System
Internationale (SI for short) system to units.
We will usually use this in class.
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Not so much fun, but useful to have. We will
post it on the web site.
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Common Prefixes

• pico 10-12 picometer , picogram, picosecond
• nano 10-9 nanometer
• micro 10-6 micrometer
• milli 10-3 millimeter
• centi 10-2 centimeter
• deci 10-1 decimeter
• deca 101 decameter
• kilo 103 kilometer
• mega 106 megameter
• giga 109 gigameter
• terra 1012 terrameter

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Temperature Units
Temperature in the US is commonly measured in
degrees Fahrenheit, often written as ºF. In most
countries temperature is measured in Celsius or
centigrade. Scientists measure temperature in
Kelvins (after Lord Kelvin, a British physicist).
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Temperature Continued
In the Fahrenheit system, water freezes at 32 ºF
at (a pressure of 1 atmosphere) and boils at 212
ºF. In the centigrade system, water freezes
at 0 ºC and boils at 100 ºC. In SI units water
freezes at 273 K and boils at 373 K.
Conversions ºC (ºF-32)x100/180
(ºF-32)x5/9 ºF (9/5) x ºC 32. K ºC 273
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Angles
Angles are commonly measured in degrees but may
also be measured in radians.
The radian is defined as the ratio of the arc
length defined by the angle to the radius of the
circle. We say that angle a in the figures
subtends A?B/R radians. There are 2? radians in
a circle and 360 degrees.
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Solid Angles
Solid angle refers to the fraction of a sphere
subtended by a cone. It is defined as the ratio
of the area on the sphere to the radius of the
sphere. The units of solid anlge are steradians.
There are 4? steradians in a sphere.
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Weird Units

• Stones
• Furlongs
• Fathoms
• Knots
• light-years
• parsecs

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Conversions
1 inch 2.54 cm 1 mile 5280 feet 1 quart
0.946 meters
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Example

• An on line dating service in France claims that
  your perfect mate is 90 cm tall. How tall is
  that in feet and inches?

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Example

• Youre interested in buying a new Chinese car and
  the sales information says it has a top speed of
  1 meter/sec. How fast is that in mph?

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Density and Volume
The SI units for density and volume are defined
so that the density of water is 1 gram/cm3 and 1
liter of water has a mass of 1 kg. How much
volume is occupied by 1 liter of water?