🟦 Units and signs
In engineering, we talk in terms of numbers a lot. But did you know that numbers on their own hold very little meaning?
Random person: "8"
You: <um, OK>
But look at what happens with the magic of units and signs:
Random person: "-8N"
You: <clearly this person needs to improve their communication skills, but I least I know they are talking about force ... and it's negative ... so perhaps they are trying to tell me something about a compressive force?>
And look at the magic that happens once we add in a symbol:
Random person: "F = -8N"
You: "Cool, F is commonly used to express force, so you must be talking about a 8N compressive force!"
In this class, we will place a high priority on the proper use of units and signs, as they play a major role in your ability to communicate as an engineer. Be the person that writes F = -8N, not the person that writes 8.
🟦 Engineering notation
For many of you, this is your first engineering course (welcome!). Engineering is a practical career that builds on mathematics and the sciences, and you have been taking those prerequisites to get to this particular point in your journey.
Reflect back on the math and science courses you have taken. Your instructors were mathematicians and scientists, and because of that background, they (probably) used scientific notation.
In this course, we will not be using scientific notation. That is, not exactly. We will be using a variant that is called engineering notation. After you see how it works, you will love it.
Please put your calculator into engineering notation mode now. For most calculators, select "mode." You will generally see three options: NORM(al) notation, SCI(entific) notation, and ENG(ineering) notation. Always put your calculator into engineering notation mode prior to working on assignments for this class (please and thank you).
Here is how it works. Consider the number 12,300.
In scientific notation, you move the decimal point to the right of the left-most number. Then, you can think of the number in terms of a coefficient and an exponent: 1.23 x 10^4. The coefficient is 11.1 and the exponent is 4.
Engineering notation works the same way, except that all of the exponents must be multiples of 3. We need to change 10^4 to 10^3. So, divide the exponent by 10 while multiplying the coefficient by 10 to get 12.3 E3. In this notation, the capital letter E stands for "exponent" and it replaces "10^" in scientific notation.
🟦 Prefixes and Order of Magnitude
In Statics, we will use the following prefixes in our problem-solving practice.
Please commit these to memory if you haven't already done so.
G = giga = E9 = billion
M = mega = E6 = million
k = kilo = E3 = thousand
m = milli = E-3 = one-thousandth
µ = micro = E-6 = one-millionthNote: the symbol is pronounced "mu" but the word "micro" is used when speaking, e.g. micrometers)
U.S. Customary Prefixes
In U.S. customary units, prefixes are rarely used. Instead, we prefer to report answers that are appropriate to the scale question, so that we avoid using engineering notation in final answers unless it is absolutely necessary.
For instance, if you were asked to report the distance between Denver and Chicago, you would use miles (not feet or inches). If you were asked to report the span of a bridge, you'd use feet (not miles or inches). And if asked to report the diameter of an orange, you'd use inches (not feet or miles).
There are a few prefixes that I would like for you to understand and use when working in U.S. customary units.
In Statics, we will often use units of force. The "pound-force" (and not the "pound-mass") unit is commonly expressed as the pound and it is often given a symbol: #Note: This symbol used to simply be called the "pound symbol" but these days it seems to have a new name: the hashtag.
Within Civil Engineering, the # unit is small compared to the forces exerted on bridges and buildings. So, we have invented a useful unit called the kilopound. It is equal to 1,000#, and we use the symbol k. In this case, the k is not a prefix -- it is simply shorthand for kilopound or 1,000 pounds (pound-forces). To make it even more complex, instead of taking time to say kilopound (three syllables), we will use the word kip when speaking about kilopounds conversationally.
In Fluid Statics (a topic that is generally covered in both Statics and Fluids courses, so you are likely to see it twice), we will need to express force per area (pressure). In U.S. Customary Units, we will use psi for "pounds per square inch" and ksi for "kips per square inch." Of course, 1 ksi = 1,000 psi.
🟦 Significant Figures
I know you want to skip past this section, but don't do it. Significant figures are an important part of effective communication. Seriously, they are more important than you realize.
Let's say that your height (in U.S. Customary Units) is 5 foot 10 inches (notationally, this is written as 5'-10").
Here's what happens if you are too precise (use too many significant figures):
Question: How tall are you?
Answer: I am 5.83456 feet.
It sounds ridiculous, right? Answering a question with too much precision makes you sound like you don't understand the point of the question, or the intent of the person asking the question. No one needs to know their height that precisely, and I would also question your ability to measure your height accurately with that much precision.
But we also don't want to be too approximate (use too few significant figures):
Question: How tall are you?
Answer: I'm about 6 feet tall.
If we wanted to round to the closest foot, this answer is mathematically correct, but it is also a lousy way to answer the question. The person asking you this question wants more precision that you provided.
The conventional way we answer the question (5'-10") teaches us about the appropriate use of significant figures in engineering communication. The answer to the question is provided in three significant figures (or close to it, anyway -- since 1 inch is equal to 1/12 foot and not 1/10 foot). As engineers, we must use numbers in a way that allows us to communicate effectively. Being too precise can be just as problematic as lacking precision.
Please report all of your final answers to three significant figures. We do not want any more precision -- and we do not want any less precision. Three is generally appropriate for final answers in this class, as well as in the world beyond this class. In this class, only final answers with three significant figures are correct: no more, no less.
That said, while we are working through the calculation, we do want to keep a reasonable amount of precision. If we round numbers too soon, we lose precision. If we do this successively, in several steps, our final answer drifts -- the rounding can cause 1% error, 2% error, or more. We need to minimize computational errors. That's just good engineering.
Here are some tips to maintain precision throughout a problem:
if it is easy and reasonable, keep all digits (or significant figures) in your calculator storage (use the STO command as needed)
let's also acknowledge that practically speaking, engineering problems have a multitude of steps and it's not always a good use of our time to keep track of every single digit (after all, engineers are not scientists; our work is full of approximations)
therefore, in order to accurately report 3 significant figures in the final answer, we will write down four significant figures throughout intermediate steps of the problem