# General Engineering Introduction/Units Dimensions/Units

Do I have to? Yes.
Aren't there conversion programs? Yes, but they only help find conversion factors. They don't help solve problems.
I have developed my own way of doing units that is easier. Most people do, but:
   * In complicated problems you will make mistakes
   * You will be unable to explain this to people
   * Others will not be patient enough to listen to your way
   * Others will not be able to find your mistakes
   * You will loose respect
When everyone does them the same way, then communication of technical information improves.
There are two standard ways of doing units. One involves tables, the other does the conversion on a single horizontal line. The line method is described below.

## Treat Units like Numbers

Units become complicated when multiplying and dividing. Treat them like numbers. Follow the same rules.
| ${\frac {2}{2}}=1$ | ${\frac {meter}{meter}}=1$ |
| $2*2=2^{2}$ | $meter*meter=meter^{2}$ |
| ${\cancel {2}}*{\frac {3}{\cancel {2}}}=3$ | ${\cancel {meter}}*{\frac {feet}{\cancel {meter}}}=feet$ |

## Find Truth, Make Identities

Find a true statement. Set it equal to 1 to create an identity and then start doing math. Change language into math (grammar)! Become an expert at word problems!

### Truth

Units start with true statements such as 5280 feet = 1 mile. Or three 3 teaspoons = 1 tablespoon. Or 3600 seconds = 1 hour. Who comes up with this? We count money in 10's but eggs in 12's. And this is just English. There are other weird units. The French tried to fix the world with the SI system of units. The goal was to multiply by 10 and have no strange numbers like 5280, 3600 or 3. SI is a tremendous success. Once you start using it, going to another system is painful. There is just one slight oddity:
    $1Newton=1killogram*{\frac {1\,\mathrm {meter} }{1\,\mathrm {seconds^{2}} }}$
"killo = 1000" is another truth, but slightly different. It is called a "prefix", but is merely equating a word and to number.
Every country has it's own "standards" organization that maintains these truths. In the United States this organization is called the "National Institute of Standards and Technology" or "NIST." NIST defines what a meter, foot, yard or mile is for the US and harmonizes with other countries. Tools that measure the physical world have to produce trusted numbers in order for science and commerce to function. This is done through a tree system that starts at NIST called "calibration." Stickers on equipment or certificates that ship with equipment mean that it has been tested to measure weight, time, length, count, etc. accurately. The calibration tests are maintained and improved by NIST.

### Identities

Suppose in wizard world of Harry Potter, 29 knuts = 1 sickle. How is math done with this truth?
Any number or any word can be multiplied by 1 and still be itself. The number 1 is some times called the "identity" because of this property.
Most engineering and science text books start off with a list of identities to be used. They are a subset of the world's possibilities. Identities are formed by putting all the numbers and words on one side of the equal sign. This can be done in two ways. For each truth, there are two identities. Here is the list of identities that can be formed from the true statements above:
| Truth | Identity #1 | Identity #2 |
| 1 mile = 5280 feet | $1={\frac {5280\,\mathrm {ft} }{1\,\mathrm {mile} }}$ | $1={\frac {1\,\mathrm {mile} }{5280\,\mathrm {ft} }}$ |
| 3 teaspoons = 1 tablespoon | $1={\frac {3\,\mathrm {teaspoons} }{1\,\mathrm {tablespoon} }}$ | $1={\frac {1\,\mathrm {tablespoon} }{3\,\mathrm {teaspoons} }}$ |
| 1 hour = 3600 seconds | $1={\frac {3600\,\mathrm {seconds} }{1\,\mathrm {hour} }}$ | $1={\frac {1\,\mathrm {hour} }{3600\,\mathrm {seconds} }}$ |
| 1 sickle = 29 knuts | $1={\frac {29\,\mathrm {knuts} }{1\,\mathrm {sickle} }}$ | $1={\frac {1\,\mathrm {sickle} }{29\,\mathrm {knuts} }}$ |
As the textbook and course progresses, there may be new truths introduced. Chemistry classes spend a lot of time with moles and then stops introducing new units. Physics likes to introduce new units constantly and summarizes them on the jacket covers. You have to be able to create the identities. New identities typically appear in a problem statement. Sometimes they are temporary.

## Word Problems

Almost all engineering problems start off as word problems. The first step is to look for new truths, or temporary truths that can be turned into identities. The problems below are related to Harry Potter in order to force students to deal with new units. (From Milner_d FortLewis.edu.)

### Hagrid

As Hagrid says, wizard money is very easy to understand. There are three coins: knuts, sickles and galleons. There are 29 knuts in one sickle and 17 sickles in one galleon. How many knuts in a galleon?
There are three steps to finding the solutions:
* Find new truths/identities
* Find what is wanted in the form of an unknown identity
* Find closest starting identity, multiply by other identities until reach destination
| Truth | Identity #1 | Identity #2 |
| 1 galleon = 17 sickle | $1={\frac {17\,\mathrm {sickles} }{1\,\mathrm {galleon} }}$ | $1={\frac {1\,\mathrm {galleon} }{17\,\mathrm {sickles} }}$ |
| ?? knuts = 1 galleon | $1={\frac {??\,\mathrm {knuts} }{1\,\mathrm {galleon} }}$ | the unknown identity |
${\frac {17\,\mathrm {\cancel {sickles}} }{1\,\mathrm {galleon} }}*{\frac {29\,\mathrm {knuts} }{1\,\mathrm {\cancel {sickle}} }}$ = $1={\frac {17*29\,\mathrm {knuts} }{1\,\mathrm {galleon} }}$
The answer also forms another identity to add to our list.

### Ron

Ron has carefully horded every knut he’s found for 10 years and now he has three huge bags. He didn’t want to count every coin so he weighed the bags and found he had 75 pounds of knuts. One knut weighs 2 ounces. 16 oz = 1 lb. How much does he have in galleons?
Sometimes extraneous information (truth that is not needed) confuses us. Other times it appears there is truth needed, but we don't know it. Here the issue is 3 bags. Are the bags equal size? Is one a lot larger than the other? We don't know. We have to hope that the problem can be solved without knowing the details of this issue.
Don't worry about needing to use every fact or truth in the world to solve a problem. Just follow the three steps. Don't try to do math in your head. This will just confuse people. Remember the goal of this system is to clearly communicate how you arrive at your answer. If other engineers can agree with you .. without having to do the unit analysis themselves, then a trust is born. This trust grows into respect and an engineer is born.
| Truth | Identity #1 | Identity #2 |
| 10 years = 3 bags | $1={\frac {3\,\mathrm {bags} }{10\,\mathrm {years} }}$ | $1={\frac {10\,\mathrm {years} }{3\,\mathrm {bags} }}$ |
| 3 bags = 75 pounds | $1={\frac {3\,\mathrm {bags} }{75\,\mathrm {pounds} }}$ | $1={\frac {75\,\mathrm {pounds} }{3\,\mathrm {bags} }}$ |
| 1 knut = 2 ounces | $1={\frac {1\,\mathrm {knut} }{2\,\mathrm {ounces} }}$ | $1={\frac {2\,\mathrm {ounces} }{1\,\mathrm {knut} }}$ |
| 16 ounces = 1 pound | $1={\frac {16\,\mathrm {ounces} }{1\,\mathrm {pound} }}$ | $1={\frac {1\,\mathrm {pound} }{16\,\mathrm {ounces} }}$ |
| ?? galleons = 3 bags | $1={\frac {??\,\mathrm {galleons} }{3\,\mathrm {bags} }}$ | the unknown identity |
${\frac {75\,\mathrm {\cancel {pounds}} }{3\,\mathrm {bags} }}*{\frac {16\,\mathrm {\cancel {ounces}} }{1\,\mathrm {\cancel {pound}} }}$ * ${\frac {1\,\mathrm {\cancel {knut}} }{2\,\mathrm {\cancel {ounces}} }}$ * ${\frac {1\,\mathrm {galleon} }{17*29\,\mathrm {\cancel {knuts}} }}$ = $1={\frac {75*16\,\mathrm {galleons} }{3*2*17*29\,\mathrm {bags} }}$
The answer above is not quite finished. The answer wanted is galleons, not galleons per bag (an identity). Furthermore we don't know if the bags are of equal size. We could assume bag size is equal, but we don't have to if we multiple the above answer by the number of bags.
${\frac {75*16\,\mathrm {galleons} }{2*17*29*{\cancel {3}}\,\mathrm {\cancel {bags}} }}$ * ${\cancel {3}}{\cancel {bags}}$ = ${\frac {75*16\,\mathrm {galleons} }{2*17*29}}$
This answer appears to be uniquely associated with Ron, so it's identity will probably not be used again unless we stumble across Ron and his money again.

### Harry

Harry is practicing flying on his Firebolt. He does 10 laps around the Quidditch field in 18 minutes. One lap of the field is 700 meters (m). How fast is he going in kilometers (km) per hour?
| Truth | Identity #1 | Identity #2 |
| 10 laps = 18 minutes | $1={\frac {18\,\mathrm {minutes} }{10\,\mathrm {laps} }}$ | $1={\frac {10\,\mathrm {laps} }{18\,\mathrm {minutes} }}$ |
| 1 lap = 700 meters | $1={\frac {1\,\mathrm {lap} }{700\,\mathrm {meters} }}$ | $1={\frac {700\,\mathrm {meters} }{1\,\mathrm {lap} }}$ |
| 1 kilo = 1000 | $1={\frac {1\,\mathrm {kilo} }{1000\,\mathrm }}$ | $1={\frac {1000\,\mathrm }{1\,\mathrm {kilo} }}$ |
| 60 minutes = 1 hour | $1={\frac {60\,\mathrm {minutes} }{1\,\mathrm {hour} }}$ | $1={\frac {1\,\mathrm {hour} }{60\,\mathrm {minutes} }}$ |
| ?? kilo meters = 1 hour | $1={\frac {??\,\mathrm {kilometers} }{1\,\mathrm {hour} }}$ | the unknown identity |
${\frac {60\,\mathrm {\cancel {minutes}} }{1\,\mathrm {hour} }}*{\frac {10\,\mathrm {\cancel {laps}} }{18\,\mathrm {\cancel {minutes}} }}$ * ${\frac {700\,\mathrm {meters} }{1\,\mathrm {\cancel {lap}} }}$ * ${\frac {1\,\mathrm {killo} }{10000\,\mathrm }}$ = $1={\frac {60*10*700\,\mathrm {killometers} }{18*1000\,\mathrm {hours} }}$

### Boomslangs

One of the most important ingredients in Polyjuice Potion (used to make you look like someone else) is dried boomslang skin. As you know, boomslangs are very small which is why boomslang skin is so expensive. It takes 32 boomslangs to make 1 teaspoon (tsp.) of dried boomslang skin. The potion calls for ½ cup (c.) of skin. 3 tsp. = 1 tablespoon (tbsp.) 16 tbsp. = 1 cup How many boomslangs have to give their lives for the recipe?
| Truth | Identity #1 | Identity #2 |
| 32 boomslangs = 1 teaspoon | $1={\frac {32\,\mathrm {boomslangs} }{1\,\mathrm {teaspoon} }}$ | $1={\frac {1\,\mathrm {teaspoon} }{32\,\mathrm {boomslangs} }}$ |
| 1 potion = 0.5 cup | $1={\frac {1\,\mathrm {potion} }{0.5\,\mathrm {cup} }}$ | $1={\frac {0.5\,\mathrm {cup} }{1\,\mathrm {potion} }}$ |
| 3 teaspoons = 1 tablespoon | $1={\frac {3\,\mathrm {teaspoons} }{1\,\mathrm {tablespoon} }}$ | $1={\frac {1\,\mathrm {tablespoon} }{3\,\mathrm {teaspoons} }}$ |
| 16 tablespoons = 1 cup | $1={\frac {16\,\mathrm {tablespoons} }{1\,\mathrm {cup} }}$ | $1={\frac {1\,\mathrm {cup} }{16\,\mathrm {tablespoons} }}$ |
| ?? boomslangs = 1 potion | $1={\frac {??\,\mathrm {boomslangs} }{1\,\mathrm {potion} }}$ | the unknown identity |
${\frac {32\,\mathrm {boomslangs} }{1\,\mathrm {\cancel {teaspoon}} }}*{\frac {3\,\mathrm {\cancel {teaspoons}} }{1\,\mathrm {\cancel {tablespoon}} }}$ * ${\frac {16\,\mathrm {\cancel {tablespoons}} }{1\,\mathrm {\cancel {cup}} }}$ * ${\frac {0.5\,\mathrm {\cancel {cup}} }{1\,\mathrm {potion} }}$ = $1={\frac {32*3*16*0.5\,\mathrm {boomslangs} }{1\,\mathrm {potion} }}$

### Polyjuice

The evil Barty Crouch Jr. had to drink at least 4 fl. oz. of Polyjuice Potion every hour in order to maintain his disguise as Mad Dog Moody. 64 fl. oz. = 1 gallon. How many days would 5 gallons of potion last him?
| Truth | Identity #1 | Identity #2 |
| 4 fluid oz = hour | $1={\frac {4\,\mathrm {fluidoz} }{1\,\mathrm {hour} }}$ | $1={\frac {1\,\mathrm {hour} }{4\,\mathrm {fluidoz} }}$ |
| 64 fluid oz = 1 gallon | $1={\frac {64\,\mathrm {fluidoz} }{1\,\mathrm {gallon} }}$ | $1={\frac {1\,\mathrm {gallon} }{64\,\mathrm {fluidoz} }}$ |
| 24 hours = 1 day | $1={\frac {24\,\mathrm {hours} }{1\,\mathrm {day} }}$ | $1={\frac {1\,\mathrm {day} }{24\,\mathrm {hours} }}$ |
| ?? days = 1 gallon | $1={\frac {??\,\mathrm {days} }{1\,\mathrm {gallon} }}$ | the unknown identity |
${\frac {1\,\mathrm {\cancel {hour}} }{4\,\mathrm {\cancel {fluidoz}} }}*{\frac {1\,\mathrm {day} }{24\,\mathrm {\cancel {hour}} }}$ * ${\frac {64\,\mathrm {\cancel {fluidoz}} }{1\,\mathrm {gallon} }}$ = $1={\frac {64\,\mathrm {days} }{4*24\,\mathrm {gallons} }}$
The question was how many days would 5 gallons last. So we need to multiply the identity above by 5 gallons to give an answer in days alone.
${\frac {64\,\mathrm {days} }{4*24\mathrm {\cancel {gallons}} }}$*$5{\cancel {gallons}}$ = ${\frac {64*5\,\mathrm {days} }{4*24}}$