Unit Conversions & Dimensional Analysis
•What are units?
•Why are they important?
•Famous Conversion Errors
•Dangers of not using units?
•An Assignment
WARNING: Avoid Dimensional Analysis if
1) You enjoy making mistakes.
2) You have a phobia of correct answers
3) You enjoy solving problems in a convoluted manner.
4) You take pleasure in “re-inventing the wheel.”
5) You are a bad human being.
Metric Mania
• How many feet long is the room?
• Who tells a ruler how long to be?
• The Royal Foot vs Standard Foot
• We need a standardized system.
What Ruler Measures Rulers?
1ft = 12in
1ft = 12in
m-k-s system
A Meter is defined by the speed of light. A meter
is defined as the length light will travel in
1/299792458 of a second.
A kilogram measures mass and it is equal to an
international prototype. Meaning there is an
object in a museum that was arbitrarily chosen to
have a mass of 1kg. All other masses (including
pounds, grams, etc.) and weight are based upon
this object.
A second is defined as the duration of
9,192,631,770 periods of the radiation of a
cesium atom between two hyperfine levels of the
ground state cesium-133 atom.
Img via BIPM
7 BASE SI Units
Unit Name and
Symbol
Dimension Symbol
What does it
measure?
Meter (m)
L
Length
Kilogram (kg)
M
Mass
Second (s)
T
Time
Ampere (A)
I
Electric Current
Kelvin (K)
Θ
Temperature
N
Amount of substance
J
Luminous Intensity
Mole (mol)
Candela (cd)
Other units are DERIVED units
m
d
F
D=
S=
P=
v
t
A
F = ma
E = mc
2
Some Units Get Special Names
Scientists are Lazy!
• A NEWTON is the force required to accelerate
a one kilogram object at a rate of one meter
per second squared.
Instead of writing (kg)(m)/(s2) physicists just use
Newton (N) instead.
• Thus 1N = 1(kg)(m)/(s2)
F = ma
Pressure = Force/Area
• Force = (kg)(m)/(s2)
• Area = m2
• Units for pressure
kg * m
2
s
2
m
=
N
2
m
=
1 Pascal
Units on One side of an Equation
must equal the units on the other in
order to be dimensionally correct!
d 1 2
= m
t 2
d
S=
t
• This formula is saying a measure of length
divided by some measure of time is equal
to half of a mass squared. Is this valid?
• Speed in miles/hour = a distance
(miles) divided by a time (hours).
This is valid.
What are units? Why are they important?
• Units give us some idea of the
magnitude of the thing we are
describing.
SI vs English Units
Unit Name and
Symbol
Murica
What does it
measure?
Meter| Kilometer
Feet | Miles
Length
Gram | Kilogram
Ounce | Pound
Mass
Second
Seconds
Time
Kelvin or Celsius
Fahrenheit
Temperature
Converting Units is Important
Countries in Red have not adopted the SI system of measurement.
Major Conversion Errors
•
•
•
•
•
•
•
NASA Mars Climate Orbiter
Gimli Glider -- 747
Space Mountain in Tokyo
Escaping 250kg Tortoise
Korean Air-MD Crash
Expensive Rice
Medical Overdose Errors
Mars Climate Orbiter: Pounds vs Newtons
Gimli Glider: fuel in lbs vs kgs
Rollercoaster Derailment: Metric Conversion
250lb Tortoise …..err 250kg Tortoise (=550lbs)!
Wild Rice: $0.39 per pound or kilogram???
Medication Overdose Errors!!!
• The FDA has estimated that
medication errors cause a death a
day and injure over a million
people a year.
• These are preventable medical
injuries.
• Unit conversion errors are a big
part of this.
What happens when we don’t use units?
Medication: A typical adult prescription of
the painkiller acetaminophen with codeine
in it is 500 _______?
(a) tablets (b) grams (c) milligrams (d) ounces (e) milliliters
Consequences of Being Incorrect: symptoms of overdose may
include cold and clammy skin, extreme sleepiness, progression to
stupor or coma, general bodily discomfort, heart attack, kidney
failure, liver failure, low blood pressure, muscle weakness, nausea,
slow heartbeat, sweating, and vomiting.
What happens when we don’t use units?
Speed Limit: The speed limit in a school
zone in Canada is 30 _______?
(a) miles/hour, (b) kilometers/hour, (c) feet/second,
(d) meters/second , (e) yards/second
Consequences of Being Incorrect: speeding in a school zone may
result in an accident, loss of life or a significant fine.
What happens when we don’t use units?
Bicycle Tire Pressure: The recommended air
pressure in many mountain bike tires is 60
_______?
(a) Pascals (b) kilopascals (c) pounds per square inch
(d) torr (e) millimeters of mercury
Consequences of Being Incorrect: underinflated tires may
puncture more easily, wear out faster, are more difficult to pedal,
and don’t stop as efficiently. Overinflated tires may blow out.
Improperly inflated tires increase the chances of a serious injury or
loss of life.
What happens when we don’t use units?
Dietetics: Many physicians recommend that
pregnant women take no more than
10,000_______ of vitamin A per day?
(a) grams (b) milligrams (c) ounces (d) international units
(e) microliters
Consequences of Being Incorrect: Excess vitamin A may give rise
to birth defects, dry skin, scaly skin, headaches, fatigue, painful
bones and loss of appetite.
What happens when we don’t use units?
Food Storage: Food scientists recommend
that produce companies store apples,
cherries, apricots, and most berries at 2
_______?
(a) Degrees Fahrenheit (b) Kelvin (c) degrees Celsius
(d) degrees Rankine (e) degrees Reaumur
Consequences of Being Incorrect: Storing these fruits at too high a
temperature will result in ethylene production and early ripening.
Storing them at too low a temperature will damage the integrity of
the fruit.
Resources and Thanks
• Sourcebook for Teaching
Science by Norman Herr
• Metric Mixups by Colorado
State University USMA
• Google Images
Your Assignment
CONCEPT CHECK: Why is it essential
that all measurements be
accompanied by units?
1.
Individuals who travel to regions of the world with poor sanitation are warned to
filter water before drinking it to remove deadly water-born pathogens that cause
diseases such as cholera or typhoid. If you were traveling in a region known to
have a polluted water supply, would you drink water that your host said said had
been heated to 100 degrees for five minutes? Explain.
1.
We saw five examples of what happens when we don’t use units, now its your
turn to create a scenario. Develop a hypothetical problem that may arise when
units are omitted in a field such as sports-casting, international trade, or an area
of your choice. Include a “consequences of being incorrect” section.