DIMENSIONAL DIMENSIONAL ANALYSISANALYSIS

How to Change Units using How to Change Units using MathMath

If these are one quart each, how many gallons are here?

Strawberries and other fruits are often sold in one quart containers. Are these quarts the same size as a quart of strawberry juice?

What does 8 quarts of strawberries make?What does 8 quarts of strawberries make?

Apothecary System used in Apothecary System used in medicinemedicine

• Based on the weight of one grain of Based on the weight of one grain of wheat.wheat.

• 20 grains make a scruple20 grains make a scruple• 3 scruples make a dram3 scruples make a dram• 8 drams make a ounce8 drams make a ounce• 12 ounces make a pound (troy pound)12 ounces make a pound (troy pound)• Does this system of constantly Does this system of constantly

changing numbers make sense?changing numbers make sense?

Lets Look at another - Units Lets Look at another - Units of Length of Length 

• 12 inches (in)= 1 foot (ft)12 inches (in)= 1 foot (ft)• 3 feet= 1 yard (yd)3 feet= 1 yard (yd)• 16-1/2 16-1/2 feetfeet= 1 rod (rd), pole, or = 1 rod (rd), pole, or

perch, depending on where you are perch, depending on where you are fromfrom

• 40 rods= 1 furlong (fur) = 660 40 rods= 1 furlong (fur) = 660 feetfeet• 8 furlongs= 1 U.S. statute mile (mi) 8 furlongs= 1 U.S. statute mile (mi) • 1 U.S. statute mile (mi) = 5280 1 U.S. statute mile (mi) = 5280 feetfeet• 6076.115 49 feet  = 1 nautical mile6076.115 49 feet  = 1 nautical mile

Even More?????Even More?????

• 14 pounds= 1 stone14 pounds= 1 stone• 2 stones= 1 quarter = 28 pounds2 stones= 1 quarter = 28 pounds• 4 quarters= 1 hundredweight = 112 4 quarters= 1 hundredweight = 112

poundspounds• 20 hundredweight= 1 Long Ton = 20 hundredweight= 1 Long Ton =

2240 pounds2240 pounds

No, Not MORENo, Not MORE

• 4 gills= 1 pint4 gills= 1 pint• 2 pints= 1 quart2 pints= 1 quart• 4 quart= 1 gallon4 quart= 1 gallon• 2 gallons= 1 peck2 gallons= 1 peck• 8 gallons (4 pecks)= 1 bushel8 gallons (4 pecks)= 1 bushel• 8 bushels= 1 quarter8 bushels= 1 quarter

AND THIS IS THE AND THIS IS THE SYSTEMSYSTEM

• Americans say it Americans say it easy????????????????easy????????????????

• How about the Metric System, it is How about the Metric System, it is based only on 10based only on 10

Why Do Units Matter?Why Do Units Matter?

• Why is it Important to always write Why is it Important to always write the Units?the Units?

• I know what the answer should be I know what the answer should be in, why do I have to write the stupid in, why do I have to write the stupid unit?unit?

Dec. 11, 1998

• The loss of the the Mars Climate Orbiter on September The loss of the the Mars Climate Orbiter on September 23, 1999, was a most unfortunate and highly avoidable 23, 1999, was a most unfortunate and highly avoidable event. event.

• The cause of the mishap has been traced to a mix-up The cause of the mishap has been traced to a mix-up over units. Preliminary findings indicated that one team over units. Preliminary findings indicated that one team used English units (e.g., inches, feet and pounds) while used English units (e.g., inches, feet and pounds) while the other used metric units for maneuvers required to the other used metric units for maneuvers required to place the spacecraft in the proper Mars orbit. place the spacecraft in the proper Mars orbit.

• The 'root cause' of the loss of the spacecraft was the The 'root cause' of the loss of the spacecraft was the failed translation of English units into metric units.failed translation of English units into metric units.

• For nearly three centuries, engineers and scientists For nearly three centuries, engineers and scientists have been struggling with English units.have been struggling with English units.

• The loss of the the Mars Climate Orbiter on September The loss of the the Mars Climate Orbiter on September 23, 1999, was a most unfortunate and highly avoidable 23, 1999, was a most unfortunate and highly avoidable event. event.

• The cause of the mishap has been traced to a mix-up The cause of the mishap has been traced to a mix-up over units. Preliminary findings indicated that one team over units. Preliminary findings indicated that one team used English units (e.g., inches, feet and pounds) while used English units (e.g., inches, feet and pounds) while the other used metric units for maneuvers required to the other used metric units for maneuvers required to place the spacecraft in the proper Mars orbit. place the spacecraft in the proper Mars orbit.

• The 'root cause' of the loss of the spacecraft was the The 'root cause' of the loss of the spacecraft was the failed translation of English units into metric units.failed translation of English units into metric units.

• For nearly three centuries, engineers and scientists For nearly three centuries, engineers and scientists have been struggling with English units.have been struggling with English units.

Air conditioner rated at 18,000 BTU per Air conditioner rated at 18,000 BTU per hour.hour.

What does BTU stand for and what does it measure?

British Thermal Units = Energy to raise or lower 1 pound of water 1 degree Fahrenheit

Seabiscuit’s races like most horse races goes for 10 furlongs. How far is that?

In his first annual In his first annual message to Congress message to Congress (1790 January), (1790 January), President George President George Washington pressed for Washington pressed for uniformity in currency, uniformity in currency, weights, and measures. weights, and measures. Secretary of State Secretary of State Thomas Jefferson urged Thomas Jefferson urged Congress to reduce the Congress to reduce the existing separate existing separate systems of dry and systems of dry and liquid measures to one liquid measures to one set of measuresset of measures..Over 200 years later, we still have separate systems. For Over 200 years later, we still have separate systems. For

example, our dry quart is larger than our liquid quart, so four cups example, our dry quart is larger than our liquid quart, so four cups of flour and 4 cups of milk of flour and 4 cups of milk shouldshould be measured with separate be measured with separate measuring devices-which differ in size! measuring devices-which differ in size!

Jefferson's second plan put before Congress was Jefferson's second plan put before Congress was to reduce every to reduce every branch [of measurement] to the same decimal ratio already branch [of measurement] to the same decimal ratio already established for coin, and thus bring the calculations of the principal established for coin, and thus bring the calculations of the principal affairs of life within the arithmetic of every man who can multiply affairs of life within the arithmetic of every man who can multiply and divide plain numbers.and divide plain numbers.

Congress took no action.  Washington urged action again in his Congress took no action.  Washington urged action again in his second annual message to Congress (1790 December 08) and the second annual message to Congress (1790 December 08) and the House referred the matter to the Senate, whose committee in this House referred the matter to the Senate, whose committee in this matter declined (1791 March 01) due to negotiations in progress matter declined (1791 March 01) due to negotiations in progress with the French and British Governments to obtain an international with the French and British Governments to obtain an international standards of measurements. standards of measurements.

Currency is based on 10.

In the same year of 1790 King Louis XVI of France was also weary of the inconsistencies of measurement that was hurting trade.

He ordered a new system of measurements be developed.

Earlier in that century, Fahrenheit and Celsius used water as the basis of their thermometers. Their thermometers were calibrated using water, which everyone had access to. This made their thermometers useful.

Everyone has access to the Earth, so this new system was to be based on the size of the Earth.

The idea was to measure a part of Earth’s surface over a few degrees, then calculate distance from Equator to North Pole.

By dividing the distance from the equator to the north pole by 10 million, a convenient length was defined.

It was a little longer than a yard, and the name would be from the Greek word, Metron,

meaning to measure.

10,000,000Equator to North PoleEquator to North Pole

The motto for the metric system was:

“For all people, for all time”

Since we have 10 Since we have 10 fingers, a number fingers, a number system based on system based on 10 was developed. 10 was developed.

4/20/01

0.5 versus 5

DON’T BE A BABY DON’T BE A BABY KILLERKILLER

• If there is no number in front of a If there is no number in front of a decimal point, you MUST put a decimal point, you MUST put a ZERO (0)ZERO (0)

• If you have .25 grams as an answer, If you have .25 grams as an answer, you must write it 0.25 grams.you must write it 0.25 grams.

• Never leave a decimal by itself, they Never leave a decimal by itself, they will wonder off and get lost.will wonder off and get lost.

CubitsYardsMilesNautical MilesFeetHandsInchesLeaguesChainsFurlongs

SI BASE UNITSQUANTITY NAME SYMBOLLength Meter mVolume Liter lMass Kilogram Kg

gram gTime second sTemperature Kelvin K

Celsius CAmount mole molMemory byte b

SI System Prefixes pico- p 10-12 1/1,000,000,000,000 nano- n 10-9 1/1,000,000,000 micro- µ 10-6 1/1,000,000 milli- m 10-3 1/1000 centi- c 10-2 1/100 Kilo- K 103 1000 Mega- M 106 1,000,000 Giga- G 109 1,000,000,000 Tetra- T 1012 1,000,000,000,000

Know these prefixes

PUTTING THEM TOGETHERPUTTING THEM TOGETHER

• 1000 meters 1000 meters 1 Kilometer1 Kilometer

• 1,000,000 bytes 1,000,000 bytes 1 Megabyte1 Megabyte

• 1/1000 Liter1/1000 Liter 1 milliliter1 milliliter

• 1/100 meter1/100 meter 1 centimeter1 centimeter

• 1,000,000,000 seconds1,000,000,000 seconds 1 1 GigasecondGigasecond

Conversion FractionsConversion Fractions

• A Conversion Fraction is a Fraction A Conversion Fraction is a Fraction that has the units in it and is equal to that has the units in it and is equal to oneone

• 1000m/1Km1000m/1Km• 100cm/1m100cm/1m• 1000mm/1m1000mm/1m• 1000g/1Kg1000g/1Kg• NOTE: The larger number always goes NOTE: The larger number always goes

with the smaller unitwith the smaller unit

Convert 0.15 meters to Convert 0.15 meters to centimeterscentimeters

0.15 meters = centimeters

Starting amount End Amount

.01

1 centi

0.15 meters = 15 meters100

= 15 centimeters

If saying 15 hundredths, just say 15 centi-

equal

Convert 5234 millimeters to Convert 5234 millimeters to metersmeters

5234 millimeters = meters

Starting amount End Amount

milli

0.001

5234 millimeters 5234 meters1000

equal

Convert 5234 millimeters to Convert 5234 millimeters to centimeterscentimeters

5234 millimeters = centimeters

Starting amount End Amount

milli

0.001

5234. millimeters

equal equal

0.01

centi

523.4 centimeters

Convert 5234 millimeters to Convert 5234 millimeters to inchesinches

5234 millimeters = inches

Starting amount End Amount

milli

0.001

equal equal

0.01

centi 2.54 cm

1 inch

1 inch

2.54 cm

Lets try somethingLets try something

How many millimeters are in 5 metersHow many millimeters are in 5 meters

5 m x 1000mm/1m = 5000mm5 m x 1000mm/1m = 5000mm

Let’s look at this:Let’s look at this:

We write our starting point first, and We write our starting point first, and then determine the fraction, there then determine the fraction, there are 1000 milli’s in one whole, so the are 1000 milli’s in one whole, so the fraction is 1000 mm/1 mfraction is 1000 mm/1 m

Let’s try another

How many Kilometers are there in 5000 millimeters?

First, remove the starting prefix5000mm x 1m/1000mm = 5 m

Second, place the new prefix

5 m x 1Km/1000m = 0.005 Km

Let’s Try Another

How many milligrams are in 5000 Kilograms?First, remove the starting prefix:

5000 Kg x 1000g/1Kg = 5000 gSecond, place the ending prefix:

5000g x 1000 mg/1g = 5,000,000 mg

LET’S LOOK AGAIN

How many meters will a car traveling 100 Km/hr travel in 1 minute?– This one is a little harder: You have to make two

sets of conversions First Convert 100 Km/hr to meters per hour

– 100 Km/hr x 1000m/1 Km = 1000000 m/hr Second you have to convert m/hr to m/min

– 1000000 m/hr x 1hr/60 min = 16667 m/min Notice: hr goes on top in the conversion because it is

on the bottom in the initial number

Some Practice Problems

I drive 5 miles to school each morning. How many inches is that? Express your answer in scientific notation.

Another one

We have seven class periods here at AHS. If you zone out 5 minutes every class period, how many hours of education will you have missed in the 180-day school year?

Again

If a car is traveling 60 miles per hour, how many inches per second is it moving?

And Yet Again

Mr Simms wants to give all of his chemistry students enough chocolate to make them goofy for the rest of the day. It's a known fact that it takes 47 chocolate chips to make a student goofy. The average cookie contains 29 chocolate chips. Mr Simms has 5 classes and each class has an average of 32 students. How many toll house cookies does Mr Simms need to make?

One Last One

If you never learn dimensional analysis, you will have to work at a fast food restaurant for the rest of your life wrapping hamburgers. Each hour you wrap 184 hamburgers. You work 8 hours per day and work 5 days a week. You get paid every 2 weeks with a salary of $840.34. How many hamburgers will you have to wrap to make your first one million dollars?